Concurrency Theoryof Cyclic and Acyclic Processes

Mark-Oliver StehrUniversit�at HamburgFachbereich InformatikArbeitsbereich Theoretische Grundlagen der Informatik

5. September 1996Mark-Oliver StehrUniversit�at HamburgFachbereich Informatik - TGIVogt-K�olln-Str. 3022527 Hamburge-mail: stehr@informatik.uni-hamburg.de2

AbstractThis work investigates the axiomatic concurrency theory proposed by Carl Adam Petrias a basis of general net theory starting with physically motivated axioms. A formulationin terms of partially ordered sets is intensionally not adopted here, in order to deal withthis theory in a more general setting, viewing causality and concurrency as pure similarityrelations. Concurrency structures, which are the models of this theory, are intended todescribe the synchronisation structure of possibly cyclic processes at an arbitrary level ofabstraction.The major result of this work is that under certain conditions we can associate exactlytwo nets (of which one is the inverse of the other) with every concurrency structure. Anappropriate elementary-net-speci�cation based upon one of these nets has a case class thatcoincides with the class of statelike cuts. In other words, under appropriate assumptionssupplementing Petri's axioms the token game is sound and complete to evolve the dynamicsof concurrency structures. KurzfassungDiese Arbeit untersucht, die von Carl Adam Petri vorgeschlagene, axiomatische Concurren-cy-Theorie als Basis der allgemeinen Netztheorie, ausgehend von physikalisch motiviertenAxiomen. Eine Formulierung mit Hilfe von partiellen Ordnungen wird absichtlich ver-mieden, um die Theorie auf einer allgemeineren Grundlage zu studieren, die Nebenl�au�gkeitund Kausalit�at als reine �Ahnlichkeitsrelationen au�a�t. Die Concurrency-Strukturen, diesich als Modelle dieser Theorie ergeben, sollen die Synchronisationsstruktur von m�oglicher-weise zyklischen Prozessen auf einer beliebigen Abstraktionsebene beschreiben.Das Hauptresultat dieser Arbeit ist, da� wir unter bestimmten Bedingungen genau zweiNetze (ein Netz und sein Inverses) mit jeder Concurrency-Struktur assoziieren k�onnen. Eingeeignetes elementares Netzsystem, das auf einem dieser Netze basiert, hat ferner eine Fal-lklasse, die mit der Klasse der zustandsartigen Schnitte identisch ist. Mit anderen Worten:Das �ubliche Markenspiel ist unter geeigneten, Petris Axiome erg�anzenden Annahmen, ko-rrekt und vollst�andig, um die Dynamik von Concurrency-Strukturen zu entwickeln.3

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Contents1 Motivation 72 Cellular Automata and Discrete Physics 83 Basic Concepts 113.1 Pragmatic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Causality and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 States and Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Determinism and Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Axioms of Concurrency Theory 165 Properties of Concurrency Structures 205.1 Partial Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Cuts and Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Local States and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Details and Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Immediate Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.6 The Structure of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.7 Propagating Concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.8 Consistent Orientations and Nets . . . . . . . . . . . . . . . . . . . . . . . . 375.9 The Flow Relation in Cyclic and Acyclic Structures . . . . . . . . . . . . . 495.10 Acyclic Concurrency Structures . . . . . . . . . . . . . . . . . . . . . . . . . 535.11 Cone Intersection Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.12 Reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.13 Elementary Net Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 Conclusion and Open Questions 677 Acknowledgements and Recent Works 69A Notation and Basic De�nitions 74B Models of Concurrency Theory 815

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1 MotivationIs it true that between two points of time (space) there is always another one? Doesthe point of time (space) p2 really exist? The �rst question addresses the adequacy ofmodelling an analogous quantity by the set of rational numbers (equipped with the conven-tional total order) instead of using a scale of integers. The fundamental di�erence betweenintegers and rationals is that rationals do not contain any jumps as they are dense. Thesecond question goes even a step further appealing to the property of the totally orderedset of rational numbers to contain gaps. The usual way to �ll these gaps is to choose thesmallest completion which are the real numbers to describe analogous quantities.From the viewpoint of measurements, which can only supply a �nite amount of informationin a �nite interval of time, it will certainly never be possible to �nd any evidence, whichcould solve our problem in favour or against the assumption that the intrinsic character ofsome physical property is actually a real number that is not rational. So for our measure-ment scale it is su�cient to employ the rationals. Real numbers, which are conventionallyused to represent physical quantities, are insofar convenient as they allow the use of a richand mathematically well-elaborated library of analytic methods (namely those for partialdi�erential equations) to predict the time evolution of dynamical systems. On the otherhand only few complex problems can be solved analytically, that is, in such a way that onecan derive the exact solution in a �nite number of steps. So it is common practice to solvecomputational problems by numerical approximations. Again a calculation yields alwaysa �nite amount of information. So rationals are certainly su�cient to represent the resultwith arbitrary accuracy.But even with the rational model we have a paradoxical situation when we try to describesmooth movements. Certainly the physical motion of an object on some scale modelled byrationals should skip no element on its path. Intuitively it should visit the elements oneafter the other. But, as every rational interval consists of an in�nite number of elements,the moving object will never leave this interval, which is certainly a contradiction with ourexperience. The conventional solution is that the position of our object must be a functionof a rational time scale. But this only postpones the actual problem to another quantity,which is the time in this case: Is it not true that starting at a �nite interval of time everypoint is visited at some instant? Again this implies that an in�nite number points have tobe visited one after the other, which is intuitively not reconcilable with the boundednessof the interval. From a di�erent point of view we can reformulate the problem as follows:Given a rational it is impossible to �nd an immediate successor, such that the temporalevolution of a physical system necessarily stops (as it does not know where to go). Butaccording to our experience this does not happen.A radical answer to this paradox is to reject even the rationals as valid representatives ofanalogous quantities and to go back to the integers where we know that every interval isactually �nite. This leads immediately to the mathematical theory of discrete dynamicalsystems which has a counterpart in computer science, namely, in the theory of cellularautomata. In fact, recently there have been many (although non-concrete) proposals totake information (in form of binary decisions) as the most fundamental concept from whichevery other aspect of physical existence has to be derived (see Wheeler 1990). Followingthis line physical systems could be simply viewed as information processing systems.7

This preceding solution to the dilemma of analogous quantities is simply to deny theirexistence with the assumption that every physical property can be described by digitalquantities. But now we are faced with the problem that the digital representation mustbe exact to derive sound results, that is, we have to consider every detail of the problemdomain, what is impossible as the discrete nature (if it really exists) of physical quantitiesis a very �ne one and exact measurements are impossible due to practical and theoreticalrestrictions. Moreover the solution to represent every detail exactly by digital quantitiesand to reformulate microscopic physical laws on that level that is believed to be the atomicone leads to a mismatch between theory and application as most of the real-world problemshave to be solved on a macrosopic level.An alternative approach towards a solution of these problems might be given by concur-rency theory, which was proposed by C.A.Petri (e.g. in Petri 1987) as a general theoryto deal with uncertainty of analogous quantities, although this work concretely refers tospace and time. With the more general interpretation of concurrency theory as a theoryof measurement (Smith 1989) the two constituting relations of causality and concurrencycan be viewed as the relations of comparability and indi�erence, respectively. For totalorders (e.g. the standard models for measurement scales: integers, rationals and reals)every two elements are comparable. The explicit articulation of indi�erence (which arisesfrom the impossibility to compare two analogous quantities due to theoretical or technicalrestrictions) is a major aspect of concurrency theory. The essential idea of the time-spaceinterpretation is to identify physical laws that are valid on every conceivable level of ab-straction such that the incompleteness of the current view of a system is accepted as anatural restriction. The question if the actual nature of analogous quantities must be de-scribed by integers, rationals or reals is meaningless in this theory, as it allows us to adoptthat view that is adequate in the context of a particular problem. The general requirementto express uncertainty introduces a further dimension into the theory, that is orthogonalto the measured quantity. Applying this idea to the time-space interpretation the exis-tence of space (or concurrency) is a necessary consequence of the temporal dimension (orcausality). On the other hand time is a necessary resource to explore the spatial dimension.Before we step deeper into the matter an overview will be given of cellular automata todetermine di�erences and similarities between these two approaches.2 Cellular Automata and Discrete PhysicsA computational approach to discrete physics is based on Cellular Automata (Feynman1982; To�oli 1988) proposed by von Neumann originally to analyze the capability ofself-reproduction within cellular space. Interestingly they also provide a very natural anduniform model of parallel computing. A cellular automata can bee seen as an in�nite,regular network (typically an n-dimensional grid) of cells each of them being an identical�nite automata with the capability to communicate with a �nite neighborhood. A localrule determines the successor state of an automaton given the states of all cells in itsneighborhood. A con�guration of a cellular automaton ( the global state) describes theindividual local state of each cell. Cellular automata evolve in discrete time steps where anew con�guration is always obtained by applying the local rule to all cells synchronously.8

Cellular automata seem to be appropriate for the description of temporally and spatiallydiscrete dynamical systems respecting the locality principle, as the speed of information ow within the Cellular Automata is limited. The physical state has to be coded into thelocal state of each cell and global conservation of certain quantities can be realized by theappropriate choice of the local rule.Cellular automata are deterministic and may be irreversible since information about thepast may get lost if two con�gurations are mapped to the same successor. Microscopicreversibility of physics (which is also necessary for quantum mechanics of closed sys-tems) can be modeled with reversible cellular automata which are forward- and backward-deterministic by de�nition (To�oli und Margolus 1990). An important result in the �eldof reversible computation is the existence of universal, reversible machines.Identifying cellular space-time with physical space-time has led to interesting applicationsin the �eld of gas and uid dynamics. Nice theoretical results presented in To�oli 1990(although in a more general setting) are that (statistical) variational principles knownfrom analytical mechanics can be derived from very weak assumptions (e.g reversibility)and that statistically a microscopic grid is capable to show full rotational symmetry on themacroscopic level. Furthermore there are some basic ideas how Lorentz-invariance couldemerge as a statistical feature.Associating the global simulation time with the local time for actual events within thecellular automata may be adequate at low velocities, but Relativity Theory requires adi�erent time-space-metric for each inertial system moving in the cellular automata, whichis determined by the Lorentz-transformation.New approaches, in particular Fredkin's program of Digital Mechanics (see Fredkin 1990),try to go a step further. The naive correspondence between physical space-time and cellularspace-time is given up, and time is made explicit by assigning it to an additional dimension.This leads to a notion of information-cones within cellular space. These are exactly thoseregions relative to a particular cell which may have an in uence on its successor state.The explicit representation of time within the formalism is also an essential feature ofconcurrency theory.Unfortunately, so far there is no competitive approach of discrete physics, which is indeedan alternative to conventional theories. Actually, discrete physics su�ers from several in-conveniences arising from lacking mathematical methods for the treatment of dynamicalsystems, as the dynamics is not continuous in the sense of analysis. In contrast to the cel-lular automata approach concurrency theory leads to a generalized concept of continuitythat allows countable and even �nite sets to be continuous. Of course a closed and ap-plicable mathematical theory (e.g. a counterpart to analysis) does not yet exist, althoughthere are promising connections to the �eld of topology, which is also the modern basis ofanalysis.As a motivation for the need to investigate a di�erent theory, some hints are presented jus-tifying the view that (conventional) cellular automata are incapable of the exact simulationof physical systems and are furthermore not adequate, also from an application-orientedpoint of view.� A �rst point is the global synchronization of all cells. There is no evidence thatthis synchronization really exists in nature and furthermore it is not even necessary9

if the local time di�ers from simulation time by explicit simulation of local clocks.Asynchronous cellular automata may provide a solution to this problem.� Moreover, cellular automata are deterministic. Even if we assume physical determin-ism, this excludes the separate analysis of subsystems or macroscopic views whichare not necessarily deterministic due to environmental in uence. So either we canonly describe closed systems without environment, or we have to choose an approachusing nondeterministic cellular automata.� Nondeterminism emerges not only from unsolved alternatives but may also occur dueto concurrency. Concurrent nondeterminism arises from the fact that it is impossibleto de�ne an objective order of events, which are spatially separated from each other,such that the exchange of signals between them is impossible. How is it possible torealize this kind of nondeterminism with cellular automata? And how is it possibleto separate these two aspects of nondeterminism?� The vision of an exact simulation of all physical phenomena with cellular automatadepends on the general assumption that a discrete and exact representation of physicsis possible and that the actual atomic units of physical quantities (e.g. the smallestunits of time and space) and corresponding local rules can be found.� Even if an appropriate description of microscopic physical laws could be found interms of a local rule for cellular automata, the applicability is not guaranteed, asthe exact simulation, which has to be carried out on the atomic level will be prac-tically infeasible from the computational point of view. Although a straightforwardrealization of parallel computers on the basis of cellular automata is conceivable, theenormous number of elements, which are necessary to represent the states of a smallvolume element of interest, can certainly not be realized with current technology.� If an appropriate cellular automata can be found, this is certainly of interest forthe theoretical foundation of physics. On the other hand, if this cellular automatacan be actually realized, the practical value of an exact simulation is not obvious.Firstly the initial conditions are mostly not known exactly, and secondly exact resultscovering all microscopic details are often not necessary for practical applications.Unfortunately the local rules are only valid on the atomic level and a naive reductionof the resolution of the cellular automata (by reducing the number of cells and statesper cell) leads to incorrect results. Altogether we recognize the general necessity fora means bridging the gap between di�erent levels of abstraction which has not yetbeen developed for cellular automata.� A major problem of cellular automata, which is deeply connected to the previousargument, is the inherent discontinuity in the theory. There was no attempt to �nda solution to the apparent contradiction between discreteness and continuity. Con-cretely, the con�guration between two time-steps is not de�ned, there are jumpsbetween con�gurations due to jumps on the total order of integers, which has beenchosen as a model of time. Of course this is also true for all other quantities repre-sented within the cellular-space. At least on higher levels of abstraction this way ofdealing with analogous quantities is inadequate to describe smooth changes that arepart of our everyday experience. 10

� Again seen from the application point of view the regular grid-structure of cellularautomata is often inappropriate for the problem domain. The structure of the prob-lem has to be coded arti�cially into the states and the local rule. In other words,to apply cellular automata the non-uniform structure of a practical problem has tobe mapped to the uniform formalism of cellular automata. As a consequence, thelacking exibility of cellular automata might lead to unnecessarily complex repre-sentations of an originally simple problem. Certainly this point is linked with theprevious ones, as it is the right abstraction which is a primary ingredient to solvepractical problems.Comparing cellular automata and Net Theory these two approaches are in a certain sensearticulations of two di�erent extremes: The cellular automata formalism is based on struc-tural uniformity and puts the whole complexity into its local rule. In the formalism of netsthe local rule (which is the token game) is quite simple and the structure of a net is used torepresent the complexity of the problem. As this work only deals with concurrency theory,it will not be possible to provide a solution to all of the problems mentioned above. Thepreceding list should be taken as a motivation for the alternative approach of Net The-ory which includes concurrency theory as an essential component. We will mainly addressthe points concerned with causality and concurrency. In particular, concurrency theorydoes not assume an apriori global synchronization. We will not deal with those problemsconnected with the notion of state space, as they cannot be captured by the notion ofconcurrency and have to be examined in a more general theory. Concerning continuity itis believed that a solution of this issue can be provided by concurrency theory (as it wasindicated in Petri und Smith 1987), although a detailed analysis of this topic is beyondthe scope of this work. Finally we come to the question how to change between di�erentlevels of abstraction of the same system. Once we have established a connection betweenconcurrency theory and the formalism of nets (and this will be done in this work), topolog-ical methods (that can be found in Fern�andez 1975) can be applied to describe continuousmappings between di�erent views.3 Basic ConceptsGeneral Net Theory is an attempt to combine the di�erent special applications and for-malisms concerning nets into a uniform framework (Petri 1980b). It can be imagined as ahierarchy containing a theory of concurrency and possibility on the lowest level followed bya theory of nets, elementary net systems and information ow. Higher level net formalismsand specializations constitute the upper levels of General Net Theory.On a tutorial at Milano in April 1989 C.A.Petri presented a \Combined Axiomatics forConcurrency, Causality and Possibility" as a basis for General Net Theory. He also pre-sented these ideas in lectures at the University of Hamburg (Petri 1988a, Petri 1988c andPetri 1989). The major aim was to give a justi�cation for Net Theory based on a combina-torial formulation of physical laws. Unfortunately work on this axiomatic system has notbeen �nished yet, but nevertheless there are some interesting parts which have alreadybeen published (Petri 1980a; Petri 1982; Petri 1987). Concurrency theory can be con-ceived as a projection (or specialization) of the combined axiomatics including causality11

and concurrency but excluding possibility. To exclude possibility means that alternativesdo not occur. So we might imagine the structures of concurrency theory as describingexactly one possibility of the system's evolution. In addition to approaches using partialorders to describe the causality structure of processes (thoroughly developed in Best undFern�andez 1988), concurrency theory (as presented in Petri 1980a and Petri 1987) allowsspatially and temporally cyclic structures such that in�nite, repetitive processes can bedescribed by �nite means. In the following a survey will be given of the pragmatic ideasand fundamental physical concepts relevant in General Net Theory.3.1 Pragmatic IdeasIn order to get a rough impression how General Net Theory tries to capture apparentlyincompatible phenomena of very di�erent disciplines some general ideas will be mentionedthat might play a role in the development of a complete theory of systems. According toPetri 1988b the carrier of the theory should be a (possibly in�nite) setX of pragmatic unitsequipped with four symmetric and irre exive binary relations: Causality (li), Concurrency(co) and Alternative (al), Togetherness (to).For a given system the elements of X are articulations of desired or observed behavior.They are called pragmatic units, as they are those elements, which have to be consideredin a certain pragmatic context. This means that it depends on the problem domain andthe desired results how detailed the concrete problem is represented. In other words, thedegree of accuracy is determined by the application and not by the theory.The causality relation (li) holds between those elements that are causally dependent ofeach other. This means one element has a de�nite e�ect upon the other one. The directionof the e�ect is not essential here. Two elements are concurrent (co), if both of them occurbut without in uencing each other. Notice that concurrency does not necessarily implysimultaneity. Two elements are alternative (al), if they are mutually exclusive, that isthe occurrence of one of them excludes the occurrence of the other one. The relationof togetherness (to) introduces subjective (that is observer-dependent) aspects into thetheory. Di�erent observers might use a di�erent words to denote elements of X , which areobjectively indistinguishable. Elements related by to are coincident, that is, they occurat the same place and time and in the same state. As to is an equivalence relation, it ispossible to consider the objective quotient structure (X=to; li=to; co=to; al=to) proceedingwithout to. This is the reason why we are not concerned with to in this work.Apart from the requirement that li, co and al are symmetric and irre exive, a majorassumption (the completeness axiom) is that between every pair of di�erent elementswe can establish li, co or al. This means we can express the relation between every twoelements within the theory. A further elementary requirement is that li and co as well asli and al are mutually exclusive. Intuitively, it is clear that causality and concurrency arenot reconcilable and, of course, two elements which are mutually exclusive (al) cannot becausally dependent of each other. Further axioms concerning al are necessary, but as itwas mentioned above only that part of the theory dealing with li and co is worked outyet. Nevertheless the relation al will not be ignored in the following sections in order togive an idea into which direction concurrency theory might have to develop.Certainly we cannot deny the relevance of these relations on the low level of microscopic12

physics (e.g between particles) as well as on the high level of planning and organization (e.g.between persons, groups). An essential point concerning the axioms of the desired theoryis their validity on every conceivable level of abstraction. This is a strong requirement, asnot every physical law can be expressed in a form, which remains invariant, if the currentview of the system is changed.Once a theory is formulated in terms of causality, concurrency and alternative, an adequateformalism is required to deal with practical problems. It might turn out that the formalismof nets and elementary net systems is appropriate, as it allows us to describe causality,concurrency and alternatives in a natural manner. The major aim of this work is toshow the adequacy of elementary net systems for a theory restricted to causality andconcurrency.3.2 Causality and TimeFrom Relativity Theory we know that time is not absolute between moving inertial sys-tems. Observers in di�erent inertial systems record events with respect to their own sub-jective time-space-metrics. But if we apply the fact of limited signal propagation velocityconsequently, we �nd that we have even less knowledge about the time at di�erent placeswithin one inertial system: Consider one inertial system with two clocks at di�erent placesA and B. Both clocks are at rest. Now we would like to know if time within the inertialsystem is the same at A and B. Clearly we cannot decide this with arbitrary accuracy,since all signals which may be exchanged between A and B are limited by the speed oflight. What is typically done is to postulate that time is the same at A and B (this cannever be refuted). That time is absolute within one inertial system leads to the convenientLorentz-transformation, which is applied between inertial systems. In Net Theory it is noteven postulated that time is objective within one inertial system. As a theory based onobjective concepts is prefered, one can argue that time is inadequate as a basic notion anda theory formulated in terms of pure causality is justi�ed.So the notion of time in General Net Theory is not an elementary one. According to Petri,time is nothing more than the state of clocks. So time has to be modeled explicitly by clockswithin the system, which are synchronized with each other. In this sense time is treated asevery other analogous quantity that could be measured. As we have seen, clocks which arespatially separated may show di�erent times, even if they are synchronized. The formalmethod to deal with this in Net Theory is based on the concept of synchronic distancedetermining the degree of synchronization between these clocks (which might depend onthe spatial distance between them). For observers within the system it is impossible to�nd an experiment that could be carried out to detect the di�erence in time at di�erentlocations, as every exchange of signals forces the two clocks to resynchronize.If time or temporal properties are mentioned in the subsequent sections, this only refersto the objective aspect of causality but not to subjective delays between events.A major result of this work is that under certain conditions exactly two choices for thearrow of time are possible of which one is the inverse of the other. Intuitively these twodirections correspond to the temporal evolution of a physical system in opposite directionsof time. 13

3.3 States and LocalityThe global state of a system is a maximal set of elements that might coexist concurrently.Physically, the global state consists of all local states on a spacelike surface in time-space.The fact that the global state is distributed in space makes it impossible for observerswithin the system, which are themselves restricted by physical laws (in particular theycannot exchange signals faster than light), to exploit the global state completely on onespace-like surface.According to the principle of locality, the temporal evolution of the global state is governedby a local rule. As there is no apriori metric de�ned on the structure of pragmatic units, itis a major di�culty to de�ne an appropriate notion of locality. Of course, this is one of thepoints, which will be addressed in this work. As the rule is a local one, a global state, thatis extended in space, is developed independently at di�erent locations such that not allevents (the application of the local rule might be seen as an event) can be totally ordered(these events are also concurrent pragmatic units). So with the presence of concurrency atotal order of events is the subjective impression of a particular observer.With the presence of alternatives (al), we would have to deal with an additional problem,namely, the structure of a branched time scale. The temporal evolution might at a certaininstant of time decide to drive the system into one of several possible directions which aremutually exclusive.3.4 Determinism and ReversibilityThe usual notion of nondeterminism has to be separated into two di�erent aspects: Non-determinism of concurrency and nondeterminism of alternatives.Nondeterminism of concurrency occurs, if two spatially separated events which are causallyindependent of each other may occur in undetermined order, when they are mapped on atotal time scale of some observer. In particular di�erent observers may �nd di�erent occur-rence sequences of events, although the causality structure is always the same. Di�erentobservers will agree on the fact that these events occur, but they may disagree about theorder of perception. The time-reversal-counterpart of concurrency is synchronization.Nondeterminism of alternatives corresponds to the indeterminate choice between pos-sibilities for future dynamical evolution. In contrast to concurrent nondeterminism thealternatives are mutually exclusive. Alternatives can be interpreted as sources of informa-tion. Immediately after solving the alternative the system contains the information whichalternative has been chosen. Note that an alternative does not necessarily lead to nondeter-minism, since the alternative may be solved within the system. If all forward-alternativesare solved within the system the system is deterministic (with respect to alternatives). Thetime-reversal-counterpart of an alternative is the backward-alternative, where two or morepossible branches of time evolution are combined. Information contained in the systemmaybe erased in this case. To avoid the production and erasure of information it is possibleto conceive a completion of our system by some environment which solves all (forward-and backward-) alternatives such that the total amount of information is conserved. Sucha system which is forward- and backward-deterministic is also called reversible, and it isexactly this reversibility which is crucial in physical systems on the microscopic level.14

Notice that in concurrency theory we concentrate on nondeterminism of concurrency, butnondeterminism due to alternatives (which are essential to construct information process-ing systems) does not occur, as we do not have any alternatives at all.3.5 ContinuityGeneral Net Theory can be conceived as a method to deal with dynamical systems in away involving only �nite or countably in�nite concepts. The major criticism of GeneralNet Theory concerning todays physical methods (which are certainly successful, we cannotdeny that) refers to the assumption that continuous change can only be achieved by meansof uncountability. Here continuity means continuity of orders (also refered to by order-completeness). Continuity of total orders was axiomatized by Dedekind (a continuousorder does contain neither gaps nor jumps) who applied this notion to construct thecontinuum of real numbers. Petri recognized that a generalization of Dedekind-continuityfrom total to partial orders yields combinatorial partial orders which are continuous butnowhere dense. The precise de�nition of generalized Dedekind-continuity (D-continuity) ofpartial orders can be found in Petri und Smith 1987; C.Fern�andez und A.Merceron 1987;Best und Fern�andez 1988.It is not intended to deal with continuity in this work, but our choice of axioms which is notexactly the set of axioms proposed by Petri should be justi�ed. Concurrency structures,as they will be introduced below, should describe continuous changes (movements) intime-space. D-continuity with respect to certain sets of concurrency axioms was analyzedin Fern�andez und Thiagarajan 1983; Best und Merceron 1985. The result was that theoriginal set of axioms proposed by Petri in Petri 1980a does not imply D-continuity. Itis the cone-intersection-property (see Petri 1987) and combinatorialness (of the temporalpartial order) which are additionally necessary to derive D-continuity. We will see thatfor acyclic structures (which have to be de�ned) our choice of axioms implies the cone-intersection-property and ensures that lines are combinatorial.Strictly D-continuity is only de�ned for partial orders. We will see that not every con-currency structure can be represented by a partial order. It is lacking some more generalconcept to deal with cyclic structures. With this concept it should be possible to extendthe de�nition of D-continuity to cyclic orders (whatever this may be).For the sake of completeness a further form of continuity is mentioned, which is crucialwhen di�erent (more or less detailed) views of a system have to be combined via mor-phisms. If we de�ne an appropriate topology on our system (for nets this can be done ina natural manner), we can allow exactly those functions as valid re�nements which arecontinuous mappings with respect to this topology. Proceeding in this way we are alwayssure that re�nements do not destroy the coherence of our structure. It is obvious that re-�nements are essential for modelling physical systems (e.g. distinction between microstatesand macrostates). 15

4 Axioms of Concurrency TheoryConcurrency theory, as it was proposed by Petri, is the theory of concurrency structures,which can be applied to describe physically realizable, possibly cyclic processes in time-space. As it was already mentioned, a slightly modi�ed version of concurrency theoryis presented here. For a historical survey of concurrency theory comparing the di�erentaxiomatic approaches M�uller 1993 is worth to read. An overview of concurrency theory isalso given in Fenske 1992.A concurrency structure CS is a triple consisting of a set X (which may be �nite orin�nite) and two binary relations li and co de�ned on X satisfying a variety of axiomswhich are given below.Scope S1 Let CS = (X; li; co)^ li; co � X �X .Although there are further interpretations of concurrency structures (see Smith 1989)concerning the representation and measurement of analogous quantities only the standardinterpretation will be considered which is the following:� X is a set of elements in time-space. There are no a priori requirements on thenature ofX concerning dimensionality, cardinality or density. Furthermore no metricis assumed on X .� li is the causality relation. It covers our intuitive notion of the causal dependencebetween two elements. The direction of causal in uence is ignored here, avoidingto break the temporal symmetry by introducing causes and e�ects on this level.Altogether we can say that li establishes the temporal dimension in X .� co is the relation of concurrency. Two elements are concurrent, if they are spatiallyseparated enough to exclude any interaction between them. This is a parallel tothe notion of space-like distance between two points in Minkowski-Space. On higherlevels of abstraction it is not necessarily only the limited speed of light, which isthe source of concurrency. Concurrency may also be realized by excluding causaldependencies with the help of certain technical boundary conditions.We will use this interpretation to give a short motivation for each of the subsequentaxioms. Sometimes it will be necessary to anticipate some results that will proven in thesubsequent sections.Axiom A1 j X j > 1. 2This axiom excludes trivial structures which are empty or contain only one element. Inthese structures there is neither causality nor concurrency. Hence they are not of interestfor us.Axiom A2 [Completeness] co [ li [ idX = X �X . 2Every two distinct elements of time-space must be concurrent or causally dependent ofeach other. There is no further possibility for the relation between them.Axiom A3 [Disjointness] co \ li = li\ idX = co \ idX = ;. 216

Causality excludes concurrency. Together with the previous axioms this establishes con-currency and causality as complementary relations. That causality and concurrency areirre exive is assumed for mathematical convenience.Axiom A4 [Symmetry] co�1 = co. 2Concurrency is symmetric. This corresponds to the assumption that there is no preferreddirection in space which could cause an asymmetry (isotropy of space). Obviously we couldequivalently require li�1 = li which ensures the symmetry of causality. Hence there is noprivileged direction of time either. So if x li y holds we can say: x and y are causallydependent, x a�ects y, y a�ects x or x has an in uence on y and y has an in uence onx. In case of x co y we simply say x is concurrent to y which already expresses a certainsymmetry.Axiom A5 [Irreducibility] ~coX = ~liX . 2This is the axiom of irreducibility. It is similar to the axiom of extensionality of elemen-tary set theory. Every two distinct elements of time-space should be distinguishable interms of their relation of concurrency as well as causality with respect to other elements:~coX = idX = ~liX . We will see that the apparently weaker axiom is su�cient to guaranteethis property in combination with the previous axioms. Physically, it will be postulatedthat there is no interior structure which could lead to a further identity of elements intime-space. A structure which does not satisfy this axiom can be successively reducedby identifying those elements which cannot be distinguished in terms of concurrency orcausality. But this has to be taken with care, as it might happen that other axioms getviolated in this reduction process.Axiom A6 [Coherence] co�X = li�X . 2The axiom of coherence requires that two elements which are connected by a �nite chainof causality-steps should also be connected by a �nite chain of concurrency-steps and viceversa. We will see that with this axiom it follows that for an arbitrary pair of elementsthere are �nite causality- and concurrency-chains connecting them. It is exactly this axiomwhich establishes the �nite character of the theory. Note that this intentionally does notexclude in�nite concurrency structures.Axiom A7 [Finiteness of concurrent neighborhood] 8x 2 X : co[x] is �nite. 2For every element x it is postulated that there are only a �nite number of elements thatcan coexist concurrently to x. This requirement corresponds to the very strong assumptionthat space is �nite. We could argue that space may extend within the temporal evolution,but there was one instant of time where space was �nite, such that space remains �nitebut may expand arbitrarily. We will see that many important properties of concurrencystructures are connected with this axiom. A further reason to introduce this axiom is toavoid di�culties arising from in�nite cuts on the level of elementary net systems.De�nition D1 [Lines and Cuts]a) Lines := Kens(liX);b) Cuts := Kens(coX). 217

Lines and cuts correspond to time-like and space-like surfaces in Minkowski-Space. A lineis a maximal clique of causality also known as world-line. Each particle or signal propagatesalong some particular world-line. A cut is a concurrency-clique of maximal extension. Itcan be conceived as a spatial snapshot of our physical system and represents the globalstate at that instant of time relative to some observer which is not necessarily at rest.Axiom A8 [K-density] 8c 2 Cuts : 8l 2 Lines : c \ l 6= ;. 2The axiom of K-density formalizes our intuitive idea that every cut meets every line: Ifwe take an arbitrary snapshot of our system, every line should appear in this snapshot.Certainly it is impossible for propagating signals to jump over some particular cut. K-density requires the existence of certain elements in the intersection of a line and a cutwithout necessarily leading to conventional density.De�nition D2 [Proximity-Relation] x P y :, liX [x] � liX [y]. 2liX [x] � liX [y] indicates that every element that has a causal in uence on x also a�ectsy. This means the causal in uence of those elements in li[x] is conveyed to y via x. Aswe have even the stronger condition liX [x] � liX [y] there are elements a�ecting y whichdo not have any in uence on x. If these elements a�ect y, this in uence must be carriedto y via some element di�erent from x. Therefore y can be conceived as the center of aninteraction directly involving all elements of P�1[y], which are the signals participatingin this interaction. Viewing an interaction between signals as a local change of states, thecenter y will be called an event and x is a local state that is changed by y.Axiom A9 [No Changes of Changes] P 2 = ;. 2With this axiom we require that a local state cannot be an event and vice versa. If xis changed by y (x P y) then y is an event and cannot be changed by some z (y P z isimpossible). This axiom is fundamental in our theory, since together with the other axiomsit induces a partition of X into local states and events. It will turn out that every elementis either located in the range of P (if it is an event) or in the domain of P (if it is a signal).Note, furthermore, that this is the �rst axiom, which introduces an asymmetry betweencausality and concurrency indicating a fundamental di�erence between time and space.De�nition D3 [Immediate Neighborhood] im := P [ P�1. 2Typically the notion of change is introduced with the help of a local neighborhood. Hereexactly the opposite way has been choosen, since the concept of change arises naturally(and seems to be more fundamental) as it was introduced above. The immediate (temporal)neighborhood im[x] of some event X contains all local states which are changed by thisevent. The immediate neighborhood im[x] of a local state x contains exactly those eventswhich change x.De�nition D4 [Details and Detail Neighborhood]a) x D y :, coX [x] � coX [y];b) dn := D [D�1. 2Formally similar to P it is possible to introduce a relation D, where x D y holds, i� everyelement which is concurrent to x is also concurrent to y, and there are elements concurrentto y but not concurrent to x. We will see later that x D y may be interpreted as \x is a18

detail of y". The physical signi�cance of details is not yet fully understood, so we do notrequire any axioms here concerning D.Axiom A10 [Coherence on Lines] 8l 2 Lines(CS) : (imjl)�l = l � l. 2For each world-line it will be required that between every pair of elements on this line thereis a �nite im-chain which is completely covered by that line. Thus taking two elementsfrom some line, the e�ect of one element on the other occurs within a �nite number of steps(changes). This indicates a further manifestation of the �niteness already mentioned inconnection with the (�rst) axiom of coherence. Again in�nite world-lines are not excludedby this axiom.The following two axioms are concerned with the postulate that each of the elements of oursystem should be capable to infer the arrow of time by local rules without ambiguity. Oncethe system evolves in some direction of time, this direction is never changed (in analogyto the conservation of momentum) and there is no part of the system which might stopthe evolution. The arrow of time which can be conceived as a relation F , which orients therelation of immediate (temporal) neighborhood in some consistent manner (all we needhere is: F [ F�1 = im, F \ F�1 = ; and F 2 � li). A complete de�nition of \consistentorientation" is not necessary here, as it is possible to motivate the following two axiomsby pure symmetry arguments.Axiom A11 [Local Transitivity of Concurrency] 8x 2 X : (cojim[x])2 � coX jim[x]. 2From the axioms of symmetry, disjointness, completeness and coherence it will be shown inthe subsequent section that concurrency cannot be a transitive relation. What we postulatewith this axiom is a local transitivity of concurrency within the immediate neighborhoodim[x] of each element x. Local transitivity can be justi�ed as follows: Assume there arethree elements a; b; c in im[x] violating local transitivity, e.g. a co b, b co c and a li c. Then,if we choose an arrow of time F requiring that a occurs before x and x occurs before c,the temporal order between b and x is ambiguous, since we have a local symmetry (aand c could be exchanged). It is this kind of ambiguity which is avoided by this axiom.Furthermore every arti�cial ordering of x and b would violate the initial assumption ofconcurrency between either b and a or b and c. Another justi�cation for the local transi-tivity of concurrency is the transitivity of simultaneity, which is the limit of concurrencyfor small distances (here the smallest conceivable distance is given by the im-relation).Axiom A12 [Local Orientability] 8x 2 X : (lijim[x])2 � coX jim[x]. 2Assume that for some element x there are three elements a; b; c in im[x] and the axiomdoes not hold. Then these three elements might be causally dependent of each other, i.e.a li b, b li c and a li c. From the (local) view of element x the arrow of time concerninga; b; c is not determined: If we require that a occurs before x and x occurs before c, forreasons of symmetry (we could exchange c and a or c and b) the temporal order betweenb and x is ambiguous. Loosely speaking, x cannot determine the arrow of time by somelocal rule. Every local orientation would be arbitrary and this is what we exclude by thisaxiom.Axiom A13 [Local Extendability] 8x 2 X : idX jim[x] � (lijim[x])2. 2This axiom postulates that every element located within the immediate neighborhood ofsome x has at least one temporal predecessor or successor, which can be locally determined.19

Given a in im[x] the set of possible successors or predecessors is simply (lijim[x])[a], whichis not empty by this axiom.This completes our list of axioms, which will serve as a starting point for the followinginvestigation. The major changes to the original axioms of concurrency theory are thefollowing: The original axiom of im-coherence im�X = X � X has been replaced by thestronger version of coherence on lines, to avoid concurrency structures, where betweentwo elements on a line there is an in�nite number of further elements. The axiom of �niteconcurrency neighborhood, 8x 2 X : co[x] is �nite, which was not present in the originalformulation, has been chosen as an additional axiom. As the later property will turn outto serve as a su�cient condition for the cone intersection property, it is not necessary toensure this property by a further axiom, which would be di�cult anyway, as the originalformulation of the cone intersection property is based on the concept of partial orders. Asone section is devoted to this subject, we will get more into detail there.ExamplesIn Fig. 1.a we see the smallest known model satisfying these axioms. As j li j is large onlyco is shown and li = coX � idX is assumed. P is also shown, although it can be derivedfrom li as it was de�ned above. Fig. 1.b shows a regular, in�nite concurrency structure,which could be imagined as the unfolding of the previous one. Fig. 2.a gives a larger modelwith its in�nite unfolding, which is also a concurrency structure, in Fig. 2.b. The excactrelations of the �nite models are given in the appendix and the axioms have been veri�edautomatically...... .....

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2223 24Figure 2: Another �nite concurrency structure and its in�nite unfolding.We start with a summary of de�nitions already introduced above:De�nition D5 Let CS = (X; li; co)^ x; y 2 X ^ li; co � X �X .a) Lines(CS) := Kens(liX);b) Cuts(CS) := Kens(coX);c) x PCS y :, x P y :, liX [x] � liX [y];d) x DCS y :, x D y :, coX [x] � coX [y];e) imCS := im := P [ P�1;f) dnCS := dn := D [D�1. 2The axioms of concurrency theory we have already discussed are collected in the followingde�nition of a concurrency structure:De�nition D6 cs(CS) :,a) j X j > 1;b) co[ li[ idX = X �X ;c) co\ li = li \ idX = co\ idX = ;;d) co�1 = co;e) ~coX = ~liX ;f) co�X = li�X ;g) 8x 2 X : co[x] is �nite;h) 8c 2 Cuts(CS) : 8l 2 Lines(CS) : c \ l 6= ;;i) P 2 = ;;j) 8l 2 Lines(CS) : (imjl)�l = l� l;k) 8x 2 X : (cojim[x])2 � coX jim[x];l) 8x 2 X : (lijim[x])2 � coX jim[x];m) 8x 2 X : idX jim[x] � (lijim[x])2. 2For everything that follows we assume that CS is a concurrency structure satisfying allthese axioms. 21

Scope S2 Let CS = (X; li; co)^ cs(CS).Concurrency was postulated to be symmetric. It trivially follows that causality must besymmetric too, such that the formal symmetry between co and li (which indicates asymmetry between time and space up to a certain degree) is not violated by this axiom.Proposition P1 li�1 = li. 2Proof Assume there are x; y 2 X such that x li y and :y li x. Then D6b and D6c requirethat y co x and :x co y. This is impossible with co = co�1. 2The following remark gives a more compact notation of ~RX , that was used in D6e topostulate the extensionality principle of concurrency theory, which is derived in the nextproposition.Remark R1 Let R � X �X ^R be symmetric.~RX = ((R�RX)[ (RX �R))X . 2Proof By de�nition a ~RX b , 8x 2 X : (a RX x , b RX x). a ~RX b is equivalentto :9x 2 X : (a RX x ^ :x RX b) _ (:a RX x ^ xRX b) which is itself equivalent to:((a R�RXb) _ (a RX �R b)). 2As a direct consequence of D6e we �nd that it is actually true that two elements areidentical if and only if their relations of concurrency and causality to all other elements areidentical. Later on we will see that this extensionality principle has a natural counterparton the level of nets.Proposition P2 ~coX = idX ^ ~liX = idX . 2Proof It is clear that ~coX � coX and ~liX � liX . As we know furthermore coX\liX = idXit is necessary that ~coX \ ~liX � idX . Together with D6e we get ~coX � idX and ~liX � idX .And �nally it is evident that idX � ~coX and idX � ~liX . 2From axiom D6f we can derive that for each arbitrary pair of elements there is a �niteco-chain as well as a �nite li-chain. Note that this is actually a very weak �niteness, sinceit does not imply that all chains between two elements are �nite.Proposition P3 co�X = X �X ^ li�X = X �X . 2Proof We prove: X �X � co�X . Assume there are x; y 2 X with :x co�X y. This impliesx li y (by co[ li[ idx = X �X). But this contradicts co�X = li�X . The proof remains validif we exchange co and li. 2The last proposition reveals an important property of concurrency structures which is thenon-transitivity of concurrency and causality (more precisely coX and liX). This point isan essential di�erence to approaches postulating transitivity of concurrency !Corollary C1a) co�X 6= coX ;b) li�X 6= liX . 2Proof coX = X�X implies li = ; (D6b, D6c) which contradicts li�X = X�X if j X j > 1(D6a). The proof for li is analogous. 222

5.1 Partial OrdersGiven an arbitrary partial order (X;�) we can derive a causality relation li = (� [ ��1)�idX and a concurrency relation co = liX � idX (the relation of disorder). In this way wecan separate a special class of concurrency structures, which can be represented by partialorders. This is a proper subclass as concurrency structures may by cyclic in general, suchthat they cannot be covered completely by the formalism of partial orders. Neverthelessis is convenient to introduce the following de�nition:De�nition D7 Let poset(X;�).cs(X;�) :, cs(X; li; co) whereli = (� [ ��1)� idX and co = liX � idX . 2It is clear that the same concurrency structure in terms of co and li is given by a partialorder and its converse.Remark R2 Let poset(X;�).cs(X;�), cs(X;��1). 2Yet it is not said whether the orientation of the partial order is somehow related to thearrow of time (which is not yet de�ned). We will discuss this point in a more generalsetting when we deal with the orientation of concurrency structures. Choosing partialorders as the fundamental structure is a conventional approach to process theory (Bestund Fern�andez 1988). Here it is intentionally tried to deal with concurrency theory inits general form, although several di�culties arise, as the proof techniques known frompartial orders cannot be simply applied (in particular one cannot exploit the nice propertyof transitivity).5.2 Cuts and LinesIn Minkowski-Space we can de�ne space-like surfaces and world-lines with the help of lightcones, such that a world-line is always contained in some cone, and a space-like surface hasto be contained in the complement of some cone. In concurrency theory similar notionsare desired, but the de�nition is not immediately obvious since, as we have seen in P3,concurrency and causality cannot be transitive. In particular they are no equivalencerelations such that the concept of equivalence classes of co and li have to be replaced bya more general idea, which can be applied to relations which are only partially transitive(transitivity with respect to a certain subset of elements). The solution is not di�cult:Cliques of a relation denote exactly those subsets of elements which are transitive withrespect to concurrency or causality. Kens are maximal cliques, that is, they cannot beextended by adding further elements.Lines are de�ned to be Kens of causality, and cuts are Kens of concurrency (more precisely,coX and liX). In our standard interpretation a cut corresponds to the complete, spatiallydistributed state. A line is a set of elements (signals and events) similar to a world-linebut note that lines as well as cuts are unordered sets.In contrast to world-lines and space-like cuts in Minkowski-Space, where between everytwo points we can �nd a further one (density), we required the axiom of K-density, whichguarantees that our structure is \dense"-enough to ensure that every line intersects with23

every cut. We may think of K-density as the actual purpose of density, although we shouldare aware of the fact that in Minkowski-Space density fails to satisfy this purpose, as it isnot K-dense (there is a world-line and a cut which do not meet each other).From the assumption that the intersection of a cut and a line is not empty we can eveninfer that they must meet in exactly one element as the following proposition shows.Proposition P4 8c 2 Cuts(CS) : 8l 2 Lines(CS) : j c \ l j = 1. 2Proof Assume there are c; l with x; y 2 c \ l and x 6= y. Then x co y and x li y whichcontradicts D6c. Thus 8c 2 Cuts(CS) : 8l 2 Lines(CS) : j c\ l j � 1 and together withD6h we get the proposition. 2That each clique can be extended to a ken is clear for �nite structures, but we have toapply Zorn's Lemma to prove this in general.Proposition P5 Let C be a clique of R. Then 9K 2 Kens(R) : C � K. 2Proof Let CL = fC 0 : C 0 is a clique of R and C � C 0g and notice that (CL;�) isa poset. Choose an arbitrary chain cl � CL such that (cl;�) is a total poset. De�nescl 2 Sup(�; cl) which is scl = (S cl) 2 CL. Since cl was arbitrary every chain cl has asupremum within CL. Applying Zorn's Lemma we �nd 9K : K 2 Max(�; CL) whichshows that K is a ken of R with C � K. 2As a simple but useful corollary we �nd that every element can be extended to a lineas well as to a cut. Furthermore every pair of elements which are concurrent or causallydependent is part of some cut or line, respectively.Corollary C2a) 8x 2 X : 9l 2 Lines(CS) : x 2 l;b) 8x 2 X : 9c 2 Cuts(CS) : x 2 c;c) 8x; y : x li y ) 9l 2 Lines(CS) : x; y 2 l;d) 8x; y : x co y ) 9c 2 Cuts(CS) : x; y 2 c. 2As every element of X is contained is some line and some cut, the following corollary isimmediate, which states that the whole structure is covered by lines as well as cuts.Corollary C3a) X = SLines(CS);b) X = SCuts(CS). 2From the axiom of �nite concurrency neighborhood D6g it immediately follows that all cutsare �nite. That in�nite lines are possible, on the other hand, is illustrated by the examplein Fig. 1.b. So the causality neighborhood li[x] of some element x is not necessarily �nite,so we recognize a further asymmetry between concurrency and causality in our formulationof the theory.Remark R3 8c 2 Cuts(CS) : c is �nite. 2Proof A direct consequence of D6g. 224

K-density as it was already required by D6h can be de�ned independently of concurrencystructures for any symmetric and re exive relation R on a set X . In contrast to K-densitywhich requires a global view of R (we have to determine kens) N-density is a local formof density, as we have to verify the property given below only within the neighborhoodof each tuple (a; b; c; d). The name N-density comes from the fact that we have to lookat those elements (a; b; c; d) that resemble the shape of the letter N with respect to co aswell as li (this is the left-hand-side of the implication below). Loosely speaking, N -densitypostulates the existence of an element in the intersection of the N of li with the N of co(the right-hand-side of the implication).De�nition D8 Let R � X �X ^R \ idX = ; ^ R�1 = R ^ S = RX � idX .a) KDense(X;R) :, 8r 2 Kens(RX) : 8s 2 Kens(SX) : r \ s 6= ;;b) NDense(X;R) :, (8a; b; c; d2 X : a R c ^ b R d ^ c R d ^ a S b ^ a S d ^ b S c)9e 2 X : c R e ^ d R e ^ a S e ^ b S e). 2The following proposition shows that K-density implies N-density. As there are relationswhich are N-dense but not K-dense, N-density could be characterized as the local aspectof K-density.Proposition P6 KDense(X;R)) NDense(X;R). 2Proof Let KDense(X;R) (1) and a; b; c; d 2 X with a R c ^ b R d ^ c R d ^ a S b ^a S d^b S c. Certainly we have some l 2 Kens(SX) with a; b 2 l and some c 2 Kens(RX)with c; d 2 c. By 1 the must be some element e with e 2 c \ l. By de�nition of Kens it isnecessary that e R d ^ e R c and e S a ^ e S b. And this is exactly the e required by ourproposition. 2The converse is not true, even not for �nite relations R as we see in Fig. 3 which gives astructure which is N-dense but not K-dense.R

ken(R)

ken(R)Figure 3: N-dense but not K-denseAs a direct consequence of D6h we have K-density and N-density with respect to concur-rency as well as causality.Remark R4a) KDense(X; co)^KDense(X; li);b) NDense(X; co)^NDense(X; li). 25

2The following example shows that the axiom of K-density is independent of the otherconcurrency axioms (even for the �nite case): Fig. 4 gives a slight modi�cation of theconcurrency structure, that was already shown in Fig. 1. It is not K-dense (as it is notN-dense: a,b,c,d constitute the N) but satis�es all other axioms. Furthermore this exampleshows that results found for (occurrence) posets in Best und Fern�andez 1988 (e.g. everyoccurrence poset is N-dense) cannot be easily transferred to concurrency structures (whichare no posets).coim

a

b

c

dFigure 4: Satis�es all axioms except for K-densityThat K-density cannot be derived from N-density (at least for in�nite structures) if wesatisfy all Concurrency Axioms except for D6h is shown by a simple model, which is a twodimensional in�nite grid with respect to the relation im. The formal de�nition is:X = f(x; y) : (x; y) 2 Z�Z ^ :(odd(x)^ odd(y))g;(x0; y0) coX (x1; y1) :, ((x0 � x1) ^ (y0 � y1)) _ ((x1 � x0) ^ (y1 � y0));co := coX � idX ;li := X �X � co� idX .im and some part of co and li is shown in Fig. 5. (X; li; co) is N-dense but not K-dense:The line l = f(x; 0) : x 2 Zg does not intersect with the cut c = f(x; 1) : x 2 Z^even(x)g.A �nite example of the last kind in not known. Hence it is an open question, if for �nitestructures we could replace the axiom of K-density by N-density without changing theclass of models.K-density plays a crucial role when building the bridge to elementary net systems. Fur-thermore K-density is one of the necessary ingredients for D-continuity. In that context itis used to ensure gap-freeness. The undesired jumps are excluded by additional require-ments. For further details concerning K-density of partial orders it is refered to Fern�andezund Thiagarajan 1985, Pl�unnecke 1985 and Best und Fern�andez 1988. General K-densityof concurrency structures, which are not partial orders, has not been investigated much.26

licoim

. . . .. . . . ....

. . . .

. . . . . . . .....

. . . .

Figure 5: N-dense but not K-dense5.3 Local States and EventsAccording to the principle of locality in one step of system evolution an element in uencesor is in uenced only by elements, which are located in the immediate neighborhood. In factwe have to deal with the immediate temporal neighborhood here, as it is the relation ofcausality from which the in uence emerges. But how is it possible to derive the immediateneighborhood from a causality relation, which apparently does not reveal any informationabout the distance between elements? It turns out that the axiom of extensionality (D6e)plays a crucial role as it guarantees that for two distinct elements x; y 2 X we are alwayssure that liX [x] 6= liX [y], which means that causality cones of two elements are alwaysdi�erent. So it is a natural assumption that the distance between two elements is com-pletely expressed by the relation of causality. Astonishingly, it is appropriate to de�ne twoelements to be immediate neighbors if and only if their causality cones are comparable.Intuitively (if we imagine the light cones in Minkowski-space) this cannot be satis�ed, iftwo elements are far away from each other. And this is exactly the way we have de�nedthe symmetric relation im := P [ P�1 of immediate neighborhood on the basis of theproximity relation P (x P y :, liX [x] � liX [y]). In this sense x P y means that thecausality cone of x is contained (as a proper subset) in the causality cone of y.It could be argued that this is a very strict requirement for immediate neighborhood, butthere seems to be no alternative. Fortunately it follows from D6j that the neighborhoodcannot be empty (C4). Indeed, it can be proved from the coherence of lines (D6j) that aconcurrency structure is coherent in terms of im, such that between every two elementswe can �nd a �nite im-chain as the following proposition shows.Proposition P7 im�X = X �X . 2Proof Certainly we have 8l 2 Lines(CS) : (imjl)�l = l�l by D6j and li�X = X�X by P3.So between every two elements x; y 2 X we can �nd a �nite li-chain (x = x0; x1; : : : ; xn =y). For every pair xi; xi+1 there is a line l 2 Lines(CS) with xi; xi+1 2 l and by D6j a�nite im-chain (xi = zi;0; zi;1; : : : ; zn = xi+1) which proves that xi im�X xi+1 and x im�X y.27

2In the original proposals for concurrency axioms it was exactly this proposition P7, whichwas required as an axiom instead of D6j. In Best und Merceron 1985 it was recognizedthat D-continuitiy does not follow with that choice of axioms. One reason is that D6j doesnot generally hold within that system, so we have chosen it as an axiom.Corollary C4 8x 2 X : im[x] 6= ;. 2As we have seen, the derivation of the immediate neighborhood from causality naturallyleads to P the proximity relation as a by-product. In the following we try to motivate ourinitial interpretation of x P y as \x is changed by y" from a di�erent point of view. Firstwe observe that P is an asymmetric and irre exive relation.Proposition P8a) P \ P�1 = ;;b) P \ idX = ;. 2Proof By D5c. 2Furthermore it is evident that P � li and im � li, as our intuitive idea of temporalneighborhood suggests.Proposition P9 P � li. 2Proof Assume x P y. By de�nition of P we have liX [x] � liX [y] and in particularx 2 liX [y] which implies x = y or x li y. But x = y is excluded by P8. 2Corollary C5 im � li. 2As every element has a neighborhood (P7), we know that this element is located in therange or in the domain of P . Those elements contained in Dom(P ) will be called S-elements or local states, and elements of Ran(P ) will be called T -elements or events. Thiswill be justi�ed later when it will turn out that depending on the element type (S or T )there will be di�erent constraints on the number of immediate neighbors (im[x]). Noticethat this de�nition indicates a certain relation to places and transitions in the formalismof nets, and later it will turn out that this is indeed the case.De�nition D9a) SCS := S := Dom(P );b) TCS := T := Ran(P ). 2Remark R5 S [ T = X . 2Proof From im�X = X �X (P7) we get (P [ P�1)�X = X �X . Then 8x 2 X : 9y 2 X :x P y _ y P x and 8x 2 X : x 2 Dom(P ) _ x 2 Ran(P ). 2Each element is either a local state or an event and not both. This is explicitly requiredby D6i. In this way we have established a partition of X into local states and events.Generally we associate local states with passive entities, which are changed by events asactive instances. Notice that this is indeed consistent with our interpretation of P .28

Remark R6 S \ T = ;. 2Proof By D6i. 2See Fig. 1 and 2 where the partition in S-elements and T -elements can be easily imaginedas P is always directed from S to T .Triangles (i.e. closed chains of size 3) of im are excluded, as the following propositionshows. This can be easily generalized by induction to the statement that closed im-cyclesof odd length do not occur.Proposition P10 im2 � imX . 2Proof Imagine our proposition does not hold. Then we can �nd an im-triangle x; y; zwith x im y ^ y im z ^ z im x. Due to P8 and D5e we have either x P y or y P x.Proceeding with the former case (the later case is analogous) we need z P y to ensurethat P 2 = ; (D6i). Finally we have to choose x P z or z P y yielding a contradiction withP 2 = ; is both cases. 2Proposition P11 8n 2 N : im2n � imX . 2In the next two lemmas some useful relations between P , co and li will be derived thatwill be applied subsequently in several proofs.The �rst lemma shows that x P y holds, i� and only if we have x li y and there is no z,such that x li z and z co y.Lemma L1 P = li� li � co. 2ProofAt �rst we show x P y ) x li y ^:x (li �co) y. Assume x P y. P � li yields x li y (P9).Assume x (li �co) y. Then 9z : x li z^z co y implies z 2 liX [x]^z 62 liX [y]. Contradictionwith x P y which is equivalent to liX [x] � liX [y].Now it is left to prove x li y^:x li � co y (1)) x P y. Assume :x P y which is equivalentto :(liX [x] � liX [y]) (2). Then there are two possibilities: :x li y (which contradicts 1) orx li y, which requires in combination with 2 that 9z : z liX x^:z liX y. This is equivalentto 9z : z liX x^ z coy. For z = x we have x co y, which is not reconcilable with x li y. Sowe conclude 9z : z li x^ z co y or equivalently x li �co y. But this is a contradiction withour assumption 1. 2Moreover, if we have x P y this requires the existence of some z, such that x co z andz li y.Lemma L2 P � co � li. 2Proof By D5c x P y is equivalent to liX [x] � liX [y]. This suggests 9z : z 62 liX [x] ^ z 2liX [y] implying (by D6c and D6b) that 9z : z 2 co[x] ^ z 2 li[y] (z = y is impossible byP9, which requires x li y). Then we have 9z : x co z ^ z li y leading to x(co � li)y. 2Concurrency structures are not only im-coherent but also coherent with respect to thecomplement of im.Proposition P12 (imX)�X = X �X . 2Proof By P3 we have co�X = X �X . From im � li we �nd co � (X �X � im). 229

That for each element we have a non-empty neighborhood can be sharpened to the state-ment that every element must have at least two neighbors. This is a minimal requirementto prevent world-lines from ending somewhere in time-space without the possibility oftemporal continuation. Regrettably we will see that this is no guarantee (Actually, weneed ASS1.).Proposition P13 8x 2 X : j im[x] j � 2. 2Proof In C4 we have seen that j im[x] j � 1. To satisfy idjim[x] � (lijim[x])2 (D6m)there must be at least one further element in im[x] otherwise (lijim[x])2 = ;. 2Finally, although this is only a minor simpli�cation, it has been found that a shorter formof the original axiom A13 is su�cient as it is formulated in A14. So the following axiomhas to be taken as an alternative to A13.Axiom A14 8x 2 X : (lijim[x])2 6= ;. 2This axiom postulates that within the immediate neighborhood there are at least two ele-ments which are causally dependent of each other. From the viewpoint of x this correspondsto the existence of at least two local directions of time (the past and the future). In somesense it is even stronger than A13 as it directly implies that the immediate neighborhoodhas at least two elements.The proof of the following proposition, which is the same as A13 shows that this apparentlyweaker axiom would also be su�cient (although A13 is prefered for its physical evidence).For every element u 2 im[x] we can �nd at least one element u0 2 im[x] within the sameimmediate neighborhood which is in uenced by u. Viewing u0 as the temporal successoror predecessor of u this proposition guarantees the local existence of a temporal direction,where the evolution of the system seen from x might continue.Proposition P14 Assume A14. Then 8x 2 X : idjim[x] � (lijim[x])2. 2Proof Choose some arbitrary x 2 X . It is su�cient to prove the claim that for every u 2im[x] we can �nd some u0 2 im[x] with u li u0. A14 suggests the existence of y; z 2 im[x]with y li z. So by D6d the claim is already satis�ed, if we choose y and z for u and u0 orvice versa. Now assume u 2 im[x] is a further element with u 6= y and u 6= z. Accordingto D6b we have y li u or y co u. In the former case the claim holds trivially, if we choosey for u0. In that later case we cannot have u co z as this is incompatible with D6k. So wemust have u li z and choosing z for u0 prove the claim. 25.4 Details and ObservablesAbove the immediate (temporal) neighborhood was introduced as a subset of the causalitycone of a particular element. Although there is no evident physical interpretation, we couldexploit the symmetry between causality and concurrency, which has been maintained bythe axioms D6b{D6h and proceed in a similar way de�ning concurrency cones coX [x] ofelements x. The formal counterparts of P and im are D and dn, respectively. We havede�ned x D y :, coX [x] � coX [y] and dn := D[D�1. x D y means that the concurrencycone of x is completely covered by the concurrency cone of y, and everything that isconcurrent to x is necessarily concurrent to y. So, if we observe x as part of a cut, we30

are always sure that y must be contained in the same cut (the converse is not necessarilytrue). This leads to a surprisingly natural interpretation of x D y as \x is a detail of y".Of course the detail relation D is asymmetric and irre exive as P , and it can be onlyestablished between concurrent elements.Proposition P15a) D \D�1 = ;;b) D \ idX = ;. 2Proof By D5d. 2Proposition P16 D � co. 2Proof By D5d. 2Detail neighborhood and immediate neighborhood are always disjunct.Corollary C6a) dn � co;b) im\ dn = ;. 2The D-relation of the smallest known concurrency structure is shown in Fig. 6.Pa)

..... .....

b)

D

Figure 6: A concurrency structure and its detail-relationThe lemmas L1 and L2 can be translated directly, as they only rely on the lower axiomswhere symmetry between causality and concurrency is preserved.Lemma L3 D = co� co � li. 2Proof Similar to L1. 2Lemma L4 D � li �co. 2Proof Similar to L2. 2In Fig. 7 we have another �nite concurrency structure (X; li; co) (more precisely given inthe appendix) with its detail-relation. It can be easily veri�ed that D2 = ; and dn�X =X � X , which suggests a further symmetry between D and P . In fact, this is true for31

some concurrency structures, but it does not hold in general. A counterexample withdn�X 6= X �X is already given by Fig. 2, where the detail relation is empty. That D2 6= ;may occur is proven by the example in Fig. 7.1

2 3 4 5 6

78910

11 12 13

1415

16

coimD

Figure 7: Details of DetailsAs a consequence, the coherence and non-emptiness of detail neighborhood (the counter-parts of P7 and C4 cannot be derived, because an axiom similar to D6j was not required forcuts. For sake of completeness, it should be mentioned that these two properties (detailaxioms) were required in Petri 1987 (for their mathematical beauty but not for physi-cal reasons) but have been given up, probably because it turned out that they are toorestrictive.What is demonstrated by Fig. 7 is that the detail-relation establishes a hierarchy, whichcan be formally described by the fact that (X;D) is a strict partial order. Indeed, thiswas already true for P but this order is degenerate, as it is a collection of pairs whichare not connected with each other. Recent ideas of Petri indicate that allowing arbitrarystrict posets of P leads to a consequent generalization of nets incorporating an interestingconcept of dimensionality into net theory. The signi�cance of similar ideas with respect toD is not yet clear.Proposition P17a) (X;D) is a strict poset;b) (X;P ) is a strict poset. 2Proof This strict poset is inherited from the poset (P(X ),�) in the de�nitions of D andP . 2The interpretation of D already designates all elements of Dom(D) as details. Lead bythe notion of observables in quantum mechanics, where due to Heisenberg's uncertaintyrelation not every detail (of a systems state) can be observed by one measurement, wemight try the following (speculating) de�nition:32

De�nition D10a) Details(CS) := Dom(D);b) Observables(CS) := Ran(D). 2Proceeding in this way we �nd the intuitively expected property of observables: Everyobservable is a local state and not an event. But as we have seen in the examples above,not every local state must be necessarily observable, and in general not every event is adetail.Proposition P18 Observables(CS)� S. 2Proof Assume x D t with t 2 T . By D5d there is some z with x li z ^ z co t. Sincet 2 T there must also be a b with b P t. Now what is the relation between x and b?According to D5d is cannot be x co b. But by L1 it cannot be x li y. Altogether this leadsto contradiction with D6b. 25.5 Immediate NeighborhoodIn this section we deal with the e�ect of axioms D6k, D6l and D6m upon the immediateneighborhood of local states and events. Remember that the purpose of these axioms wasthat within the space-time of locally interacting elements it should be possible for eachelement with its restricted view to infer in which temporal direction it has to drive thesystem's evolution. As boundary conditions it was required that there is no ambiguity inthe local time orientation rule the element applies (i.e. there is at most one solution) andthere is always a direction in which time evolves (i.e. there is at least one solution). In anycase one should exclude a local time reversal, that is, from the local view of an element thedirection into which time goes should be di�erent from that where it has come from. Theglobal evolution of a system (which could be imagined as the movement of the global staterepresented by a cut through time-space) will be uniquely (except for non-determinism dueto concurrency) determined by this local evolution rule. The question, if this is su�cientto guarantee the unique existence of a global arrow of time, will be addressed separatelylater on.First we recognize that the axiom requiring local transitivity of concurrency (D6k) estab-lishes an equivalence relation in the immediate neighborhood of each element.Proposition P19 8x 2 X : coX is an equivalence on im[x]. 2Proof That coX is re exive on im[x] and symmetric is clear. That coX jim[x] is transitiveis ensured by (coX jim[x])2 � coX jim[x] which follows directly from D6k. 2To exclude local freezing of temporal evolution it is necessary to ensure that causality doesnot vanish in the scope of the immediate neighborhood.Remark R7 8x 2 X : lijim[x] 6= ;. 2Proof We know that im[x] 6= ; (C4). By D6m there are a; b 2 im[x] with a li b ^ b li awhich together with D6c implies lijim[x] 6= ;. 2By the help of the equivalence relation introduced above we can decompose the immedi-ate neighborhood of each element x into a number of equivalence classes establishing a33

partition on im[x]. Note that x itself is not contained in im[x], and each of these classesis a clique of coX . Furthermore two equivalence classes are not covered by a common cut.Remark R8 8g; g0 2 im[x]=coX : (g 6= g0 ) :9c 2 Cuts(CS) : g; g0 � c). 2Proof Let g; g0 2 im[x]=coX . Then 8x 2 g : 8x0 2 g0 : x co x0 implies g = g0. Contradic-tion. 2Hence we can conceive each of these local equivalence classes as representing (local aspectsof) di�erent global states, which arrive/leave the considered element from/in di�erentdirections of time. As we have con�ned ourselves to models without branching time (byexcluding alternatives on this level), we should expect that only two equivalence classesexist (corresponding to temporal predecessor and successor states), what is easily provedby the following proposition.Proposition P20 8x 2 X : j im[x]=coX j = 2. 2Proof D6m ensures that there are a; b 2 im[x] with a li b. This implies j im[x]=coX j � 2.(lijim[x])2 � coX jim[x] (D6l) shows that j im[x]=coX j � 2. 2Applying D6l to local states leads to the the fact that each state has exactly two neighbors.As we know that neighbors of states are always events, we could denote one of them asthe sender of a signal and the other one as the receiver.Proposition P21 Let s 2 S. Then j im[s] j = 2. 2Proof P13 already shows j im[s] j � 2. So there are x; y 2 im[s] with x 6= y. Now assumethere is an additional z 2 im[s] with z 6= x and z 6= y. s 2 S implies s P x^ s P y^ s P z.Then by L1 we must have x li z, y li z and x li y. Using (lijim[s])2 � coX jim[s] (D6l) wederive a contradiction since we have x li z and z li y but neither x co y nor x = y. 2The immediate neighborhood is �nite. This corresponds directly to the locality principleknown from physics that temporal evolution should be governed by local laws. Noticethe parallel to cellular automata, where �niteness of neighborhood is motivated in thesame manner. Here neighborhood-�niteness emerges trivially from �niteness of all cuts(although lines may be in�nite). A more general approach to concurrency theory mightrequire the following proposition as an axiom instead of cut-�niteness, but this may leadto the problem of reachability of an in�nite cut on the level of elementary net systems.Proposition P22 8x 2 X : im[x] is �nite. 2Proof Assume the contrary: There is an element x with in�nite neighborhood j im[x] j 62N. Then P20 gives us im[x]=coX = fE1; E2g such that E1 or E2 must be in�nite. Assumej E1 j 62 N then there is an in�nite cut c 2 Cuts(CS) with E1 � c. Contradiction with R3.2Lemma L5 X is countable. 2Proof If X is �nite this is clear. So assume X is in�nite. Fix an arbitrary element x. P22implies that Nj := j imjX [x] j is �nite for every j 2 N. Therefore for every j we can �nda function dj : N[0; : : : ; Nj]! imjX [x]. Furthermore without loss of generality we assumei <= j implies 8k 2 N[0; : : : ; Ni] : di(k) = dj(k). Combining all dj for j 2 N we canestablish an enumeration of all elements of X , that is, a surjective function d : N ! Xde�ned by d(j) := dj(j). d is well-de�ned as for every j we have j 2 Dom(dj) because34

j <= Nj (remember that due to D6j there is an in�nite acyclic im-chain containing x).According to P7 every z 2 X is contained in imj [x] for some j 2 N such that d is surjectiveon X . 2Of course as it is already suggested by D6j the elements on a line are also countable:Corollary C7 Let l 2 Lines(CS).Then l is countable. 25.6 The Structure of LinesSo far we have seen lines as unordered sets of elements. With D6j it is postulated that everytwo elements on a line are connected via a �nite im-chain, which is completely covered bythat particular line. Due to the properties of the proximity relation P (which is necessaryfor im) these chains consist of local states and events appearing in an alternating fashion.So on a line between (to be de�ned in terms of im) two events there is always a state andvice versa.Considering the intersection of the immediate neighborhood of an element with a linecontaining this element gives us a natural vehicle to classify elements with respect to thatline into four categories:De�nition D11 Let l 2 Lines(CS).a) IsolatedPointsCS(l) := fx : x 2 l ^ j (imjl)[x] j = 0g;b) EndPointsCS(l) := fx : x 2 l ^ j (imjl)[x] j = 1g;c) MidPointsCS(l) := fx : x 2 l ^ j (imjl)[x] j = 2g;d) BranchedPointsCS (l) := fx : x 2 l ^ j (imjl)[x] j � 3g. 2As our intuitive notion of world-lines suggests, elements on a line are neither isolated norbranched.Lemma L6 Let l 2 Lines(CS). Then IsolatedPointsCS (l) = ;. 2Proof Let x 2 l. If l \ im[x] = ; then D6j cannot be satis�ed. 2Lemma L7 Let l 2 Lines(CS). Then BranchedPointsCS (l) = ;. 2Proof Let x 2 l and assume j l \ im[x] j � 3. Then there are a; b; c 2 im[x] with a li b^b li c ^ a li c. This directly contradicts (lijim[x])2 � coX (D6l). 2Moreover, lines should have no endpoints, as this could lead to partially dead systems.Certainly D6m is a necessary condition to ensure this, but is it really su�cient (in com-bination with the other axioms)? The following assumption has not been proved. On theother hand, it is believed that it holds, as no counterexample has been constructed. In anycase it is physically justi�ed, and, if it turns out that it cannot be proved, we should takethis as an axiom, e.g. as a stronger form of D6m. Regrettably we cannot do without thisassumption, as it is heavily linked with P24 and P28, which themselves provide a essentialconnection to the formalism of elementary net systems.Assumption ASS1 Let l 2 Lines(CS). Then EndPointsCS(l) = ;. 235

Altogether we come the the conclusion that the elements constituting lines are all mid-points, i.e. they have exactly two neighbors on every line.Proposition P23 Let l 2 Lines(CS). Then MidPointsCS (l) = l. 2Once we have established x P y we can be always sure to �nd a z with z co x and z P y.This is a crucial result following from ASS1.Proposition P24 P � co �P . 2Proof We will show x P y ) 9z0 : x co z0^z0 P y. Assuming x P y we get liX [x] � liX [y](by D5c) such that 9z : z li y ^ z co x. Then there must be a line l 2 Lines(CS) withz; y 2 l. Applying ASS1 there must be z0; z00 2 l with z0 im y ^ z00 im y. It is clear thatx 6= z0^x 6= z00 since x 62 l and x co z^z li z0^z li z00. It is necessary that z0 co x_z00 co yotherwise we have a line l0 2 Lines(CS) with x; y; z0; z00 2 l0 with (imjl0)[z] = fx; z0; z00gviolating ASS1. D6i requires that z0 P y and z00 P y. So we have x co z0 ^ z0 P y orx co z00 ^ z00 P y. 2One way to �nd a proof for ASS1 might involve the previous proposition: It might bepossible to prove P24 without using ASS1 and then to derive ASS1 from it.5.7 Propagating ConcurrencyIt was already mentioned that the global time evolution could be seen as the movementof cuts though time-space. So from a given cut representing the current state we have toderive new cuts, which correspond to predecessor or successor states, and all this has tobe done on the basis of local rules, such that only the immediate neighborhood of thoseelements is involved, which contribute to the evolution at that instant of time.Concretely, if we have an arbitrary element which is concurrent to an event, we can prop-agate concurrency in such a way that our element is concurrent to all elements in theimmediate neighborhood of that event.Proposition P25 Let t 2 T . Then x co t) (8y 2 im[t] : x co y). 2Proof Let x co t and assume there is some y 2 im[t] with x liX y. x = y implies x im t,contradicting x co t. t 2 T requires y P t. Then we get a direct contradiction with L1since we have y li t and y li � co t. 2Interestingly this proposition has a counterpart concerning causality and local states: Forall elements that are causally related to a state-element the causality can be extended toits immediate neighborhood.Proposition P26 Let s 2 S. Then x li s) (8y 2 im[s] : x liX y). 2Proof Similar to the previous proof let x li s and assume there is some y 2 im[s] withx co y. s 2 S implies s P y. s li y and s li �co y together with L1 leads to :(s P y). Acontradiction. 2As a direct consequence, given a local state on a line, that line must necessarily containits immediate neighborhood.Proposition P27 Let s 2 S ^ l 2 Lines(CS)^ s 2 l. Then im[s] � l. 236

Proof Combining j im[s] j = 2 (P7) and j l \ im[s] j = 2 (P23) we immediately getim[s] = l \ im[s]. 2A more complicated but essential propagation rule for concurrency can be derived fromASS1: Given an event t and an arbitrary element x, which is concurrent to all elements inone equivalence class of t (remember the de�nition of local equivalence classes in section5.5), concurrency can be propagated in such a way that element x is also concurrent toevent t.Proposition P28 Let t 2 T ^E 2 im[t]=coX .Then (8e 2 E : x co e)) x co t. 2Proof According to P20 we can write im[t]=coX = fE1; E2g where E1 [ E2 = im[t] ^E1 \ E2 = ;. Let x 2 X such that 8e 2 E1 : x co e. Now assume x li t. First observethat there is some line l 2 Lines(CS) with t; x 2 l. D6j additionally requires that a �niteim-chain (x = x0; : : : ; xn = t) is contained completely in l (fx0; : : : ; xng � l). As x co eholds for all e 2 E1 the line l must pass through some s 2 im[t] � E1 implying s 2 E2.Now observe that t is an endpoint of l in the sense that there is no further element ofim[t]�fsg contained in that line (im[t]� fsg \ l = ;). This is a direct contradiction withASS1. 2The restriction that we have to consider events (and their immediate neighborhood) is evennot necessary, such that a slightly more general form of this proposition is also possible:Remark R9 Let z 2 X ^E 2 im[z]=coX .Then (8e 2 E : x co e)) x co z. 2Proof Combine P25 (for z 2 T ) and P28 (for z 2 S). 2Applying these propagation rules it is an easy task to rule out lines that are shorter thatfour elements. This lower bound cannot be improved as we can see from Fig. 1 where wecan identify two lines with exactly four elements.Proposition P29 Let l 2 Lines(CS). Then j l j � 4. 2Proof Let l 2 Lines(CS) (1). By D6a and P3 we have j l j � 2. If j l j = 2 we havel = fa; bg and as a im b we can assume a 2 S and b 2 T . Applying P26 to a with the factthat im[a] = fb; cg we �nd that even fa; b; cg is a clique of liX . Hence we conclude j l j > 2.Now assume j l j = 3 and l = fa; b; cg. Without loss of generality we can assume a im band b im c (by D6j). Then by D9 and P8 there are two possibilities, either a 2 S, b 2 T ,c 2 S (case (2)) or a 2 T , b 2 S and c 2 T (case (3)). D6m suggests that there must bean element d 2 im[fcg] with d 6= b and b li d. b li c and c li d is clear by C5. In case 2 wehave c 2 S and assuming a co d leads to a co c (applying P26) which contradicts a li c Incase 3 we have c 2 T and assuming a co d leads to b co d (by P26), again a contradiction.So in both cases we have a li d and fa; b; c; dg is a clique of liX , which is not reconcilablewith our initial assumption 1 that l = fa; b; cg 2 Lines(CS). 25.8 Consistent Orientations and NetsIn this section a link will be established between the formalism of nets and concurrencystructures. A major characteristic of nets is the partition of net elements into two sorts:active transitions and passive places. Given a concurrency structure CS = (X; li; co) events37

TCS and local states SCS were already introduced constituting a partition of X , and wewill naturally identify transitions and places with events and local states, respectively. Afurther feature of nets is the possibility of expressing the symmetric relations of causalityand concurrency by means of a single directed ow relation. The term ow relation alreadyindicates that a certain direction of time is chosen, in which the ow (of the distributedstate in time-space) is oriented. On the level of concurrency structures, however, we had noprivileged arrow of time. So we expect that given a concurrency structure we �nd di�erentnets with di�erent orientations of the ow relation. Later, when the dynamics of nets isinvestigated, we will see that certain elementary net systems based on these nets are insome sense equivalent to the underlying concurrency structure.We start this section with the de�nition of a consistent orientation, which is a relationF orienting each pair of immediate neighbors x im y in exactly one direction (that isF [ F�1 = im and F \ F�1 = ;) and satisfying some further conditions that will bementioned immediately. From the viewpoint of some element x 2 X we interpret thatpart of the neighborhood im[x] that is given by F�1[x] as those elements that have adirect e�ect upon x and F [x] as the elements that are directly a�ected by x. In otherwords, F [x] and F�1[x] are immediate temporal successors and predecessors, respectively.An additional condition of consistent orientation is given by F �F � li. This ensures thatthe relation of causality holds between an element y that directly a�ects x (y F x) andan element z that is directly a�ected by x (x F z). We are sure that in this case y and zmust be (indirectly) causally dependent, as there is only one element, namely x, locatedbetween them. On the other hand, if there are two elements y; z 2 �xF that both a�ectx, they can do this only concurrently and similarly y; z 2 x�F which are directly a�ectedby x must be concurrent. The last two conditions are expressed by F �F�1 � coX andF�1 �F � coX .De�nition D12 Let F � X �X and Y � X .F is a consistent orientation on Y in CS :,a) F \ F�1 = ;;b) F [ F�1 = imjY ;c) F �F � li;d) F �F�1 � coX ;e) F�1 �F � coX . 2For technical reasons, the de�nition of consistent orientation will be needed with respectto a subset Y of X . This corresponds to a partial consistent orientation of a concurrencystructure, which will be useful for the inductive approximation of a (total) consistentorientation.De�nition D13 F is a consistent orientation on CS :,F is a consistent orientation on X in CS. 2An example for a consistent orientation F of the concurrency structures given in Fig. 1 isshown in Fig. 8.Remark R10 Let Z � Y .F is a consistent orientation on Y ) F jZ is a consistent orientation on Z. 238

..... .....

coF

a)

b)

T]SFigure 8: A consistent orientation FAn equivalent formulation of the notion of consistent orientation is given by the followingremark by simply reversing the implications of the de�nition D12.Remark R11 Let F � X �X .F is a consistent orientation on CS ,a) :(x F y ^ x F�1 y);b) x im y , x F y _ x F�1y;c) x im y im z ^ x li z ) x F y F z _ x F�1 y F�1 z;d) x im y im z ^ x coX z ) x F y F�1 z _ x F�1 y F z. 2So far is has not been proved that such a consistent orientation really exists. It is mainlythis question that is to be addressed in the remainder of this section. As mentioned above,a set of nets Nets(CS) will be associated with every concurrency structure CS and themajor problem is what we can say about the cardinality of this set.De�nition D14 Nets(CS) := f(S; T; F ) : F is a consistent orientation on CS g. 2Although we do not know if Nets(CS) 6= ;, we can easily derive some essential propertiesof those objects that might be contained in Nets(CS). First some trivial remarks followingimmediately from the de�nition:Remark R12 Let (S; T; F ) 2 Nets(CS)^ x 2 X .Then 8y 2 �xF : 8z 2 x�F : y li z. 2Proof Immediately from F 2 � li (D12c). 2Remark R13 Let (S; T; F ) 2 Nets(CS)^ x 2 X .a) 8y; z 2 �xF : y coX z;b) 8y; z 2 x�F : y coX z. 2Proof Immediately from F �F�1 � coX and F�1 �F � coX (D12). 2Lemma L8 im[x] = �xF [ x�F . 2Proof Clear by D12 as F [ F�1 = im. 239

Proposition P30 im[x]=coX = f�xF ; x�F g. 2Proof By R12, R13 and L8. 2In fact, elements of Nets(CS) satisfy the de�nition of a net, as it was already anticipatedabove.Proposition P31 Let N 2 Nets(CS). Then N is a net. 2Proof Let N = (S; T; F ) ^ N 2 Nets(CS). S \ T = ; has been found in R6. So allwe have to prove is: F � (S � T ) [ (T � S). But this is clear because P � (S � T ) andF � im = P [ P�1. 2All nets associated with CS turn out to be connected and pure.Proposition P32 Let N 2 Nets(CS). Then N is connected. 2Proof Let (S; T; F ) 2 Nets(CS). From im�X = X � X (P7) and F [ F�1 = im weimmediately �nd (F [ F�1)�X = X �X . 2Proposition P33 Let N 2 Nets(CS). Then N is pure. 2Proof Immediately from F \ F�1 = ; (D12a). 2The elements of these nets are directly a�ected by at least one element, and they directlya�ect at least one further element. So the future as well as the past of an element is neverempty.Proposition P34 Let (S; T; F ) 2 Nets(CS).Then 8x 2 X : j �xF j � 1 ^ j x�F j � 1. 2Proof Let x 2 X . Assume x�F = ;. C4 and D6m require that there are y; z 2 im[x]such that y li z. From our assumption it follows that y F x and z F x, but this wouldcontradict F �F�1 � coX . The situation for �xF = ; is similar. 2Local states or places have exactly one immediate temporal predecessor and one successor.Proposition P35 Let (S; T; F ) 2 Nets(CS).Then 8s 2 S : j �sX j = j s�X j = 1. 2Proof Directly from P21 and P34. 2Events or transitions are necessarily equipped with more than one predecessor and morethan one successor.Proposition P36 Let (S; T; F ) 2 Nets(CS).Then 8t 2 T : j �tX j > 1 ^ j t�X j > 1. 2Proof Let t 2 T . Assume s 2 �tX and s0 2 t�X (this is possible by P34). Certainly wehave s P t and s0 P t. P24 implies 9z : s co z ^ z P t and 9z0 : s0 co z0 ^ z0 P t such thatD12 leads to z 2 �tX and z0 2 t�X . 2A �rst hint about the cardinality of Nets(CS) in given by the next proposition. If a netN is contained in this set, this is also true for the inverse net of N . So, if Nets(CS) is�nite, it must be of even cardinality.Proposition P37 (S; T; F ) 2 Nets(CS), (S; T; F�1) 2 Nets(CS). 2Proof It is su�cient to show: F is a consistent orientation , F 0 := F�1 is a consistentorientation. And this follows directly from D12:40

a) F [ F�1 = F 0 [ F 0�1 = im,b) F \ F�1 = F 0 \ F 0�1 = ;,c) F 2 � li, F 02 � li�1 = li,d) F �F�1 = F 0�1 �F 0 � coX ,e) F�1 �F = F 0 �F 0�1 � coX . 2In the following a proposition is prepared, which shows that Nets(CS) must contain atleast one element, if a certain condition (ASS2) is satis�ed. To state this condition ina compact form and to simplify the proof two abbreviations are introduced. Given anarbitrary im-chain A we say that (the unordered set of) two concurrent elements, whichare separated by only one element in A, constitute a co-root. CoRootsCS(A) is simplythe set of co-roots on A. If B is an im-cycle, the idea is similar, but we have to take intoaccount also the two neighbors of the �rst (which is the same as the last) element leadingto the de�nition of AllCoRootsCS(B).De�nition D15Let A = (a0; : : : ; an) be an im-chain and B = (b0; : : : ; bn) be an im-cycle.a) CoRootsCS(A) := fi 2 N[1; n� 1] : ai�1 coX ai+1g;b) AllCoRootsCS(B) := fi 2 N[0; n� 1] : b(i�1)modn coX b(i+1)modng. 2For an illustrating example look at Fig. 9, where we see an im-cycle A = (1; 2; 3; 4; 5; 6; 7; 8; 1)and the following relations: 1 li 3, 2 li 4, 3 co 5, 4 li 6, 5 li 7, 6 li 8, 7 li 1. Further elementsand relations are not of interest. Conceiving A as an im-chain we see that CoRoots(A) =f 3g. Viewing A as an im-cycle we �nd the same result for AllCoRoots(A) = f3g. Choos-ing a di�erent im-cycle B = (4; 5; 6; 7; 8; 1; 2; 3; 4) (on the same set of elements) we getCoRoots(B) = ;, but AllCoRoots(B) = f0g.licoim

S

T]

1 2

3 4

5

67

8 Figure 9: An im-cycle with odd co-rootsThe subsequent remark should clarify the relation between these two de�nitions.Remark R14 Let A = (a0; : : : ; an) be an im-cycle.a) a1 li an�1 ) AllCoRootsCA(A) = CoRootsCS (A);b) a1 co an�1 ) AllCoRootsCA(A) = CoRootsCS(A) [ f0g;41

2Dealing with concurrency in analogy to information as a owing quantity co-roots areintended to mark potential sources and sinks of concurrency. The following assumption,states that on every im-cycle the cardinality of co-roots is even. Loosely speaking, thisseems to be a necessary condition for the conservation of concurrency on an im-cycle(whatever this may mean), if we argue that for every source there must be a sink ofconcurrency, and vice versa.Assumption ASS2 Let A be an im-cycle.Then j AllCoRootsCS(A) j is even. 2The next de�nition will be helpful in subsequent proofs. It simply de�nes the conceptof consistent orientation with respect to im-chains similar to D12. Note that in generalimjA 6= imjSet(A) if A is a im-chain. Hence the following notion is generally not equivalentto a consistent orientation of A.De�nition D16 Let F � X �X and A be an im-chain.F is an A-orientation in CS :,a) F \ F�1 = ;;b) F [ F�1 = imjA;c) ai F ai+1 F ai+2 ) ai li ai+2;d) ai F�1 ai+1 F�1 ai+2 ) ai li ai+2;e) ai F ai+1 F�1 ai+2 ) ai co ai+2;f) ai F�1 ai+1 F ai+2 ) ai co ai+2. 2Certainly a consistent orientation F of CS implies that F is an orientation on everyim-chain.Remark R15 Let A be an im-chain.Then F is a consistent orientation on X ) F jA is an A-orientation. 2Proof Use R11. 2From the fact that CoRootsCS(A) is of even cardinality for a given im-chain A we canintuitively conclude that an A-orientation F �xed between the �rst two elements of thechain can be propagated along the chain and undergoes an even number of reversals (onefor each co-root), such that at the end of the chain we again �nd the original orientation.If j CoRootsCS (A) j is odd, the last two elements of the chain are oriented in the oppositedirection.Lemma L9 Let A = (a0; : : : ; an) be an im-chain with n � 2 andF be a A-orientation in CS.a) j CoRootsCS(A) j is even ) a0 F a1 , an�1 F an;b) j CoRootsCS(A) j is odd ) a0 F a1 , an�1 F�1 an. 2Proof De�ne rY := j CoRootsCS(Y ) j and proceed by induction over rA. For rA = 0 wehave either ai F ai+1 for all i; i+1 2 N[0; n] or ai F�1 ai+1 for all i; i+1 2 N[0; n]. So L9a issatis�ed and L9b holds trivially. Now we show that our lemma is also valid for rA = i > 042

if it holds for rA = i � 1: For rA = i there is a pair ak�1; ak+1 2 A with ak�1 coX ak+1.Choose the largest k with this property. Now decompose A into B = (b0; : : : ; bn) andC = (c0; : : : ; cm) such that A = (b0; : : : ; bn�1 = ak�1; bn = ak = c0; c1 = ak+1; : : : ; cm) andnotice that rC = 0, bn�1 coX c1 and rB = rA � 1.If rA is odd then rB is even and we conclude b0 F b1 , bn�1 F bn (using L9a withrA = i� 1) and bn�1 F bn , c0 F�1 c1 (from R11) and c0 F c1 , cn�1 F cn (from L9a).Altogether this yields b0 F b1 , cn�1 F�1 cn which is actually L9b.If rA > 0 is even then rB is odd and we �nd b0 F b1 , bn�1 F�1 bn (applying L9b withrA = i� 1) and bn�1 F bn , c0 F�1 c1 (from R11) and c0 F c1 , cn�1 F cn (from L9a).Altogether this yields b0 F b1 , cn�1 F cn which is L9a. 2Combining the previous lemma with the assumption that on im-cycles there is always aneven number of co-roots we can �nd the following lemma concerning F in the neighborhoodof the �rst and last element of an im-cycle.Lemma L10 Let A = (a0; : : : ; an) be an im-cycle with n � 2 andF be an A-orientation in CS.a) an�1 li a1 ) an�1 F (an = a0) F a1 _ an�1 F�1 (an = a0) F�1 a1;b) an�1 coX a1 ) an�1 F (an = a0) F�1 a1 _ an�1 F�1 (an = a0) F a1. 2Proof First apply ASS2 to A and observe that AllCoRoots(A) is even. We distinguishtwo cases: Either an�1 li a1 (1) or an�1 coX a1 (2). In case 1 we have that CoRoots(A) iseven and in case 2 CoRoots(A) is odd (see R14). Applying L9 yields a0 F a1 , an�1 F anin case 1 and a0 F a1 , an�1 F�1 an in case 2. 2With these lemmas we are prepared to give a procedure to construct a consistent orienta-tion on an arbitrary large im-coherent subset ofX (that is a subset Y � X with im�Y = Y )and to prove that this procedure yields a (total) consistent orientation in the limit.Scope S3The subsequent construction is guided by an enumeration of all elements ofX , which musthave the property that it preserves the coherence of im, that is, the set of all elementsenumerated up to a certain index should be im-coherent (X is im-coherent i� im�X =X � X). This is ensured by the additional condition that a newly enumerated elementshould be the im-neighbor of a previously enumerated one.De�nition D17a) e is an enumeration of CS :,e : N! X and e is surjective on X and total on N or N[0; n] for some n;b) e is a coherent enumeration of CS :,8i 2 Dom(e) : i > 0) 9j 2 Dom(e) : j < i ^ e(j) im e(i). 2The following construction inductively de�nes several sets, which are indexed over thedomain of an arbitrary coherent and injective enumeration, and �nally de�nes F in termsof these sets (D18g). Notice that there is some further arbitrariness in this construction(as we have the possibility to choose between alternatives in particular at D18b), such thatwe can not exclude the possibility that the construction yields di�erent results in terms43

of F . But this does not matter, as our �rst goal is only to prove the existence of at leastone consistent orientation.Construction D18a) Fix a coherent injective enumeration e of CS and de�ne ei := e(i);b) Choose D0, E0 such that fD0; E0g = im[e0]=coX ;c) F0 := DF0 := (D0 � fe0g)[ (fe0g �E0);d) Choose Di, Ei such that fDi; Eig = im[ei]=coX and9j : j < i ^ (ej Fj ei ^ ej 2 Di) _ (ei Fj ej ^ ej 2 Ei);e) DFi := (Di � feig)[ (feig � Ei);f) Fi := Fi�1 [DFi;g) F := SfFi : i 2 Dom(e)g. 2The proof that F is well-de�ned by this construction will follow after some auxiliaryde�nitions and lemmas. But �rst the idea behind this construction will be informallysketched.Starting with an orientation of the neighborhood of e0 we will propagate this orientationthrough the whole structure with the help of our enumeration e. In every step i we orientei and its immediate neighborhood im[ei] in some locally consistent manner by DFi, andwe hope that collecting all local orientations yields a (global) orientation of X . Let us gostep by step through the construction: In D18b we simply name the two equivalence classesof im[e0] with respect to coX . By convention D0 should denote the immediate temporalpredecessors of e0, and E0 contains its immediate temporal successors. D18c establishesF0 to be consistent with this choice, that is F�10 [e0] = D0 and F0[e0] = E0. D18d similarlyde�nes the equivalence classes of im[ei] to be Di and Ei, but now with the boundarycondition that this choice is consistent with a previous orientation already establishedby Fj for j < i: That is if ei is already chosen to be the successor of some ej , then ejmust be contained in the predecessors Di of ei. Otherwise, if ei is already a predecessorof some ej , we want ej to be contained in the successors Ei of ei. D18e again expressesthe convention that Di and Ei are predecessors and successors, respectively. D18f collectsall local orientations DFj in Fi, which have be found up to and including step i. FinallyD18g yields the smallest set F containing all Fj constructed during the procedure, whichmay be (countably) in�nitely many.The subsequent list of properties concerning the previous construction can be easily veri-�ed.Remark R16a) i � j ) DFi � Fj ;b) i � j ) Fi � Fj ;c) Fi � F ;d) DFi \DF�1i = ;;e) DF 2i � li;f) DFi �DF�1i � coX ;g) DF�1i �DFi � coX ;h) DFi [DF�1i =l (im[ei]� feig);i) DFi [DF�1i = imj(imX [ei]); 44

j) DFi � im;k) F [ F�1 � im;l) Fi = SfDFj : j 2 N[0; i]g;m) ei DFj ek ) i = j _ j = k. 2In the next de�nition Xi is the set of all elements and their im-neighbors reached by theenumeration up to and including step i. Notice that the intention of a coherent enumerationwas just to guarantee that Xi is im-coherent.De�nition D19 Xi := SfimX [ek] : k 2 N[0; i]g. 2The in�nite sequence X0; X1; : : : is a subset chain and X is the smallest set containing allXi.Remark R17a) e0; : : : ; ei 2 Xi;b) ei im ej ) ei 2 Xj ^ ej 2 Xi;c) X = SfXi : i 2 Ng;d) i � j ) Xi � Xj . 2By the help of Xi we recognize a further important property of DFi and Fi: Xi is exactlythat subset of X up to which the orientation has been propagated though the structureup to step i.Lemma L11a) (x DFi y _ x DF�1i y)) x; y 2 Xi;b) Fi [ F�1i = imjXi;c) Fi [ F�1i = (Fi [ F�1i )jXi. 2Proofa) Assume ek DFi ej or ek DF�1i ej . It follows that ek im ej which implies ek 2 Xjand ej 2 Xk because ej 2 Xj and ek 2 Xk. By D18e we have DFi = (Di � feig) [(feig�Ei) which means that either k = i or j = i. So we conclude that ek 2 Xi andej 2 Xi in both cases.b) By R16l Fi [ F�1i = SfDFj [ DF�1j : j 2 N[0; i]g and by R16i DFi [ DF�1i =imj(imX [ei]). It follows that Fi [ F�1i = Sfimj(imX [ej ]) : j 2 N[0; i]g= imjXi.c) From L11b by (Fi [ F�1i )jXi = (imjXi)jXi = imjXi = Fi [ F�1i . 2As promised, it follows the proof that F is well-de�ned in our construction, which meansin this case that the construction always succeeds to yield some F . This is not immediatelyclear, as, in particular, it is not evident that the condition of D18d can be always satis�ed.Lemma L12a) Di,Ei are well-de�ned; 45

b) DFi,Fi,F are well-de�ned. 2Proofa) Remember P20. It guarantees that we can always choose Di,Ei such that fDi; Eig =im[ei]=coX . But what about the boundary condition in D18d: 9j : j < i^ (ej Fj ei^ej 2 Di) _ (ei Fj ej ^ ej 2 Ei). First in D17b we required that there is some j suchthat j < i ^ ej im ei. This means that ei 2 Xj and of course ej 2 Xj . By L11b(Fj [ F�1j = imjXj) we have ei Fj ej or ej Fj ei. Furthermore ej 2 im[ei] such thatin case ej Fj ei we can choose Di such that ej 2 Di and otherwise we must haveei Fj ej and we choose Ei such that ej 2 Ei. Hence the condition in D18d can alwaysbe satis�ed.b) Follows immediately from L11c as F ,Fi,DFi are de�ned in terms of Di,Ei. 2Unfortunately, the straightforward proof that the result of construction D18 is indeed aconsistent orientation, which follows now, is technical and might obscure the actual idea,which is the following: If every im-cycle has an even number of co-roots, it can be orientedconsistently. If every im-cycle can be oriented consistently, this is also true for the whole setX . That im-cycles have to be considered comes from the observation that, if a concurrencystructure exists, which contains the structure of Fig. 9 as a subset (remember that thiswas assumed not to be the case by ASS2, as we have exactly one co-root on the im-cycleshown), it is impossible to �nd a consistent orientation.The following two lemmas prove the �rst two necessary conditions D12b and D12a for Fito be a consistent orientation on Xi.Lemma L13 8x; y 2 Xi : (x im y , x Fi y _ x F�1i y). 2Proof This is equivalent to imjXi = (Fi [ F�1i )jXi. With (Fi [ F�1i )jXi = Fi [ F�1i(L11c) we �nd Fi [ F�1i = imjXi which is L11b. 2The proof of the next lemma is simple but cumbersome, as there are several cases, that haveto be distinguished. The proof is an indirect one and has roughly the following structure:First we identify the �rst step in our construction that violates our lemma. We show thatthere must exist at least two im-paths from e0 to that location, where the violation takesplace. Intuitively these to paths represent contradicting orientation constraints carried toan element from di�erent sides. Combining these two im-paths to a cycle we apply ASS2(or L10 to be more precise) to derive a contradiction.Lemma L14 8x; y 2 Xi : :(x Fi y ^ x F�1i y). 2Proof Indirect: Assume Fi \ F�1i 6= ; for some i. We will choose the smallest such i inorder to ensure that Fi�1 \ F�1i�1 = ;. This means there are ej ; ek 2 Xi such that ej Fi ekand ek Fi ej . As DFi \DF�1i = ; we can distinguish two cases: Either ej Fi�1 ek (1) orek Fi�1 ej (2). Observe that in both cases we have ej ; ek 2 Xi�1 (3) by L11b.First we deal with case 1: If ej Fi�1 ek then there is some l with l < i (4) and ej DFl ek ,which implies either l = j (5) or l = k (6). Now consider ek Fi ej . From ej Fi�1 ek andFi�1 \ F�1i�1 = ; we infer ek DFi ej , which implies either i = j (7) (if ek 2 Di) or i = k(8) (if ej 2 Ei). 46

Now we come to the case 2, which is completely symmetric to 1: ek Fi�1 ej implies thatthere is some l with l < i (9) and ek DFl ej . Again this implies either l = j (10) or l = k(11). From ej Fi ek together with ek Fi�1 ej and Fi�1 \ F�1i�1 = ; we derive ej DFi ek .This means either i = j (12) (if ek 2 Ei) or i = k (13) (if ej 2 Di) holds.For the case 1 as well as for the case 2 we found that (ek; ej) 2l DFi such that the conditionof D18d must have been satis�ed: There is some m with m < i and em Fm ei ^ em 2 Di(14) or ei Fm em ^ em 2 Ei (15).Our aim is to derive a contradiction for every conceivable combination of cases introducedabove:a) For (5 ^ 7), (6 ^ 8), (10 ^ 12) and (11 ^ 13) this is easily done: These combinationsare impossible because they imply i = l which contradicts either 4 or 9.b) (1 ^ 5 ^ 8 ^ 14). ej Fi�1 ek , l = j, i = k, em Fm ei.em 2 Di and (ej = el) 2 Ei implies em li (ej = el).c) (1 ^ 6 ^ 7 ^ 14). ej Fi�1 ek , l = k, i = j, em Fm ei.em 2 Di and (ek = el) 2 Di implies em co (ek = el).d) (1 ^ 5 ^ 8 ^ 15). ej Fi�1 ek , l = j, i = k, ei Fm em.em 2 Ei and (el = ej) 2 Ei implies (el = ej) co em.e) (1 ^ 6 ^ 7 ^ 15). ej Fi�1 ek , l = k, i = j, ei Fm em.em 2 Ei and (ek = el) 2 Di implies em li (ek = el).f) (2 ^ 10 ^ 13 ^ 14). ek Fi�1 ej , l = j, i = k, em Fm ei.em 2 Di and (ej = el) 2 Di implies em co (ej = el).g) (2 ^ 11 ^12 ^ 14). ek Fi�1 ej , l = k, i = j, em Fm ei.em 2 Di and (ek = el) 2 Ei implies em li (ek = el).h) (2 ^ 10 ^ 13 ^ 15). ej Fi�1 ek, l = k, i = j, em Fm ei.em 2 Ei and (ej = el) 2 Di implies em li (ej = el).i) (2 ^ 11 ^12 ^ 15). ek Fi�1 ej , l = k, i = j, ei Fm em.em 2 Ei and (ek = el) 2 Ei implies em co (ek = el).Except for the �rst combination (which was already found to be impossible) we observe(using the fact that Fm � Fi�1 from m < i):a) em co el , em Fi�1 ei Fi�1 el _ em F�1i�1 ei F�1i�1 el (16);b) em li el , em Fi�1 eiF�1i�1 el _ em F�1i�1 eiFi�1 el (17).As el; em; ei 2 Xi�1 (remember 3 and m < i) there are two (imjXi�1)-chains A,Bwith A = (ei; el; : : : ; e0) and B = (ei; em; : : : ; e0). Exploiting the condition of D17bwe can even choose A,B such that p < q ^ Ap = er ^ Aq = es ) r > s and p <q ^ Bp = er ^ Bq = es ) r > s. Combining A and B we construct a (imjXi�1)-cycleC = (cn)n2I = (ei; el; : : : ; e0; : : : ; em; ei). Note that i is the maximal index necessary toconstruct C, that is, :9p > i : ep 2 C (18).Now de�ne F 0 := Fi�1jC and notice that it has the following properties:a) F 0 \ F 0�1 = ;. By our initial assumption Fi�1 \ F�1i�1 = ;.b) F 0 [ F 0�1 = imjC. Use L11b.c) cn F 0 cn+1 F 0 cn+2 ) cn li cn+2. To see this assume there are ep; eq; er 2 C withep F 0 eq F 0 er and eq 6= ei but ep co er. It is immediately clear that q 6= i. By ourconstruction we have either ep DFq eq DF�1q er or ep DF�1q eq DFq er. If q < i we47

get a contradiction with Fi�1 \ F�1i�1 = ;, because DFq � Fi�1. So we are left withq > i, but this is impossible, because i is the maximal index appearing in C (see 18).d) cn F 0�1 cn+1 F 0�1 cn+2 ) cn li cn+2. The proof is similar to the previous one.e) cn F 0 cn+1 F 0�1 cn+2 ) cn co cn+2. Assume ep; eq; er 2 C with ep F 0 eq F 0�1 er,ep 6= er and eq 6= ei but ep li er. It is clear that q 6= i. According to our constructionwe have either ep DFq eq DFq er or er DFq eq DFq ep. As above q < i and q > ilead to contradiction.f) cn F 0�1 cn+1 F 0 cn+2 ) cn co cn+2. The proof is similar to the previous one.With these facts we immediately conclude that F 0 is a C-orientation, and, as C is anim-cycle, we can apply L10, which states that in case of cn�1 li c1 we have cn�1 F 0 (cn =c0) F 0 c1 _ cn�1 F 0�1 (cn = c0) F 0�1 c1 contradicting 17 and in case of cn�1 co c1 we havecn�1 F 0 (cn = c0) F 0�1 c1 _ cn�1 F 0�1 (cn = c0) F 0 c1 contradicting 16. 2Now it is left to prove D12c and D12d, which can be done directly for F with respect toX by the next two lemmas.Lemma L15 x im y im z ^ x li z ) x F y F z _ x F�1 y F�1 z. 2Proof For el; ej ; ek 2 Xi assume el im ej im ek and el li ek. First it is clear that l 6= j,j 6= kand l 6= k. By D18e we have DFj = (Dj�fejg)[ (fejg�Ej) with fDj; Ejg = im[ej]=coX .el li ek requires either el 2 Dj^ek 2 Ej or ek 2 Dj^el 2 Ej . Hence either el DFj ej DFj ekor ek DFj ei DFj el holds and DFj � F proves our lemma. 2Lemma L16 x im y im z ^ x co z ) x F y F�1 z _ x F�1 y F z. 2Proof For el; ej ; ek 2 Xi assume el im ej im ek and el co ek. As above l 6= j,j 6= k andl 6= k. Now D18e yields DFj = (Dj � fejg) [ (fejg � Ej) with fDj ; Ejg = im[ej]=coX .el co ek implies either el; ek 2 Dj or el; ek 2 Ej . This means either el DFj ej DF�1j ek orel DF�1j ej DFj ek holds and with DFj � F this completes the proof. 2Finally we can state our �rst important conclusion that F is indeed a consistent orienta-tion.Lemma L17 F is a consistent orientation on CS. 2Proof Finite or trans�nite induction over i 2 Dom(e) with F = SfFi : i 2 Dom(e)gapplied to L13 and L14 yields F [ F�1 = im and F \ F�1 = ;. In combination with L15and L16 this shows that F is a consistent orientation on X as D12b is satis�ed. 22 S3This directly implies that we can �nd at least one net associated with an arbitrary con-currency structure.Corollary C8 j Nets(CS) j � 1. 2Proof We have (S; T; F ) 2 Nets(CS) with F constructed in D18 as consistent orientation(L17). 2As with every consistent orientation F we have immediately a further consistent orien-tation F�1 there must be at least two nets in Nets(CS), and we can even prove thatthere cannot be more: This can be derived from the fact that every concurrency structureis coherent, such that local choice of the orientation between two neighbors propagatesthrough the whole structure and determines the (total) orientation uniquely.48

Lemma L18 j Nets(CS) j � 2. 2Proof Assume j Nets(CS) j > 2. Then there are F; F�1; F 0 2 Nets(CS) with F 0 6=F^F 0 6= F�1. Hence there are a; b; c; d 2 X with a F b^c F d and b F 0 a^c F 0 d. By P7 it isclear that there is an im-chain A = (a0; : : : ; an) with a0 = a^ a1 = b^ an�1 = c^ an = d.Furthermore CoRoots(A) is either even or odd. So applying L9 to F and F 0 (noticethat this is possible due to R15) yields either a F b , c F d and a F 0 b , c F 0 d ora F b, c F�1 d and a F 0 b, c F 0�1 d. In both cases we have a contradiction with ourinitial choice of a; b; c; d. 2Proposition P38 j Nets(CS) j = 2. 2Proof By C8, L18 and P37. 2Altogether, the formalism of nets has several advantages compared to concurrency struc-tures which include the explicit di�erentiation between active and passive elements, theexplicit choice of an arrow of time, which simpli�es reasoning and the reduction of thedescriptional complexity (generally the ow relation is much smaller than causality orconcurrency relations). Later we will recognize a further convenience of the net formalismconcerned with the meaning of the token game in elementary net systems.One might wonder why concurrency theory does not choose the level of nets as a basis toformulate the axioms. One reason is that from the viewpoint of constructing an axiomaticsystem it is natural to identify atomic notions (that are notions of which is is believedthat a further regress to simpler concepts is impossible or not appropriate). An additionaladvantage emerges, if these notions are easy to understand and can be found on verydi�erent levels of abstraction such that there is no dissent about they properties. In thecase of concurrency theory these atomic notions are concurrency and causality de�nedon some uniform set. To assume as less as possible concerning the structure carrying theaxioms helps to make every postulate explicit by an additional axiom (as simple as itmight be). It should be the set of axioms, which then leads to the actual complexity ofthe matter. That the �eld reveals a certain simplicity by the use of a convenient notation(as it seems to be for concurrency theory on the level of nets and elementary net systems)might be a hint to practical applicability of the theory.5.9 The Flow Relation in Cyclic and Acyclic StructuresThe ow relation with its property to be a consistent orientation is a useful vehicle toreason about concurrency structures, as it provides a lingual means to refer to the past orfuture with respect to a given element.The �rst important property of the ow relation arises from the fact that lines are im-coherent. As one might already expect every line is oriented by the ow relation in exactlyone direction. This is implied by the following proposition, which states that given twoelements x; y on a line they are connected by a chain of the ow relation.Proposition P39 Let l 2 Lines(CS) and A be an acyclic imjl-chain.Then A is either an F jl-chain or an F�1jl-chain. 2Proof Choose x; y 2 l arbitrarily. De�ne AC(x; y) := fC : j C j > 1 ^ C = (x; : : : ; y) isan acyclic imjl-chain g. Now we proceed by induction over the size of some C 2 AC(x; y)49

to prove H(C) :, (C is either an F -chain or an F�1-chain).For j C j = 2 we have C = (x; y) and either x F y of y F x (remember D12a and D12b).For j C j = n we assume that H(C 0) is already proved for all C 2 AC(x; y) with j C 0 j < n.So we can write C = (x = x0; : : : ; xn�2; xn�1) where C 0 = (x = x0; : : : ; xn�2) is eitheran F -chain or an F�1-chain. In the former case it is impossible that xn�1 F xn�2 asthis together with xn�3 F xn�2 violates D12d. So we have to conclude xn�2 F xn�1 suchthat C is actually an F -chain. Otherwise, if C 0 is an F�1-chain, a similar argument yieldsxn�2 F�1 xn�1 such that C is an F�1-chain. This proves H(C). 2Corollary C9 Let l 2 Lines(CS) ^ (S; T; F ) 2 Nets(CS).Then (F jl)�l [ (F�1jl)�l = l � l. 2Proof It is su�cient to prove: For every pair x; y 2 l we have x (F jl)�l y or y (F jl)�l x.This is trivial for x = y and follows from D6j and P39 if x 6= y. 2Given two elements x and y with x li y there is certainly a line l 2 Lines(CS) withx; y 2 l. Then by the above proposition there is an F -chain (x; : : :; y) or (y; : : : ; x). In theformer case we say that x occurs before y (x is located in the past of y and in the latercase x occurs after y (x is located in the future of x). Notice that in general these twopossibilities are not mutually exclusive, such that these concepts are not always helpfulas past and future may have a nonempty intersection (which will be the case for cyclicstructures to be introduced below) and may even cover the whole set X .Given a particular line the immediate temporal successor and predecessor exists and isunique on that line. This is what the following lemma states.Lemma L19 Let l 2 Lines(CS)^ (S; T; F ) 2 Nets(CS)^ y 2 l.Then 91x; z 2 l : x F y F z. 2Proof By P23 we can de�ne x; z by fx; zg = (imjl)[y]. Of course, we have x; y; z 2 l. ByR11 there are two possibilities: Either x F y F z or z F y F x. So we have (exchanging xand z in the latter case) 9x; z 2 l : x F y F z. It is left to show that x and z are unique.Assume there is a further x0 2 l with x0 F y. With D12 this implies x0 co x which is acontradiction with x; x0 2 l. Similarly, the existence of a further z0 2 l implies z0 co z,again a contradiction. 2For every in�nite line there is an in�nite, acyclic F -chain (more precisely, it is !!-in�nite)that is exactly covered by that line. For a �nite line, on the other hand, we can always�nd an F -cycle, such that every element of that line is contained in it.Proposition P40 Let l 2 Lines(CS)^ (S; T; F ) 2 Nets(CS).a) l is in�nite ) 9A : A is an !!-in�nite acyclic F -chain ^ Set(A) = l;b) l is �nite ) 9A : A is an F -cycle ^ Set(A) = l. 2Proof We will construct an !!-in�nite F -chain A = (: : : ; a�1; a0; a1; : : :). We start withi = j = 0 choosing some arbitrary ai = aj 2 l. L19 gives us a unique successor ai+1 2 lwith ai F ai+1 and a unique predecessor aj�1 2 l with aj�1 F aj . Carrying out this stepfor all i; j 2 N yields an A, which satis�es Set(A) = l, as due to C9 for every element x 2 lthere is a �nite F - or F�1-chain from a0 to x. As predecessors and successors are unique(according to L19), A is either an in�nite acyclic F -chain, or A is not acyclic, which means50

it contains a (�nite) cycle B with Set(B) = l. If l is in�nite, the later case is excluded,so A must be an acyclic F -chain. If l is �nite the former case is impossible, so B is theF -cycle required by the second part of our proposition. 2The preceding observation that �nite lines correspond to F -cycles and in�nite lines canbe conceived as in�nite, acyclic F -chains leads to the following natural de�nition: A con-currency structure is cyclic, if and only if all lines are �nite, and acyclic, if and only if alllines are in�nite.De�nition D20a) CS is cyclic :, 8l 2 Lines(CS) : l is �nite;b) CS is acyclic :, 8l 2 Lines(CS) : l is in�nite. 2An immediate question is: Are there partially cyclic concurrency structures, that is, struc-tures, where some but not all lines are �nite? It will be immediately shown that they donot exist. This property is deeply connected with the safety of the dynamics, and it willbe necessary to derive the reachability results in the subsequent section.First a lemma is proposed which shows that it is su�cient to have one �nite line to ensurethat the whole structure is �nite. The proof is not di�cult as the strong axiom D6g,which states every element has a �nite concurrency neighborhood, can be exploited. Aninteresting question is, if it is possible to derive the same result from weaker assumptions,e.g. the �niteness of all cuts, as this might lead to an even more general formulation ofconcurrency theory.Lemma L20 CS is not acyclic ) CS is �nite. 2Proof Assume CS is not acyclic. Then there is some �nite line l 2 Lines(CS). Further-more, we have Cuts(CS) = fc : c\ l = fxg^x 2 lg by K-density P4. Obviously c\ l = fxgimplies c � coX [x] which is �nite by D6g. So Cuts(CS) must be also �nite, as l is �nite.Using X = SCuts(CS) we �nd that even X is �nite, as every c 2 Cuts(CS) is �nite byR3. 2An essential result that the notions of cyclic and �nite structures are equivalent followsdirectly from this lemma.Remark R18 CS is cyclic , CS is �nite. 2Proof CS is �nite ) CS is cyclic: If CS is �nite, all lines are �nite.CS is cyclic ) CS is �nite: If CS is cyclic, it is not acyclic and L20 can be applied. 2The anticipated fact that partially cyclic structures cannot exist is equivalent to the ob-servation that the notions of acyclic and cyclic structures are complementary.Proposition P41 CS is acyclic , CS is not cyclic. 2Proof CS is cyclic ) CS is not acyclic: If CS is cyclic all lines l 2 Lines(CS) are �niteand there is no in�nite line. Hence CS is not acyclic.CS is not acyclic ) CS is cyclic: With L20 we derive that X is �nite, such that of courseall line l 2 Lines(CS) are �nite, too. 2Given an F -cycle in a cyclic concurrency structure we assume that this F -cycle meetsevery cut at least once. This property does not follow directly from K-density, as not51

every F -cycle must correspond to a line. Actually, an F -cycle might intersect a particularcut several times, which means, of course, that it cannot be a line in this case.Assumption ASS3 Let A be an F -cycle and c 2 Cuts(CS).Then Set(A)\ c 6= ;. 2The idea of the following lemma is similar to L20 exploiting ASS3 instead of K-density.Lemma L21 9A : A is an F -cycle ) CS is �nite. 2Proof Let A be an arbitrary F -cycle. By ASS3 for every cut c 2 Cuts(CS) there is somex 2 c \ Set(A). As c � coX [x] is �nite by D6g, X = SCuts(CS) and Cuts(CS) = fc :c \ Set(A) 6= ;g by ASS3, we conclude that X is �nite. 2With this lemma it is possible to derive the next proposition, which gives a further jus-ti�cation for the de�nition D20 providing a direct link to corresponding notions in theformalism of nets.Proposition P42 Let N 2 Nets(CS).a) N is acyclic , CS is acyclic;b) N is cyclic ) CS is cyclic. 2Proof N is acyclic ) CS is acyclic: Let N be acyclic and assume CS is not acyclic.Then P41 indicates that CS is cyclic. Hence it must contain some F -cycle A (by P40).But this contradicts the assumption that N is acyclic.CS is acyclic ) N is acyclic: Due to P41 we can equivalently prove: N is not acyclic )CS is cyclic. If N is not acyclic there is some F -cycle and L21 suggests that CS is �nite,which implies that CS is cyclic by R18.N is cyclic) CS is cyclic: IfN is cyclic, it is not acyclic and we can use the same argumentas for the previous implication. 2We conjecture that it is even possible to prove the converse of the second part of theproposition, but this will be not necessary for the subsequent line of reasoning.With the following de�nition we prepare a useful notation that will be employed, whenthe subject of reachability of cuts by dynamical evolution is addressed. For a clique c wewant to denote the set of those elements, which are concurrent to all elements in c, byCO(c).De�nition D21 Let c be a clique of coX .CO(c) := Tfco[x] : x 2 cg. 2Remark R19 Let c be a clique of coX .a) CO(c)\ c = ;;b) CO(c) is �nite;c) c 2 Cuts(CS)) CO(c) = ;. 2Proof That CO(c) is �nite follows from D6g. 252

Intuitively, if we partially �x a cut in c then CO(c) describes the degree of freedom ofthat part of the cut, which might be evolved using the concurrency propagation rules.Expressing this restriction in other words, every cut containing c must be covered byc [ CO(c). Interestingly, every cut c has a �nite degree of freedom CO(c).The following assumption is essential, when we want to propagate a cut containing asubset c, which remains �xed. We have seen in C9 that every causality relation betweentwo elements is accompanied by an F - or F�1-chain between these two elements. The�rst part of this assumption states an even stronger condition, which is that for any twoelements in CO(c), that are is causality relation, there is an F - or F�1 chain betweenthem, which is completely covered by CO(c). This means the causality is realized by anF -chain within the degree of freedom. On the other hand and this is the second part ofthe assumption, if x and y are identical or concurrent, such an F -chain within CO(c) hasto be excluded: This corresponds to the situation that for a cut that is partially frozen inc two concurrent or identical elements should not be reachable by an F -chain within thedegree of freedom CO(c).Assumption ASS4 Let c be a clique of coX ^ x; y 2 CO(c).a) x li y )9A : A = (x; : : :; y) or A = (y; : : : ; x) is an F jCO(c)-chain;b) x coX y ):9A : A = (x; : : : ; y) or A = (y; : : : ; x) is an F jCO(c)-chain ^ size(A) > 1. 2It immediately follows that every F -chain contained in CO(c) must be a clique of liX .Remark R20 Let c be a clique of coX and A be an F jCO(c)-chain.Then A is a clique of liX . 2Proof If there are x; y 2 A with x co y we would have a contradiction with ASS4b. 2With the last assumption it is possible to prove that all nets associated with a concurrencystructure are simple.Proposition P43 Let N 2 Nets(CS). Then N is simple. 2Proof Imagine N is not simple. Then there are two elements x; x0 2 X with x0 6= x and�xF = �x0F and x�F = x0�F . First note that x; x0 2 T is impossible by P35 which excludesbranched S-elements. So we conclude that x; x0 2 S which means that we can �nd t; t0 2 Twith x; x0 2 t�F and x; x0 2 �t0F . Certainly s co s0 holds (remember D12d and D12e).By D6e there must be a z with s co z and s0 li z (without loss of generality). As s li zand s; z 2 co[s0] we can apply ASS4a which requires the existence of an F jco[s0]-chainA = (s; : : : ; z) or A = (z; : : : ; s). But in any case we must have im[s] \ Set(A) 6= ; suchthat either t 2 A or t0 2 A. But this is not reconcilable with an F jco[s0] because t; t0 2 li[s].25.10 Acyclic Concurrency StructuresIt was mentioned that the conventional way to deal with concurrency and causality isbased on partially ordered sets (X;<) where the causality relation is derived from � by53

li =< [ <�1 and concurrency is the irre exive complement. That every acyclic concur-rency structure can be represented in the formalism of posets is proved in the followingproposition. Given a consistent orientation F the corresponding poset is simply (X;F �X).The fact that the converse poset also represents a concurrency structure coincides withour result that there are two consistent orientations F and F�1.Proposition P44 Let CS be acyclic and (S; T; F ) 2 Nets(CS).Then li = F+ [ (F�1)+. 2Proof Assume the proposition does not hold. Then there is an acyclic F -chain A = (x =a0; : : : ; an = y) with x co y. Let us take the shortest one. It follows that Set(A) � fxgand Set(A)� fyg are cliques of liX . This implies x; y 2 S, otherwise we could apply P25to x or y such that we �nd a shorter A. Moreover we are sure that n > 2 (Otherwise wecould derive x li y). Let l 2 Lines(CS) with Set(A)�fxg � l. By P23 there is some z 2 lwith fz; a2g = im[a1]. It should be clear that z 2 �a1F (z 2 a1�F would imply z co a2contradicting z; a2 2 l). Now we are prepared to apply ASS4a: As z li y and z; y 2 co[a0]this yields an F jco[a0]-chain B = (y; : : : ; z) (The reverse F jco[a0]-chain B = (z; : : : ; y)is excluded, because z 2 S and P35 and a0 li a1 implies a1 =2 B). Combining A and Bgives an F -cycle C = (z; : : : ; y; : : : ; z) contradicting the fact that F - is acyclic on X whichfollows from the precondition that CS is acyclic via P42. 2It will be immediately proved that all doubly in�nite F -chains are lines. So the class ofacyclic concurrency structures has the advantage that lines can be directly determinedonce the ow relation is given.Corollary C10 Let CS be acyclic and (S; T; F ) 2 Nets(CS).Then A is an !!-in�nite F -chain ) Set(A) 2 Lines(CS). 2Proof According to the previous proposition, Set(A) is a clique of liX , which can beextended to a line L with Set(A) � L. Assume Set(A) =2 Lines(CS). Then Set(A) 6= Land there is an x 2 L � Set(A). Clearly Set(A) � li[x]. Now choose some y 2 Set(A).As x; y 2 L D6j requires the existance of an im-chain B = (x = x0; x1; : : : ; xn = y) suchthat Set(B) � L. Let i be the minimal index such that xi =2 Set(A). It follows that i � 1,xi+1 2 Set(B) and xi im xi+1. Furthermore, xi+1 has neighbors on A, say u; v 2 Set(A)with u F xi+1 F v implying u 6= v as F is a consistent orientation. So there are threeneighbors fu; v; xig � im[xi+1] all of them being contained in L, a contradiction with L7.2K-density, formulated in terms of the ow relation, yields for acyclic concurrency structuresthe following proposition.Proposition P45 Let CS be acyclic ^ N = (S; T; F ) 2 Nets(CS)^v; x 2 c 2 Cuts(CS)^ A;B be F -chains with A = (u; : : :; v)^B = (x; : : :; y).Then 8C = (u; : : : ; y) : C is an F -chain ) Set(C)\ c 6= ;. 2Proof Assume Set(C) \ c = ;. Extending C to an !!-in�nite F -chain D yields a lineSet(D) 2 Lines(CS) according to C10. Now observe that (F�1)�X [u] \ c = ;. Otherwisechoosing some z 2 (F�1�X [u]\ c we could construct an F -chain (z; : : : ; u; : : : ; v) implyingz li v (by P44) which contradicts z; v 2 c. Similarly we have F �X [y]\ c = ;. So we are surethat Set(D) \ c = ;, but by D6h we must have Set(D)\ c 6= ;. Contradiction. 2Restricting concurrency theory to acyclic structures simpli�es several proofs, as the tran-54

sitivity of the order relation can be exploited. On the other hand, cyclic concurrencystructures are �nite, which is a simpli�cation of a di�erent kind. As a consequence it willbe sometimes convenient to carry out a proof separately for these two cases.5.11 Cone Intersection PropertyIn Petri 1987 the cone intersection property was formulated for acyclic concurrency struc-tures in the following way: Given two elements their past cones as well as their futurecones of causality should have a nonempty intersection. More precisely, for every pair x; yit is required that F+[x] \ F+[y] 6= ; and (F+)�1[x] \ (F+)�1[y] 6= ;. Note that it isnot essential, if the consistent orientation F or its inverse is used to verify this property.As concurrency theory should not cover only acyclic structures but also cyclic ones, andthe existence of a consistent orientation is no apriori postulate, it was recognized that adi�erent formulation of this property is necessary.The major reason for the cone intersection property is to ensure continuity for thoseprocesses described by concurrency structures. Unfortunately D-continuity has not beende�ned for cyclic structures, but in case of acyclic structures the relevance of the coneintersection property for D-continuity is known. Although the examination of D-continuityis beyond the scope of this treatment, it is believed that the axioms (together with ourassumptions) are strong enough to ensure this important property. Furthermore, it shouldbe mentioned, that for the reachability results and the link to elementary net systemspresented in the subsequent sections the cone intersection property is not essential.First we prove that the cone intersection property follows almost directly from our axiomof �nite concurrency neighborhood (D6g) for the class of acyclic concurrency structures.Proposition P46 Let CS be acyclic ^ x; y 2 X with x co y.a) F �X [x] \ F �X [y] 6= ;;b) (F �X)�1[x]\ (F �X)�1[y] 6= ;. 2Proof Using P42 we �nd that F is acyclic. Assume our proposition is not satis�ed, thatis, there are x; y 2 X with F �X [x]\F �X [y] = ;. Choose two arbitrary lines lx; ly 2 Lines(CS)with x 2 lx and y 2 ly. As all lines l 2 Lines(CS) are in�nite, we can apply P40 to �nd twoin�nite F -chains Ax = (x; : : :) and Ay = (y; : : :). By our assumption Set(Ax)\Set(Ay) = ;holds, and we claim that Set(Ay) � co[x], which is in�nite, contradicting D6g. To see thatthe claim holds notice that there cannot be any z 2 Ay with x li z, as C9 would require theexistence of an F -chain Az = (x; : : : ; z) or Az = (z; : : : ; x). Both cases are not reconcilablewith our assumption that F �X [x]\ F �X [y] = ;. 2As we have already mentioned, the axiom D6g seems to be very strong, and a concurrencytheory yielding similar results with a weaker postulate might be desired. Then it is anatural question how to ensure the cone intersection property in terms of concurrency andcausality without referring to the ow relation. One way to achieve this is to choose thefollowing formulation, which was proposed in Petri 1987 as a su�cient condition for thecone intersection property, as an axiom, although even this might be too strong.55

Axiom A15 [Su�cient Condition for Cone Intersection Property]8x; y 2 X : x co y )(9u; v; p; q : u li v ^ u; v 2 li[x]\ li[y]^ x; y 2 co[p]\ co[q]^u co p ^ v co q ^ u li q ^ v li p). 2This axiom states that every two concurrent elements x and y have some common elementsu and v that are causally dependent with additional the requirement that u a�ects x aswell as y and v is a�ected by x and y (the direction of this in uence is essential here). Thetwo elements p and q are used for technical reasons to ensure that u and v are located inopposite directions of time from the viewpoint of x and y. The physical aim of the coneintersection property is to avoid partially frozen systems, that are systems where in somecomponents the time evolution might stop although in other parts the dynamic continueswithout being a�ected. Loosely speaking, the cone intersection property states that x andy have been synchronized at u and will resynchronize at v (or vice versa).First a simple but useful lemma is proposed exploiting the nice property of transitivity ofthe partial order associated with an acyclic concurrency structure.Lemma L22 Let CS be acyclic.Then x co y ^ x li z ^ y li z ) x F+ z ^ yF+z _ z F+ y ^ z F+ x. 2Proof As F+ is transitive x F+ zF+y would imply x F+ y which itself implies x li yaccording to P44. Contradiction with x co y. Similarly, y F+ z F+ x implies y F+ x andy li x. 2Successive application of this lemma and the axiom A15 gives the cone intersection prop-erty for acyclic concurrency structures, which is the same as P46, but the proof does notrefer to D6g (at least not directly).Proposition P47 Let CS be acyclic ^ x; y 2 X ^ x co y ^ A15.a) F �X [x] \ F �X [y] 6= ;;b) (F �X)�1[x]\ (F �X)�1[y] 6= ;. 2Proof We apply A15 in connection with P44. So there are u; v; p; q satisfying the conditionof A15. As p li v, we have either p F+ v or v F+ p. We will only deal with p F+ v, asthe latter case is simply the F -reversal counterpart. In the following we make repetitiveuse of the fact L22. So p F+ v implies u F+ v due to p co u. With u F+ v we can onlychoose u F+ q due to v co q. With q co y this implies u F+ y. Similarly, x F+ v followsfrom p F+ v and p co x. Furthermore, we have u F+ x by u F+ q with x co q and �nallyy F+ v by p F+ v and p co y. Altogether, we have necessarily v 2 F �X [x] and v 2 F �X [y]proving the �rst part and u 2 (F �X)�1[x] and u 2 (F �X)�1[y] which yields the second partof our proposition. 2The cone intersection property for cyclic concurrency structures has not been de�nedyet, although an immediate idea is to introduce it exactly as in the acyclic case (thede�nition is P46) with the help of some consistent orientation. If we proceed in this way,an acyclic concurrency structure generated by the unfolding of a cyclic one simply inheritsthe cone intersection property, such that the unfolding of a concurrency structure is againa concurrency structure. Strictly, we have not de�ned the concept of unfolding, but fromthe examples it should be clear what is meant.56

With this general de�nition we could also examine if the cone intersection property isensured by the list of axioms proposed for a concurrency structure. This establishes adirect relation to P42, where it was conjectured that it might be possible to prove: CS iscyclic ) N is cyclic. Once we know that N is cyclic, we have F+ = X�X and F+[x] = Xfor every x 2 X . Hence the cone intersection property would be satis�ed.5.12 ReachabilityThis section the question of dynamics in concurrency structures is addressed. First it isshown that conceiving cuts of local states (which will be called S-cuts) as markings ofa net associated with the concurrency structure the common rule for the token game iscompletely equivalent to the propagation rules for concurrency. This means conceiving anS-cut as a marking and �ring some transitions according to the rules of the token gamewe �nd a �nal marking which is always an S-cut. Furthermore it will be proved, and thiswill be more di�cult and involves several conjectures proposed in the previous sections,that given two S-cuts interpreted as markings there is always a way applying the rules ofthe token game to reach one marking from the other. Note that, as the full and transitivereachability relation in nets is chosen as a basis for this treatment, transitions may be �redforward as well as backward.As stated above the class of S-cuts contains exactly those cuts consisting of local statesonly.De�nition D22 SCuts(CS) := Cuts(CS)\ P(S). 2Before considering cuts we focus our attention on the behavior of concurrency cliquesinterpreted as markings, when the �ring rule is applied. We �nd that given a concurrencyclique and �ring a transition that is enabled at the corresponding marking, we again obtaina clique of concurrency and, furthermore, the intermediate state, where the transition hasjust consumed the input tokens but not produced output yet, constitutes also a clique ofconcurrency. Of course, those tokens not touched by the selected transition are containedin each of these cliques. Actually, the following lemma is slightly more general, as itallows also events to be contained in these cliques and states the above implication alsoin its backward direction, which corresponds to the situation that the transition is �redbackwards.Lemma L23 Let (S; T; F ) 2 Nets(CS)^ t 2 T ^t 2 c2 ^ c1 = c2 � ftg [ �tF ^ c3 = c2 � ftg [ t�F .Then c1 is a clique of coX , c2 is a clique of coX , c3 is a clique of coX . 2Proof De�ne c := c2 � ftg. First we prove: c2 is a clique of coX ) (c1 is a clique of coX^ c3 is a clique of coX). Assume c2 is a clique coX . Then by P25 we have 8x 2 c : 8y 2im[t] : x co y such that c1 and c3 are cliques of coX .Now we show: c1 is a clique of coX ) c2 is a clique of coX . Assume c1 is a clique of coX .Then for every x 2 c we can apply P28 as the condition 8y 2 �tF : x co y is satis�ed (andP30 holds). So we infer 8x 2 c : x co t and conclude that c2 is a clique of coX . That c3 isa clique of coX ) c2 is a clique of coX is proved similar to the previous implications. 2The previous lemma can be extended to cuts in the following sense: Given a cut interpretedas a marking and �ring a transition, that is enabled, again yields a cut. The only point,57

which is left to be proved, is that the maximality of a clique is preserved under the �ringrule.Lemma L24 Let (S; T; F ) 2 Nets(CS)^ t 2 T ^t 2 c2 ^ c1 = c2 � ftg [ �tF ^ c3 = c2 � ftg [ t�F .Then c1 2 Cuts(CS), c2 2 Cuts(CS), c3 2 Cuts(CS). 2Proof By L23 it is already clear that c1 2 Cuts(CS) ) c2 is a clique of coX andc2 2 Cuts(CS) ) c1 is a clique of coX such that all we have to prove the maximality ofc2 resp. c1 concerning co.We start with: c1 2 Cuts(CS)) c2 2 Cuts(CS). Assume c2 is not maximal with respectto co. Then 9c02 2 Cuts(CS) : c2 � c02. With c01 := c02 � ftg [ �tF applying L23 yields thatc01 is a clique of coX . Obviously we have c1 � c01 but this contradicts c1 2 Cuts(CS).Similarly, we show: c2 2 Cuts(CS) ) c1 2 Cuts(CS). Again assume 9c01 2 Cuts(CS) :c1 � c01. De�ne c02 := c01 � �tF [ ftg. L23 implies that c02 is a clique of coX . But c2 � c02 isnot consistent with c2 2 Cuts(CS).Exchanging �tF and t�F directly leads to the proof for c2 2 Cuts(CS), c3 2 Cuts(CS).2By the help of this lemma one can construct an S-cut from every cut, that may containsome events, by �ring the corresponding transitions.Lemma L25 Let c 2 Cuts(CS). Then 9c0 2 SCuts(CS) : c \ S � c0. 2Proof Let cT = c \ T and cS = c \ S. We construct the set c0 := cS [ fs : 9t 2 cT :s 2 t�Xg (�tX would also do). With L24 it is clear that c0 2 Cuts(CS) and in particularc0 2 SCuts(CS). 2So we can easily prove that the class of S-cuts cannot be empty, which ensures that wecan always �nd some initial marking, such that the formalism of elementary net systemswill be applicable.Lemma L26 SCuts(CS) 6= ;. 2Proof Choose some x 2 X . Certainly there is a cut c 2 Cuts(CS) such that x 2 c.Applying L25 to c gives us a c0 2 SCuts(CS). 2For every transition we can �nd an S-cut, at which it is enabled, and a di�erent S-cut,at which it is reverse enabled. Although this observation is trivial, it yields together withthe following results a nice and desired property of elementary net systems.Remark R21 Let t 2 T .a) 9c 2 SCuts(CS) : (�tX � c);b) 9c 2 SCuts(CS) : (t�X � c). 2Proof Let c1 = �tX (or c1 = t�X). Then c1 is a clique of coX and it can be extended toa cut c2 2 Cuts(CS) with c1 � c2. Applying L25 we can construct c 2 SCuts(CS) withc � c2. 2The remainder of this section is concerned with the question if it is possible to establishthe reachability relation between two arbitrary S-cuts conceived as the markings of a netassociated with a concurrency structure. For this purpose we introduce two abbreviations:58

[s0 * c]N and [s0 + c]N . s0 is a place and c is a set of places. Although these de�nitions areindependent of the underlying concurrency structure, we will use this notation always withthe assumption that c is an S-cut, which should indicate the following meaning: [s0 * c]Nis the set of net elements that can be obtained by looking into the future of s0, until wereach the speci�ed cut c. Similarly, [s0 + c]N refers to the past of s0 up to c.De�nition D23 Let N = (S; T; F ) be a net ^c � S ^ s0 2 S.a) [s0 * c]N is the smallest set satisfyings0 2 [s0 * c]N and x 2 [s0 * c]N ^ x =2 c) x�F � [s0 * c]N .b) [s0 + c]N is the smallest set satisfyings0 2 [s0 + c]N and x 2 [s0 + c]N ^ x =2 c) �xF � [s0 + c]N . 2Remark R22 [s0 * c]N and [s0 + c]N are well-de�ned. 2The following list of properties follows immediately from the de�nitions.Remark R23 Let N = (S; T; F ) be a net and s0 2 S ^ c � S.a) [s0 * c]N = [s0 + c]N 0 where N 0 = (S; T; F�1);b) 8t 2 [s0 * c]N \ T : t�F � [s0 * c]N ;c) 8t 2 [s0 + c]N \ T : �tF � [s0 + c]N ;d) 8s 2 [s0 * c]N \ S � c : s0�F � [s0 * c]N ;e) 8s 2 [s0 + c]N \ S � c : �s0F � [s0 + c]N ;f) x 2 [s0 * c]N ) x = s0 _ 9y 2 [s0 * c]N � c : y F x;g) x 2 [s0 + c]N ) x = s0 _ 9y 2 [s0 + c]N � c : x F y;h) t 2 T ^ �tF � c) t =2 [s0 * c]N ;i) t 2 T ^ t�F � c) t =2 [s0 + s]N ; 2Inductively we prove that for every x 2 [s0 * c]N we �nd an F -chain from s0 to x thatis completely contained in [s0 * c]N . Additionally, if x is located on c, this is the onlyelement on our chain that might meet c (intuitively the chain ends as soon as it meetsc). Otherwise our chain does not intersect with c, which is, loosely speaking, the situationthat x is located before c, or c is located before s0 and cannot be reached by a �nite chain,which might be the case for acyclic structures.Proposition P48 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 Cuts(CS).a) 8x 2 [s0 * c]N : 9A : A = (s0; : : : ; x) is an F j[s0 * c]N -chain ^ Set(A) \ c = fxg \ c;b) 8x 2 [s0 + c]N : 9A : A = (x; : : :; s0) is an F j[s0 + c]N -chain ^ Set(A) \ c = fxg \ c.2Proof De�ne H(x) :, 9A : A = (s0; : : : ; x) is an F j[s0 * c]N -chain ^ Set(A) \ c =fxg \ c. Obviously H(s0) holds if we simply choose A = (s0) such that x = s0. AssumingH(x) holds and x =2 c we prove H(y) for all y 2 x�F . H(x) yields an F j[s0 * c]N -chainAx = (s0; : : : ; x) with Set(Ax) \ c = fxg \ c. As y 2 [s0 * c]N we append y to A whichgives an F j[s0 * c]N -chain Ay = (s0; : : : ; x; y). From x =2 c we infer Set(Ax) \ c = ; suchthat Set(Ay) \ c = fyg \ c. 259

Furthermore, it will be show that, if [s0 * c]N is �nite, it must contain some element yof c (intuitively the extension of [s0 * c]N into the future is restricted by c), which itselfimplies that an F -chain exists within [s0 * c]N from s0 to y. The last point again impliesthat [s0 * c]N is �nite, because due to k-density there is no way to circumvent the cutc. Altogether this cyclic chain of implications establishes an equivalence between thesestatements enumerated in the next proposition.Proposition P49 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 Cuts(CS).Then the following statements are equivalent:a) [s0 * c]N is �nite;b) c\ [s0 * c]N 6= ;;c) 9y 2 c : 9B : B = (s0; : : : ; y) is an F j[s0 * c]N -chain;d) 9x 2 c : 9A : A = (s0; : : : ; x) is an F -chain. 2Proof [s0 * c]N is �nite ) c \ [s0 * c]N 6= ;: Assuming c \ [s0 * c]N = ; we show that[s0 * c]N is in�nite. If CS is acyclic it is in�nite by R18, and, as for every x 2 X we havex�F 6= ;, we get an in�nite [s0 * c]N . Otherwise CS is cyclic and �nite. Then we choosea line l 2 Lines(CS) with s0 2 l and by D6h there is some z 2 l \ c. As by P40 there isalso an F -cycle A = (s0; : : : ; z; : : : ; s0) with Set(A) = l, we certainly have z 2 [s0 * c]N . Asz 2 c\ [s0 * c]N the implication holds.c \ [s0 * c]N 6= ; ) 9x 2 c : 9A : A = (s0; : : : ; x) is an F j[s0 * c]N -chain: By P48a givenx 2 c \ [s0 * c]N there is always an F j[s0 * c]N -chain A = (s0; : : : ; x).9x 2 c : 9B : B = (s0; : : : ; x) is an F j[s0 * c]N -chain ) 9y 2 c : 9A : A = (s0 =a0; a1 : : : ; an = y) is an F -chain: Choose A = B.9y 2 c : 9B : B = (s0; : : : ; y) is an F -chain ) [s0 * c]N is �nite: For cyclic CS this isimmediately clear by R18. So assume CS is acyclic. Let B = (s0; : : : ; y) be an F -chain withy 2 c and assume [s0 * c]N is in�nite. Then by P22 there must be an in�nite F j[s0 * c]N -chain C = (s0; : : :) with Set(C) \ c = ;. Furthermore, we choose an in�nite F�1-chainD = (s0; : : :). P44 suggests that there is a line lB 2 Lines(CS) with Set(D)[Set(B) � lB.Similarly, lC = Set(D)[Set(C) constitutes a line lC 2 Lines(CS). As lB \ c = fyg by P4and there must be some z with lC \ c = fzg, we conclude that z 2 Set(C). But this is acontradiction with Set(C)\ c = ;. 2Generally every statement concerning [s0 * c]N has a counterpart involving [s0 + c]N , as itcan easily seen by reversal of the ow relation.Proposition P50 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 Cuts(CS).Then the following statements are equivalent:a) [s0 + c]N is �nite;b) c\ [s0 + c]N 6= ;;c) 9y 2 c : 9B : B = (y; : : : ; s0) is an F j[s0 + c]N -chain;d) 9x 2 c : 9A : A = (x; : : :; s0) is an F -chain. 2Proof Analogous to P49. 260

Once we know that our concurrency structure is acyclic and that [s0 * c]N is �nite, wecan be sure that every transition-element occurring (directly) after c cannot be containedin [s0 * c]N . In other words, there is no chance starting at s0 to bypass the cut, if wefollow the ow relation. Notice that this proposition cannot be generally satis�ed in cyclicstructures.Proposition P51 Let CS be acyclic ^ N = (S; T; F ) 2 Nets(CS)^s0 2 S ^ c 2 SCuts(CS).a) [s0 * c]N is �nite ^s 2 c ^ t 2 s�F ) t =2 [s0 * c]N ;b) [s0 + c]N is �nite ^s 2 c ^ t 2 �sF ) t =2 [s0 + c]N . 2Proof Let [s0 * c]N be �nite, s 2 c, t 2 s�F and suppose t 2 [s0 * c]N . Then P48a suggeststhere is an F j[s0 * c]N -chain A = (s0; : : : ; t) with Set(A) \ c = ;. Furthermore as thereis some x 2 c \ [s0 * c]N by P49 we can apply P48a again which yields an F j[s0 * c]N -chain B = (s0; : : : ; x) with Set(B) \ c = fxg. Now observe that the preconditions ofP45 are satis�ed (we have two F -chains B = (s0; : : : ; x) and D = (s; t) with x; s 2 c).So we conclude that all F -chains A = (s0; : : : ; t) must meet c, that is, Set(A) \ c 6= ;.Contradiction. 2Given s0 and a cut c there are two possibilities, which are not mutually exclusive in cyclicstructures: s0 is located before c, which implies that [s0 * c]N is �nite. Or s0 occurs afterc, which means that [s0 + c]N is �nite.Lemma L27 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 SCuts(CS).Then [s0 * c]N is �nite _ [s0 + c]N is �nite. 2Proof As c 2 Cuts(CS) there is some x 2 c with s0 liX x. This implies by C9 that thereis an F -chain A = (s0; : : : ; x) or A = (x; : : :; s0). Applying P49 in the former case or P50in the later case proves the lemma. 2We have seen that, if [s0 * c]N is �nite, it must be restricted by c. As c is a S-cut, wecan �nd a transition with a postset covered by c. c, interpreted as a marking, enables thistransition in backward direction. Similarly, if [s0 + c]N is �nite, c enables some transition inforward direction. In both cases this transition is also contained in [s0 * c]N resp. [s0 + c]N .This proposition gives a �rst hint how the �niteness of either [s0 * c]N or [s0 + c]N may beexploited to decrease the distance between a cut c and a part s0 of some other cut.Lemma L28 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 SCuts(CS).a) [s0 * c]N is �nite ) 9t 2 T \ [s0 * c]N : t�F � c;b) [s0 + c]N is �nite ) 9t 2 T \ [s0 + c]N : �tF � c. 2Proof Assume [s0 * c]N is �nite and 8t 2 [s0 * c]N \ T : :(t�F � c). We conclude that8t 2 [s0 * c]N\T : 9s 2 t�F : s =2 c and with the additional fact that 8s 2 [s0 * c]N\S : s =2c ) s�F � [s0 * c]N we can construct an in�nite F j[s0 * c]N -chain A = (s0 = a0; a1; : : :)with ai =2 c for all i. Now either Set(A) is in�nite, which contradicts our initial assumptionthat [s0 * c]N is �nite, or A is not acyclic, which means it contains an F -cycle B withSet(B) � Set(A) implying Set(B) \ c = ;. But this is not reconcilable with ASS3. Asimilar proof shows the second part of the lemma. 261

With the previous lemma we can immediately derive a �rst reachability result: Imaginethe situation, where we have an arbitrary cut c, which will be interpreted as a markingand some place s0, we want to reach (that is, it should be contained in some �nal markingc0 that is reachable from c) by �ring transitions. That this is indeed possible is clear bythe following argument: We know that [s0 * c]N or [s0 + c]N is �nite. Repeated applicationof the previous lemma gives us a sequence of transitions to �re (which moves c towardss0) successively reducing the cardinality of either [s0 * c]N of [s0 + c]N . Finally there willbe no transition left between s0 and the current marking, which means that s0 is actuallyreached. As only transitions contained in either [s0 * c]N or [s0 + c]N are used, we can eveninfer a stricter reachability RN 0 instead of RN where N 0 is derived from N by removingunnecessary transitions, that have not been �red in this procedure.Lemma L29 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 SCuts(CS).a) [s0 + c]N is �nite ) 9c0 2 SCuts(CS) : s0 2 c0 ^ c RN 0 c0where T 0 := T \ [s0 + c]N , F 0 := (F jS [ T 0) and N 0 := (S; T 0; F 0).b) [s0 * c]N is �nite ) 9c0 2 SCuts(CS) : s0 2 c0 ^ c RN 0 c0where T 0 := T \ [s0 * c]N , F 0 := (F jS [ T 0) and N 0 := (S; T 0; F 0); 2Proof We prove the �rst part of the lemma by induction. Assume [s0 + c]N is �nite andde�ne H(n) :, (j [s0 + c]N \ T j = n ) 9c0 2 SCuts(CS) : s0 2 c0 ^ c RN 0 c0). H(0) issatis�ed as j [s0 + c]N \ T j = 0 implies [s0 + c]N = fs0g and s0 2 c such that we can simplychoose c0 = c. Assuming H(n � 1) holds for n > 0 we have to prove H(n): L28 gives usa t 2 T 0 with �tF � c. L24 yields a cut c00 2 SCuts(CS), which is c00 = (c � �tF ) [ t�Fwith c RN c00, and as t 2 T 0 we also have c RN 0 c00. Applying H(n� 1) to c00 (note thatj [s0 + c00]N \ T � ftg j = n � 1) yields a c0 2 SCuts(CS) with s0 2 c0 and c00 RN 0 c0. Bytransitivity of RN 0 we conclude c RN 0 c0. 2Remark R24 Let N = (S; T; F ) and N 0 = (S; T 0; F 0) be nets withT 0 � T and F 0 = F j(S [ T 0).Then RN 0 � RN . 2With the fact that additional transitions can never diminish the reachability and that atleast one of the two sets [s0 + c]N or [s0 * c]N is �nite we get a �rst important reachabilityresult.Lemma L30 Let N = (S; T; F ) 2 Nets(CS)^ s0 2 S ^ c 2 SCuts(CS).Then 9c0 2 SCuts(CS) : s0 2 c0 ^ c RN c0. 2Proof By L27 and L29 with the fact that R0N � RN (R24). 2Given two cuts the distance between two cuts can be measured in terms of the cardinalityof their intersection. If the overlapping is large we assume that we do not need much moree�ort to reach one from the other, such that the distance is short. If the intersection isempty, then we assume the two cuts are far away from each other.From this point of view the preceding lemma provides a method to decrease the distancebetween two cuts, which are disjunct initially. Unfortunately this method does not gener-ally succeed for two cuts, which have a non-empty intersection, as for �nite structures both[s0 + c]N and [s0 * c]N are �nite such that there is no global coordination determining onwhich way (for a cycle there are two major possibilities) these two cuts should approach62

each other. A solution to this problem is to start with the assumption that we have twocuts c and c0 which intersect each other and to allow only those movements covered bythe degree of freedom CO(c\ c0). That is, speaking in terms of the token game, we simply�x those tokens in c \ c0 and let c and c0 approach each other using the same method weemployed for our �rst reachability result.Scope S4 Let N = (S; T; F ) 2 Nets(CS)^ c; c0 2 SCuts(CS)^c \ c0 6= ; ^ s0 2 c0 � c^ Z = CO(c\ c0).We start with two overlapping S-cuts c and c0 and the goal is to decrease the distancebetween c and c0 (that is to increase their overlapping) interpreted as markings by applyingthe rules of the token game. As we want to achieve this within the degree of freedom Z(that is we do not want to �re any transitions not contained in Z) we have to prove that[s0 + c]N or [s0 * c]N is �nite and completely covered by Z.Scope S5 As c 2 Cuts(CS) given s0 there must be some x 2 c with x liX s0. ASS4asuggests that there is an F jZ-chain A = (x; : : : ; s0) or A = (s0; : : : ; x). For the followingwe assume that we are faced with the former case.We will immediately show that in this case [s0 + c]N is necessarily contained in Z. Thefollowing lemma is even stronger than this claim but is not di�cult to prove it by structuralinduction guided by the de�nition of [s0 + c]N .Lemma L31 8y 2 [s0 + c]N : 9x 2 c\ liX [y] : 9A : A = (x; : : :; y) is an F jZ-chain. 2Proof We prove this by induction over [s0 + c]N . For y = s0 the lemma holds by theassumption of 5.12. Assuming the lemma is proved for y =2 c we show that it also holds fory0 2 �yF . So we have an F jZ-chain A = (x; : : :; y) for some x 2 c \ liX [y]. If y 2 S thereis an unique y0 and A has the form A = (s; : : : ; y0; y), which implies y0 2 Z. Otherwisey 2 T and we exploit P25 to infer y0 2 Z. As c 2 Cuts(CS) we �nd a x0 2 c withx0 li y0. Applying ASS4a with the fact that x0; y0 2 Z requires that either an F jZ-chainA0 = (x0; : : : ; y0) or an F jZ-chain A0 = (y0; : : : ; x0) exists. Let us consider the latter case�rst: If y0 2 T then y 2 S and A = (x; : : : ; y0; y) otherwise if y0 2 S then y 2 T andA0 = (y0; y; : : : ; x0). Combining A and A0 we construct a further F jZ-chain B = (x; : : :; x0)which yields a contradiction with ASS4b because x co x0. Hence we are sure that we canonly have an F jZ-chain A0 = (x0; : : : ; y0). 2The anticipated claim immediately follows from this.Lemma L32 [s0 + c]N � Z. 2Proof Immediately from L31. 22 S5That [s0 + c]N or [s0 * c]N (which can be proved similarly in case of A = (s0; : : : ; x)) iscontained in Z is not enough. As reachability means that only a �nite number of transitionsmay be �red we must ensure that one of these two sets is �nite and has the additionallyproperty to be covered Z.Lemma L33 ([s0 + c]N is �nite ^[s0 + c]N � Z) _ ([s0 * c]N is �nite ^[s0 * c]N � Z). 2Proof With A = (x; : : :; s0) in 5.12 we have found that [s0 + c]N � Z. Applying P49we additionally �nd that [s0 + c]N is �nite. The other possibility is A = (s0; : : : ; x) which63

implies that [s0 * c]N is �nite by P49 and we can proceed analogously to L31 to prove[s0 * c]N � Z. 2With the preceding lemma we derive the desired reachability result: Given S-cuts c and c0with non-empty intersection and an arbitrary s0 contained in c0 but not in c we can �nd afurther S-cut c00 that is reachable from c and additionally covers s0. Those tokens containedin the intersection of c and c0 are again contained in c00, that is, it is not necessary to movethem.Lemma L34 9c00 2 SCuts(CS) : s0 2 c00 ^ c\ c0 � c00 ^ c RN c00. 2Proof Assume we have a �nite [s0 + c]N � Z. Otherwise we must have a �nite [s0 * c]N �Z according to L33 and an analogous proof succeeds. L29 gives us a c00 2 SCuts(CS) withs0 2 c00 and c RN 0 c00, where N 0 = (S; T 0; F 0), T 0 = T \ [s0 + c]N and F 0 = F j(S [ T ).Certainly c RN 0 c00 implies c RN c00, so it is only left to prove that c \ c0 � c00: For everyt 2 [s0 + c]N \ T we have im[t] \ (c \ c0) = ; by L32. This means that for all m;m0 withm rN 0 m0 we have (c \ c0) � m , (c \ c0) � m0 (tokens in c \ c0 are not touched) andli�ting this property to RN 0 we �nd with c RN 0 c00 that (c \ c0) � c (which is certainlytrue) implies (c\ c0) � c00. 22 S5Notice that the previous result indeed decreases the distance between c and c0 as c iscarried over to c00 and the intersection of c00 and c is strictly greater than the intersectionof c0 and c. At least the element s0 is added to the set.Corollary C11 Let N 2 Nets(CS)^ c; c0 2 SCuts(CS).Then 9c00 2 SCuts(CS) : c\ c0 � c00 \ c0 ^ c RN c00. 2Proof We distinguish two cases: Either c\ c0 = ; or c\ c0 6= ;. In the �rst case we applyL30: We choose a s0 2 c0 and the lemma gives us a c00 2 SCuts(CS) with c R c00 ands0 2 c00. Note that c \ c0 = ; � fs0g � c00 \ c0 trivially holds. In the latter case we havec \ c0 6= ; and we apply L34: Choosing s0 2 c0 � c yields a c00 2 SCuts(CS) with s0 2 c00,c \ c0 � c00 and c R c00. This implies c \ c0 � c00 \ c0 and even c \ c0 � c00 \ c0 becauses0 =2 c \ c0 but s0 2 c00 \ c0. 2Successive decrease of the distance between two cuts �nally identi�es them after a �nitenumber of steps. Hence our main result concerning the reachability between two arbitraryS-cuts is established.Proposition P52 Let N 2 Nets(CS)^ c; c0 2 SCuts(CS).Then c RN c0. 2Proof To prove this by (reverse) induction over j c \ c0 j (remember that j c\ c0 j is�nite by R3) we de�ne H(n) :, (j c \ c0 j = n ) c RN c0). H(j c0 j) holds becausen = j c0 j implies c = c0 and c RN c0 holds trivially. Assuming H(i) holds for all i withn < i � j c0 j we prove H(n): Given c; c0 2 SCuts(CS) we apply C11, which yields ac00 2 SCuts(CS) satisfying j c00 \ c0 j > j c \ c0 j = n and c RN c00. So we can apply H(i)for some i = j c00 \ c0 j > n which proves c00 RN c0. By transitivity of RN we concludec RN c0. 264

5.13 Elementary Net SystemsThe reachability results derived in the previous section indicate that elementary net sys-tems are an adequate formalism yielding a concise representation of concurrency struc-tures. We simply choose an arbitrary net associated with the concurrency structure andan arbitrary S-cut as initial case. Notice that the initial case has no further signi�cancebeyond the fact that it represents the case class of the elementary net systems.Scope S6 Let (S; T; F ) 2 Nets(CS)^ c0 2 SCuts(CS)^NS = (S; T; F; c0).As all reachable cases are S-cuts in the concurrency structure, which have been requiredto be �nite, we can identify the case class with the sequential case class, where only singletransitions are �red to generate new cases.Proposition P53 SeqCaseClass(NS) = CaseClass(NS). 2Proof It is su�cient to show SRN = RN . SRN � RN is immediately clear. RN � SRNfollows from rN � SRN by transitive closure. Given c; c0 � S with c rN c0 there is anevent E with c[E > c0. As c0 2 SCuts(CS) we are sure that c0 is �nite (R3), and fromP22 we infer that all cases c 2 CaseClass(CS) must be �nite, too. So we can write E asE = ft0; : : : ; tng and we �nd a �nite sequence (c = c0; : : : ; cn+1 = c0) with ci[ti > ci+1 fori 2 [0; n] proving c SRN c0. 2Stated in a di�erent form, the reachability result of the previous section simply correspondsto the identi�cation of the case class of the elementary net systems with the class of S-cutsof the underlying concurrency structure.Proposition P54 CaseClass(NS) = SCuts(CS). 2Proof Let NS = (S; T; F; c0) with c0 2 SCuts(CS).CaseClass(NS) � SCuts(CS): By P53 we can equivalently prove SeqCaseClass(NS)�SCuts(CS). We have c0 2 SCuts(CS) and according to L24 we cannot leave the class ofSCuts(CS) by �ring single transitions forward or backward.SCuts(CS) � CaseClass(NS): It is su�cient to show that for every c 2 SCuts(CS) wehave c0 RN c which can be directly derived using P52 and the fact that c0 2 SCuts(CS).2There is actually no information lost in carrying out the step from concurrency structuresto elementary net systems, because the concurrency relation between local states canalways be reconstructed from the the case class. The full concurrency relation can thenbe easily derived using the propagation rules (in particular P28), and �nally the relationof causality emerges as the irre exive complement.Remark R25 Let s; s0 2 S.Then s coXs0 , 9c 2 CaseClass(NS) : s; s0 2 c. 2The nice and desirable property that the elementary net systems is safe as well as secure isa direct consequence of the fact that immediate temporal predecessors are always causallyrelated to the immediate temporal successors. The property of being safe can be seen asa natural property of physical systems, where every aspect of causality should be coveredby the causality relation li and should not emerge from contacts one the level of thetoken game (that is, a cut should not have a causal in uence on itself). Security canbe interpreted in the following way: Transitions resp. events should only wait for their65

preconditions to be satis�ed and become activated without care of their postconditions,which are anyway located in the future and therefore not accessible.Proposition P55 NS is safe and secure. 2Proof To violate the condition that NS is secure it is necessary to have a transitiont 2 T and a case c 2 CaseClass(NS) with s; s0 2 c such that s F t ^ t F s0. But thisimplies s li s0 (by R12) which is impossible since c 2 SCuts(CS) according to P54. Thisproves that NS is secure which also implies the weaker property that NS is safe. 2That every transition's preconditions (and postconditions) are covered by some S-cutshows, as every S-cut is reachable, that every transition is involved in the dynamics ofour elementary net systems and may eventually �re. This implies that no transition issuper uous, which is also true for every place, as every single local state may be extendedto a (reachable) S-cut.Proposition P56 NS is proper. 2Proof From the fact that N is connected we can derive T = ProperT (NS) ) S =ProperS(NS). So it is su�cient to prove T = ProperT (NS). For every t 2 T we know that�TF is a clique of coX and can be extended to a cuts c 2 SCuts(CS) with �tF � c. By P54we have c 2 CaseClass(CS), such that, as NS is safe, there is some c0 2 CaseClass(CS)with c[t > c0 proving that t 2 ProperT (NS). 22 S62 S6The possibility to describe a concurrency structure by an elementary net systems yields asimple and compact notation and facilitates reasoning about the dynamics, as the usualtoken game can be applied. This is demonstrated by the following two examples in Fig.10, which are elementary net systems that correspond to the concurrency structures ofFig. 1.a and Fig. 2.a. Of course the inverse nets or other cuts as initial markings wouldalso be possible.1

2

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12Figure 10: Concurrency structures as elementary net systems66

6 Conclusion and Open QuestionsIn this treatment we started out from the basic notions of causality and concurrencyas binary relations one some uniform set. In our standard interpretation concurrencystructures are intended for the representation of acyclic as well as cyclic processes intime-space excluding the existence of state-space (that would be spanned by alternatives).Several mainly physically motivated postulates have been introduced as axioms that aconcurrency structure should satisfy in order to conform to physical experience. After theclari�cation of elementary properties of concurrency and causality following from theseaxioms a considerable amount of e�ort has been spent to build a bridge from concurrencytheory to the formalism of nets. That the suggested correspondence between a concurrencystructure and a set of two nets of which one is the inverse of the other is reasonable isjusti�ed �nally by the major result that the class of S-cuts exactly coincides with the caseclass of an elementary net systems based upon one of these nets.In the beginning it was mentioned that our set of concurrency axioms is a slight mod-i�cation of the original theory. At �rst, the property of coherence on lines (D6f), whichreplaces the original axiom of im-coherence im�X = X �X , is necessary to show that theclass of S-cuts is connected in terms of the reachability-relation. If there is a line, wherebetween two elements there is an in�nite number of further elements (as it was shown tobe possible in Best und Merceron 1985), there is no �nite �ring sequence (on the level ofelementary net systems), that could establish a link between two S-cuts, if each of themcontains one of these elements. The second modi�cation of the axioms is a restriction tostructures with a �nite concurrency neighborhood (8x 2 X : co[x] is �nite), which seemsto be even stronger than the requirement for �nite cuts. None of these restrictions have bepresent in the original concurrency theory and it is not clear if they are really necessary.Formally this additional axiom helps us to derive the existence of exactly two classes ofconcurrency structures, namely cyclic and acyclic ones. Furthermore the axiom leads tothe cone-intersection-property for acyclic structures, which is di�cult to formalize withoutan underlying partial order.A �rst hint that concurrency theory can be formulated without assuming an underlyingpartial order was already given in Petri 1980a. To our knowledge a consequent examinationof this approach does not exist. The idea that the introduction of a consistent orientationon a concurrency structure yields exactly two nets was stated in Petri 1988b without proofsand a connection between concurrency theory and elementary net systems, in particularthat the case-class might coincide with the class of S-cuts, has already been conjecturedin Petri 1980a.Unfortunately, we have not succeeded in proving all of these properties mentioned abovedirectly from the axioms that have been chosen as a starting point. Indeed some assump-tions have been necessary to derive the �nal results. Hopefully, these assumptions canbe proved in near future directly from the proposed axioms (maybe with additional butmore elementary assumptions), but, if this is not the case, one has to pose the questionif these axioms are really strong enough. For the moment we have to conceive these as-sumptions as additional axioms, which may be not independent of the other ones. It isbelieved that some of these assumptions, in paticular ASS3 and ASS4 as not so restrictingas it might seem. They might be satis�ed for cyclic models derived from acyclic ones, by67

a folding which is large enough in the temporal direction. Apart from investigating theseassumptions there are several further interesting questions worth to study.An immediate question emerging from the preceding section is whether the initial casein elementary net systems is really necessary to reconstruct the underlying concurrencystructure. It might be possible to prove that under certain conditions (e.g. the requirementfor proper and secure elementary net systems) there is only one possible case class for agiven net, which could be called the natural case class of that net.A second interesting aspect of concurrency theory which should be addressed in a moregeneral framework is the notion of cyclic, partial orders which could be de�ned in sucha way that not a full but only a modest transitivity is required which just prevents theorder relation from getting meaningless (that is the case if every element is before everyother one). Once an appropriate de�nition of cyclic, partial orders is found it might give amore convenient formal basis for concurrency theory. In fact an alternative and interestingapproach for dealing with cyclic structures has been proposed in Petri 1980a on the basisof so-called separation quadruples which are simply unordered pairs of unordered pairsffu; vg; fx; ygg such that u and v separate x and y on some line. The set of all separationquadruples can be used as an alternative representation of concurrency structures. Cer-tainly it provides a uni�ed frame to cope with cyclic as well as acyclic structures, but asthis set is quite large, it is not clear, if it really facilitates reasoning about concurrencytheory.An important and probably the most ingenious concept that is related to concurrency the-ory is the notion of D-continuity that is very di�erent from usual approach to continuity.A �rst goal might be to give an adequate de�nition of D-continuity, that is appropriate forcyclic as well as acyclic structures (a formalism of generalized orders will certainly help-ful here). A second step is to examine if the axioms of concurrency theory really implyD-continuity or if some further requirements are necessary. But not only horizontal con-tinuity (that is continuity within one level of abstraction) is an import issue. The verticalcontinuity (which is a topological continuity between di�erent views) is also worth to in-vestigate. In particular the relation between these apparently di�erent forms of continuitywould be interesting.A further direction of future investigation is certainly driven by the demand for the ap-plication of concurrency theory to real-world problems. So far it is even not clear howto describe elementary experiments of classical mechanics using concurrency theory or ifit is necessary to extend the theory in some way to achieve this. Certainly the connec-tion between D-continuity and continuous movements has to be exploited here. Related tothese questions is obviously the necessity to represent analogous quantities like position,velocity, acceleration and their measurement. Is it really possible to provide a solutionwithin concurrency theory?We have seen that the in�nite two-dimensional grid is not K-dense. Interestingly, fold-ing this grid to a torus in di�erent ways yields the special subclass of �nite and regularconcurrency structures called cycloids. Cycloids have many nice mathematical propertiesand are conjectured to serve as a basis to solve technical safety and security problemsin repetitive dynamical systems. The most astonishing feature of the class of cycliods isprobably that cycloids can be transformed to other cycloids with the help of the Lorentz-transformation known from special relativity theory. The number of events and local68

states turn out to be Lorentz-invariants. So the Lorentz-transformation corresponds toa retiming-transformation and might have applications in the �eld of data ow systems.A �nal and this is probably the most di�cult demand is to include alternatives into thetheory to contribute to the needs of information processing. So far concurrency theoryonly covers the aspect of synchronization, but for the storage of concrete bits and theirtransformation we need a means to express the dimension of state-space. Once informationprocessing is captured by the theory, it might help to cope with general physics, if this isviewed as an informational process. A formally elegant way to introduce state-space is tode�ne a relation al (for mutually exclusive alternatives) which might share up to a certaindegree many properties of co. At some level then the theory must mirror the fundamentaldi�erence between concurrency and alternative. Certainly there will be several physicalrestrictions to be formulated as axioms (We must look out for physical laws, which remaininvariant, if we change the level of abstraction.), and some of them might be motivatedfrom the �eld of quantum mechanics. The wave/particle-duality of quantum-mechanics,for instance, might indicate that di�erent possibilities for future temporal evolution canbe taken concurrently (for waves) as well as in strict alternative fashion (for particles).This observation could motivate the possibility of non-empty intersection between al andco. The general aim might be to develop a deterministic and reversible theory (for closedsystems) such that on the level of nets, which are already equipped with a means toexpress alternatives (by branched places), a deterministic and reversible dynamics (thatis all forward and backward alternatives are solved) emerges with the usual token game.As a theory including alternatives and concurrency simultaneously will be quite complex,a �rst approach might be to consider structures with causality and alternatives but ex-cluding concurrency. Certainly this is not su�cient for information processing purposes,as we have no concurrent ows of information, which could interact with each other (as weknow it from the theory of information- ow-graphs). But nevertheless we can pose severalquestions: Does this theory of alternatives simply correspond to a dualization of concur-rency theory and their associated nets? What is the meaning of K-density in this context?What about the Lorentz-transformation mentioned above? Is there some counterpart forit in this theory of alternatives, and has it any physical signi�cance?7 Acknowledgements and Recent WorksThis work is a revised version of the author's \Studienarbeit" (Stehr 1993) �nished inDecember 1993 at the Computer Science Department of the University of Hamburg. Thesupervisor was Prof. Dr. Jozef Gruska (Slovak Academy of Science) a major focus ofhis interests being the relation between physics and computer science. He was open fornumerous fruitful discussions and contributed to this work with competent advice. Somemodi�cations introduced in this revised version are due to Olaf Kummer, who wrotea \Diplomarbeit" (Kummer 1996) on this subject, where many of the ideas have beenfurther elaborated. In particular a proof of ASS1 has been given there. All of the otherassumptions ASS2, ASS3 and ASS4 have been found to be independent of the axioms.It has been furthermore shown that ASS3 and ASS4 can be deduced from appearentlymore elementary assumptions about concurrency and causality. Also the proofs based onASS2 has been optimized. The di�culties with cyclic concurrency structures initiated the69

axiomization and investigation of (partial) cyclic orders in Stehr 1996, which are of similarmathematical generality as partial orders. Concurrency theory on the basis of cyclic ordersis certainly possible, but has not been studied so far.

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ReferencesBest und Fern�andez 1988: Eike Best and Cesar Fern�andez. \Nonsequential Pro-cesses. A Petri Net View", volume 13 of \ETACS Series: Monographs in Computer Sci-ence". Springer-Verlag, Berlin (1988).Best und Merceron 1985: E. Best and A. Merceron. Concurrency Axioms and D-Continuous Posets. In G. Rozenberg, editor, \Advances in Petri Nets 1984", volume188 of \Lecture Notes in Computer Science", pages 32{47. Springer-Verlag (1985).C.Fern�andez und A.Merceron 1987: C.Fern�andez and A.Merceron. Some Remarkson D-Continuity. In G.Rozenberg H.Voss, H.J.Genrich, editor, \Concurrency andNets", pages 171{186. Springer-Verlag (1987).Fenske 1992: Uwe Fenske. Concurrency Theorie. Unterlagen der Arbeitsgruppe "All-gemeine Netztheorie" and der Universit�at Hamburg (February 1992).Fern�andez und Thiagarajan 1983: C. Fern�andez and P.S. Thiagarajan. A Note onD-Continuous Causal Nets. In G.Rozenberg A.Pagnoni, editor, \Applications andTheory of Petri Nets", volume 66 of \Lecture Notes in Computer Science", pages 86{97.Springer-Verlag (1983).Fern�andez und Thiagarajan 1985: C. Fern�andez and P.S. Thiagarajan. A LatticeTheoretic View of K-Density. In G. Rozenberg, editor, \Advances in Petri Nets 1984",volume 188 of \Lecture Notes in Computer Science", pages 139{153. Springer-Verlag(1985).Fern�andez 1975: Cesar Fern�andez. Net Topology I and II. GMD-Reports ISF-75-9,ISF-76-2, Gesellschaft f�ur Mathematik und Datenverarbeitung mbH, Bonn (1975).Feynman 1982: Richard P. Feynman. Simulating Physics with Computers. Interna-tional Journal of Theoretical Physics 21, 467{488 (1982).Fredkin 1990: Edward Fredkin. Digital Mechanics: An Informational Process Basedon Reversible Universal Cellular Automata. In Howard Gutowitz, editor, \CellularAutomata: Theory and Experiment". Elsevier Science Publishers B.V. (1990).Kummer 1996: Olaf Kummer. Axiomensysteme f�ur die Theorie der Nebenl�au�gkeit.Diplomarbeit, Universit�at Hamburg, Fachbereich Informatik (1996). Available viahttp://www.informatik.uni-hamburg.de/TGI.M�uller 1993: Hartmut M�uller. Geschichte und Entwicklung der Concurrency Theorie.Diplomarbeit, Fachbereich Informatik der Universit�at Hamburg (1993).Petri und Smith 1987: Carl Adam Petri and Einar Smith. Concurrency and Conti-nuity. In G. Rozenberg, editor, \Advances in Petri Nets 1987", volume 266 of \LectureNotes in Computer Science", pages 273{292, Berlin (1987). Springer-Verlag.Petri 1980a: C.A. Petri. Concurrency. In W. Brauer, editor, \Net Theory and Appli-cations", volume 84 of \Lecture Notes in Computer Science", pages 251{276. Springer-Verlag (1980). 71

Petri 1980b: Carl Adam Petri. Introduction to General Net Theory. In W. Brauer,editor, \Net Theory and Applications", number 84 in Lecture Notes in Computer Science,pages 1{19, Berlin (1980). Springer-Verlag.Petri 1982: Carl Adam Petri. State-Transition Structures in Physics and Computa-tion. International Journal of Theoretical Physics 21(12), 979{992 (1982).Petri 1987: Carl Adam Petri. Concurrency Theory. In W. Brauer and G. Rozen-berg, editors, \Petri Nets: Central Models and Their Properties. Advances in Petri Nets1986, Part I", volume 254 of \Lecture Notes in Computer Science", pages 4{24, Berlin(1987). Springer-Verlag.Petri 1988a: Carl Adam Petri. Concurrency Theorie. Vorlesungsunterlagen, Univer-sit�at Hamburg, Fachbereich Informatik (1988).Petri 1988b: Carl Adam Petri. Material zur Vorlesung Allgemeine Netztheorie. Vor-lesungsunterlagen, Universit�at Hamburg, Fachbereich Informatik (1988).Petri 1988c: Carl Adam Petri. Zug�ange zu einer formalen Pragmatik. Vorlesungsun-terlagen, Universit�at Hamburg, Fachbereich Informatik (1988).Petri 1989: Carl Adam Petri. Vollst�andige Signalordnung (Die Mailand-Papiere). Vor-lesungsunterlagen, Universit�at Hamburg, Fachbereich Informatik (1989).Pl�unnecke 1985: H. Pl�unnecke. K-Density, N-Density, and Finiteness Properties. InG. Rozenberg, editor, \Advances in Petri Nets 1984", volume 188 of \Lecture Notesin Computer Science", pages 392{412. Springer-Verlag (1985).Smith 1989: Einar Smith. Zur Bedeutung der Concurrency-Theorie f�ur den Aufbauhochverteilter Systeme. Dissertation, Fachbereich Informatik der Universit�at Hamburg(1989).Stehr 1993: Mark-Oliver Stehr. Physically Motivated Axiomatic Concurrency The-ory { A Posetless Approach. Studienarbeit, Universit�at Hamburg, Fachbereich Informatik(1993).Stehr 1996: Mark-Oliver Stehr. Zyklische Ordnungen { Axiome und einfache Eigen-schaften. Diplomarbeit, Universit�at Hamburg, Fachbereich Informatik (1996). Availablevia http://www.informatik.uni-hamburg.de/TGI.To�oli und Margolus 1990: Tommaso Toffoli and Norman H. Margolus. Invert-ible Cellular Automata: A Review. Technical Report, MIT Laboratory for ComputerScience (1990).To�oli 1988: Tommaso Toffoli, editor. \Cellular Automata Machines as Physics Em-ulators" (1988).To�oli 1990: Tommaso Toffoli. How Cheap Can Mechanics' First Principles Be ? InWojciech H. Zurek, editor, \Complexity Theory and the Physics of Information",Santa Fe Institute Studies in the Sciences of Complexity, pages 301{318. Addison-Wesley(1990). 72

Wheeler 1990: John Archibald Wheeler. Information, Physics, Quantum: The Searchfor Links. InWojciech H. Zurek, editor, \Complexity Theory and the Physics of Infor-mation", Santa Fe Institute Studies in the Sciences of Complexity, pages 4{28. Addison-Wesley (1990).

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A Notation and Basic De�nitionsWe will use �rst order predicate logic with the usual symbols :,^,_,),, and 9, 8. Forbetter readability we will often use natural language to formulate logical statements. Inparticular relations and functions are frequently notated by special symbols or in naturallanguage to remember the meaning of their arguments. In any case the notation shouldbe precise enough to avoid loss of formal accuracy.SetsWe will assume the usal axioms of elementary set theory based on the binary element-predicate 2 only which postulate the existance of certain sets. If H(x) is a predicate withx as free variable then sets will be constructed with the notation fx : H(x)g applying thecomprehension axiom. Furthermore we will use the following conventional abbreviations:De�nition D24a) A = B :, 8a : (a 2 A, a 2 B);b) A [B := fx : x 2 A _ x 2 Bg;c) A \B := fx : x 2 A ^ x 2 Bg;d) A := fa : a 62 Ag;e) B � A := B \A;f) A � B :, A [ B = B;g) A � B :, A � B ^A 6= B;h) P(A) := fB : B � Ag;i) AX := X �A;j) SA := fx : 9a 2 A : x 2 ag;k) TA := fx : 8a 2 A : x 2 ag. 2The usal order and operations on natural numbers and integers are assumed.De�nition D25a) N := f0; 1; 2; : : :g;b) N[a; b] := fx : a � x ^ x � bg.c) Z := f: : : ; 2;�1; 0; 1; 2; : : :g; 2Tuples and RelationsTuples are de�ned in the conventional way. They are denoted by (a0; a1; : : : ; an). A relationis a subset of a cartesian product of sets:De�nition D26 A0 �A1 � A2 � : : :� An :=f(a0; a1; a2; : : : ; an) : a0 2 A0 ^ a1 2 A1 ^ a2 2 A2 ^ : : :^ an 2 Ang. 274

In particular we are interested in binary relations S;R � X�X . Hence we use the followingabbreviations:De�nition D27a) a R b :, (a; b) 2 R;b) IdA := f(a; a) : a 2 Ag;c) R�1 := f(b; a) : aRbg;d) RX := X �X � Re) R[A] := fb : 9a 2 A : aRbg;f) Dom(R) := fa : 9b : aRbg (domain of R);g) Ran(R) := fb : 9a : aRbg (range of R);h) Base(R) := Dom(R)[ Ran(R) (base of R);i) RX := R [ idX ;j) bRX := R [R�1 [ IdX ;k) R�S := f(a; c) : 9b : (aRb^ bSc)g;l) R � S := S �R;m) R1 := R; Rn+1 := Rn �R;n) R+ := Sn2NRn+1;o) R�X := R+ [ IdX ;p) l R := R [R�1;q) RjX := R \ (X �X);r) a ~RXb :, RX [a] = RX [b]. 2De�nition D28a) R is re exive on X :, IdX � R;b) R is irre exive on X :, IdX \ R = ;;c) R is symmetric :, R = R�1;d) R is asymmetric :, R\ R�1 = ;;e) R is antisymmetric on X :, R \ R�1 � IdX ;f) R is transitive :, R �R � R;g) R is a similarity on X :, RjX is re exive on X and symmetric;h) R is an equivalence on X :, RjX is re exive on X , symmetric and transitive. 2De�nition D29a) R is cyclic on X :, R+ = X �X ;b) R is acyclic on X :, R+ \ idX = ;. 2De�nition D30a) (X;R) is a poset :, R is transitive, re exive on X and antisymmetric;b) (X;R) is a strict poset :, R is transitive, irre exive on X and asymmetric. 2De�nition D31 Let (X;R) be a poset. 75

a) (X;R) is total :, X �X � R [ R�1.b) Y is a chain of (X;R) :, Y � X and (Y;R) is total.c) Chains(X;R) := f(Y;R) : Y is a chain of (X;R) g. 2De�nition D32 Let (X;R) be a poset and Y � X .a) Max(R; Y ) := fy : y 2 Y ^ :9z 2 Y : (y R z ^ z 6= y);b) Min(R; Y ) := fy : y 2 Y ^ :9z 2 Y : (z R y ^ z 6= y);c) Max(R) := Max(R;Base(R));d) Min(R) := Min(R;Base(R)); 2De�nition D33 Let (X;R) be a poset and Y � X .a) UpperBounds(R; Y ) := fx : 8y 2 Y : y R xg;b) LowerBounds(R; Y ) := fx : 8y 2 Y : x R yg;c) Sup(R; Y ) := Min(R;UpperBounds(R; Y ));d) Inf(R; Y ) := Max(R;LowerBounds(R; Y )). 2Axiom A16 [Zorn's Lemma] Let (X;R) be a poset.(8Y 2 Chains(X;R) : Sup(R; Y ) 6= ;)) (Max(R) 6= ;). 2De�nition D34 Let R � X �X be a similarity and S � X .a) S is a clique of R :, S � S � R;b) Cliques(R) := fS : S is a clique of R g;c) S is a ken of R :, S 2Max(�; Cliques(R));d) Kens(R) := fS : S is a ken of R g. 2De�nition D35 Let R be an equivalence on X .a) [x]R := fx0 : x0Rxg (the equivalence class of x with respect to R);b) X=R := f[x]R : x 2 Xg (X modulo R). 2De�nition D36 Let R � X �X be an equivalence on X and S � X �X .S=R := f([a]R; [b]R) : a S bg (S modulo R). 2FunctionsDe�nition D37 F : A! B :,F � A� B and 8(a; b) 2 F : 8(c; d) 2 F : (a = c) b = d)(F is a (partial) function from A to B). 2De�nition D38 F is a function :, 9A;B : F : A! B. 2De�nition D39 Let F be a function. 76

a) y = F (x) :, (x; y) 2 F ;b) F�1(y) := fx : y = f(x)g. 2De�nition D40 Let F be a function.a) F is total on A :, Dom(F ) = A;b) F is surjective on B :, Ran(F ) = B;c) F is injective :, function(F ) ^ function(F�1);d) F is bijective on B :, F is surjective on B and injective. 2SequencesDe�nition D41 Let F be a function.a) F is a �nite sequence :, Dom(F ) � N[0; n] for some n;b) F is a !-sequence :, Dom(F ) = N;c) F is a !!-sequence :, Dom(F ) = Z;d) F is a sequence :, F is a �nite, !-, or !!-sequence. 2We will denote a sequence F by (xi)i2I where I = Dom(F ) and xi = F (i). A �nitesequence F may also be notated as (x0; x1; : : : ; xn), a !-sequence as (x0; x1; : : :) and a!!-sequence as (: : : ; x�1; x0; x1; : : :).De�nition D42 Let F = (xi)i2I be a sequence.a) Index(F ) := Dom(F ) the index set of F ;b) Set(F ) := Ran(F ) the range of F ;c) x 2 F :, x 2 Set(F ) x is contained in F .d) j F j = j I j the size of F ;e) First(F ) = F (z) if fzg = Min(�; I)) the �rst element of F ;f) Last(F ) = F (z) if fzg = Max(�; I)) the last element of F ; 2De�nition D43 Let F be a sequence.a) F is a R-chain :, 8i; j 2 Index(F ) : i+ 1 = j ) F (i) R F (j);b) F is a R-cycle :, F is a R-chain and Last(F ) = First(F ). 2De�nition D44 Let F be an R-chain.a) F is cyclic :, Last(F ) = First(F );b) F is acyclic :, 8i; j 2 Index(F ) : i 6= j ) F (i) 6= F (j). 2De�nition D45 Let R; S � X �X and F be a R-chain.SjF = f(Fi; Fj) : j = i+ 1 ^ Fi S Fjg. 277

NetsDe�nition D46 (S; T; F ) is a net :,S \ T = ; ^ F � (S � T )[ (T � S). 2De�nition D47 Let N = (S; T; F ) be a net.a) SN := S (the set of S-elements or places);b) TN := N (the set of T -elements or transitions);c) FN := F (the ow relation);d) XN := S [ T (the set of net elements). 2De�nition D48 Let N = (S; T; F ) be a net and Y � XN .a) Y �F := F [Y ] (the postset of Y with respect to F );b) �YF := F�1[Y ] (the preset of Y with respect to F ). 2De�nition D49 Let N = (S; T; F ) be a net and X = XN .a) N is primitive :, 8x 2 X : �xF [ x�F 6= ;;b) N is pure :, 8x 2 X : (�xF \ x�F = ;);c) N is simple :, 8x; y 2 X : (�xF = �yF ^ x�F = y�F ) x = y);d) N is connected :, (F [ F�1)+ = X �X ;e) N is cyclic :, F+ = X �X ;f) N is acyclic :, F+ \ idX = ;; 2De�nition D50 Let N = (S; T; F ) and N 0 = (S 0; T 0; F 0) be nets.a) N 0 is a subnet of N :,S 0 � S ^ T 0 � T ^F 0 = F \ ((S 0 � T 0) [ (T 0 � S 0)); 2De�nition D51 Let N = (S; T; F ) be a net.a) t and t0 are independent in N :,(�tF [ t�F ) \ (�t0F [ t0�F ). 2De�nition D52 Let N = (S; T; F ) be a net.a) M is a marking in N :, M � S;b) E is an event in N :, E � T and E 6= ;;c) (M [E > M 0)N :,M;M 0 � S and E � T and8t; t0 2 E : (t 6= t0 ) t�F \ t0�F = ; ^ �tF \ �t0F = ;) ^(S t�F : t 2 E = M 0 �M ^S �tF : t 2 E = M �M 0)(M 0 is reachable from M by the step E in N);78

d) E is a step in N :, 9M;M 0 :M [E > M 0;e) Steps(N) := fE : E is a step in N g;f) M rN M 0 :, 9E :M [E > M 0 (M 0 is reachable from M in one step in N);g) M RNM 0 :,M (rN [ rN�1)�P(S) M 0 (M 0 is reachable from M in N).h) M srN M 0 :, 9t 2 T : M [ftg > M 0(M 0 is sequentially reachable from M in one step in N);i) M SRNM 0 :,M (srN [ srN�1)�P(S) M 0(M 0 is sequentially reachable from M in N). 2Fundamental SituationsDe�nition D53 Let N = (S; T; F ) be a net and t; t0 2 T and M � S.a) t is enabled at M :, �tF �M ^ t�F \M = ;;b) t is reverse enabled at M :, t�F �M ^ �tM \M = ;;c) t has a contact at M :, �tF �M ^ t�F \M 6= ;;d) t has a reverse contact at M :, t�F �M ^ �tF \M 6= ;;e) t and t0 are in con ict at M :,t and t0 are enabled at M and (t�F [ �tF ) \ (t0�F [ �t0F ) 6= ;;f) t and t0 are in reverse con ict at M :,t and t' are reverse enabled at M and (t�F [ �tF ) \ (t0�F [ �t0F ) 6= ;;g) t has a transjunction at M :, �tF \M 6= ; ^ t�F \M 6= ;. 2Elementary Net SystemsDe�nition D54 (S; T; F; C) is an elementary net systems (ENS) :,(S; T; F ) is a net and C � S. 2De�nition D55Let NS = (S; T; F; C) be an elementary net system and N = (S; T; F ).a) CaseClass(NS) := [C]RN (the case class of NS);b) SeqCaseClass(NS) := [C]SRN (the sequential case class of NS). 2De�nition D56 Let NS = (S; T; F; C) be an elementary net system.a) ProperS(NS) := (SCaseClass(NS))� (TCaseClass(NS))(the proper S-elements of NS);b) ProperT (NS) := Sfe : 9m;m0 2 CaseClass(NS) :m[e > m0g(the proper T-elements of NS);c) NS is proper :, S = ProperS(NS)^ T = ProperT (NS).79

2De�nition D57 Let NS = (S; T; F; C0) be an elementary net system.a) NS is safe :, 8M 2 CaseClass(NS) : :9t 2 T :(t has a contact or a reverse contact at M);b) NS is secure :, NS is safe and 8M 2 CaseClass(NS) : :9t 2 T :(t has a transjunction at M). 2

80

B Models of Concurrency TheoryThe concurrency structure of �g. 1.a:X = f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12gco = l f(9; 1); (9; 2); (9; 3); (10; 3); (10; 4); (10; 5); (10; 9); (11; 5); (11; 6); (11; 7);(11; 10); (12; 1); (12; 7); (12; 8); (12; 9); (12; 11)gli = l f(2; 1); (3; 1); (3; 2); (4; 1); (4; 2); (4; 3); (5; 1); (5; 2); (5; 3); (5; 4); (6; 1); (6; 2);(6; 3); (6; 4); (6; 5); (7; 1); (7; 2); (7; 3); (7; 4); (7; 5); (7; 6); (8; 1); (8; 2); (8; 3);(8; 4); (8; 5); (8; 6); (8; 7); (9; 4); (9; 5); (9; 6); (9; 7); (9; 8); (10; 1); (10; 2); (10; 6);(10; 7); (10; 8); (11; 1); (11; 2); (11; 3); (11; 4); (11; 8); (11; 9); (12; 2); (12; 3);(12; 4); (12; 5); (12; 6); (12; 10)g~coX = f(1; 1); (2; 2); (3; 3); (4; 4); (5; 5); (6; 6); (7; 7); (8; 8); (9; 9); (10; 10); (11; 11); (12; 12)g~liX = f(1; 1); (2; 2); (3; 3); (4; 4); (5; 5); (6; 6); (7; 7); (8; 8); (9; 9); (10; 10); (11; 11); (12; 12)gCuts(CS) = ff1; 9; 12g; f2; 9g; f3; 9; 10g; f4; 10g; f5; 10; 11g; f6; 11g; f7; 11; 12g; f8; 12ggLines(CS) = ff1; 2; 3; 4; 5; 6; 7; 8g; f1; 2; 3; 4; 8; 11g; f1; 2; 6; 7; 8; 10g; f2; 3; 4; 5; 6; 12g;f2; 6; 10; 12g; f4; 5; 6; 7; 8; 9g; f4; 8; 9; 11ggPCS = f(1; 2); (1; 8); (3; 2); (3; 4); (5; 4); (5; 6); (7; 6); (7; 8); (9; 4); (9; 8); (10; 2); (10; 6);(11; 4); (11; 8); (12; 2); (12; 6)gDCS = f(1; 9); (1; 12); (2; 9); (3; 9); (3; 10); (4; 10); (5; 10); (5; 11); (6; 11); (7; 11); (7; 12);(8; 12)gDom(P ) = f1; 3; 5; 7; 9; 10; 11; 12gRan(P ) = f2; 4; 6; 8gDom(D) = f1; 2; 3; 4; 5; 6; 7; 8gRan(D) = f9; 10; 11; 12gThe concurrency structure of �g. 2.a:X = f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24gco = l f(2; 1); (3; 1); (3; 2); (4; 1); (4; 2); (4; 3); (5; 2); (5; 4); (6; 1); (6; 3); (6; 5);(9; 8); (10; 1); (10; 3); (10; 5); (10; 7); (10; 8); (10; 9); (11; 8); (11; 10);(12; 7); (12; 9); (12; 11); (13; 8); (13; 10); (13; 12); (14; 8); (14; 10); (14; 12);(14; 13); (15; 7); (15; 9); (15; 11); (15; 13); (15; 14); (16; 7); (16; 9); (16; 11);(16; 13); (16; 14); (16; 15); (17; 14); (17; 16); (18; 13); (18; 15); (18; 17); (19; 3);(19; 4); (19; 14); (19; 16); (19; 18); (20; 1); (20; 2); (20; 14); (20; 16); (20; 18);(20; 19); (21; 3); (21; 4); (21; 13); (21; 15); (21; 17); (21; 19); (21; 20); (22; 1);(22; 2); (22; 13); (22; 15); (22; 17); (22; 19); (22; 20); (22; 21); (23; 3); (23; 4);(23; 20); (23; 22); (24; 1); (24; 2); (24; 19); (24; 21); (24; 23)gli = l f(5; 1); (5; 3); (6; 2); (6; 4); (7; 1); (7; 3); (7; 5); (8; 1); (8; 3); (8; 5); (9; 2); (9; 4);(9; 6); (10; 2); (10; 4); (10; 6); (11; 1); (11; 2); (11; 3); (11; 4); (11; 5); (11; 6);81

(11; 7); (11; 9); (12; 1); (12; 2); (12; 3); (12; 4); (12; 5); (12; 6); (12; 8); (12; 10);(13; 1); (13; 2); (13; 3); (13; 4); (13; 5); (13; 6); (13; 7); (13; 9); (13; 11); (14; 1);(14; 2); (14; 3); (14; 4); (14; 5); (14; 6); (14; 7); (14; 9); (14; 11); (15; 1); (15; 2);(15; 3); (15; 4); (15; 5); (15; 6); (15; 8); (15; 10); (15; 12); (16; 1); (16; 2); (16; 3);(16; 4); (16; 5); (16; 6); (16; 8); (16; 10); (16; 12); (17; 1); (17; 2); (17; 3); (17; 4);(17; 5); (17; 6); (17; 7); (17; 8); (17; 9); (17; 10); (17; 11); (17; 12); (17; 13); (17; 15);(18; 1); (18; 2); (18; 3); (18; 4); (18; 5); (18; 6); (18; 7); (18; 8); (18; 9); (18; 10);(18; 11); (18; 12); (18; 14); (18; 16); (19; 1); (19; 2); (19; 5); (19; 6); (19; 7); (19; 8);(19; 9); (19; 10); (19; 11); (19; 12); (19; 13); (19; 15); (19; 17); (20; 3); (20; 4); (20; 5);(20; 6); (20; 7); (20; 8); (20; 9); (20; 10); (20; 11); (20; 12); (20; 13); (20; 15); (20; 17);(21; 1); (21; 2); (21; 5); (21; 6); (21; 7); (21; 8); (21; 9); (21; 10); (21; 11); (21; 12);(21; 14); (21; 16); (21; 18); (22; 3); (22; 4); (22; 5); (22; 6); (22; 7); (22; 8); (22; 9);(22; 10); (22; 11); (22; 12); (22; 14); (22; 16); (22; 18); (23; 1); (23; 2); (23; 5); (23; 6);(23; 7); (23; 8); (23; 9); (23; 10); (23; 11); (23; 12); (23; 13); (23; 14); (23; 15); (23; 16);(23; 17); (23; 18); (23; 19); (23; 21); (24; 3); (24; 4); (24; 5); (24; 6); (24; 7); (24; 8);(24; 9); (24; 10); (24; 11); (24; 12); (24; 13); (24; 14); (24; 15); (24; 16); (24; 17);(24; 18); (24; 20); (24; 22)g~coX = f(1; 1); (2; 2); (3; 3); (4; 4); (5; 5); (6; 6); (7; 7); (8; 8); (9; 9); (10; 10);(11; 11); (12; 12); (13; 13); (14; 14); (15; 15); (16; 16); (17; 17); (18; 18); (19; 19);(20; 20); (21; 21); (22; 22); (23; 23); (24; 24)g~liX = f(1; 1); (2; 2); (3; 3); (4; 4); (5; 5); (6; 6); (7; 7); (8; 8); (9; 9); (10; 10); (11; 11);(12; 12); (13; 13); (14; 14); (15; 15); (16; 16); (17; 17); (18; 18); (19; 19); (20; 20);(21; 21); (22; 22); (23; 23); (24; 24)gCuts(CS) = ff1; 2; 3; 4g; f1; 2; 20; 22g; f1; 2; 24g; f1; 3; 6g; f1; 3; 9; 10g; f2; 4; 5g;f2; 4; 7; 8g; f3; 4; 19; 21g; f3; 4; 23g; f5; 6g; f5; 9; 10g; f6; 7; 8g; f7; 8; 9; 10g;f7; 9; 12g; f7; 9; 15; 16g; f8; 10; 11g; f8; 10; 13; 14g; f11; 12g; f11; 15; 16g;f12; 13; 14g; f13; 14; 15; 16g; f13; 15; 18g; f13; 15; 21; 22g; f14; 16; 17g;f14; 16; 19; 20g; f17; 18g; f17; 21; 22g; f18; 19; 20g; f19; 20; 21; 22g;f19; 21; 24g; f20; 22; 23g; f23; 24ggLines(CS) = ff1; 5; 7; 11; 13; 17; 19; 23g; f1; 5; 7; 11; 14; 18; 21; 23g;f1; 5; 8; 12; 15; 17; 19; 23g; f1; 5; 8; 12; 16; 18; 21; 23g; f2; 6; 9; 11; 13; 17; 19; 23g;f2; 6; 9; 11; 14; 18; 21; 23g; f2; 6; 10; 12; 15; 17; 19; 23g; f2; 6; 10; 12; 16; 18; 21; 23g;f3; 5; 7; 11; 13; 17; 20; 24g; f3; 5; 7; 11; 14; 18; 22; 24g; f3; 5; 8; 12; 15; 17; 20; 24g;f3; 5; 8; 12; 16; 18; 22; 24g; f4; 6; 9; 11; 13; 17; 20; 24g; f4; 6; 9; 11; 14; 18; 22; 24g;f4; 6; 10; 12; 15; 17; 20; 24g; f4; 6; 10; 12; 16; 18; 22; 24ggPCS = f(1; 5); (1; 23); (2; 6); (2; 23); (3; 5); (3; 24); (4; 6); (4; 24); (7; 5); (7; 11);(8; 5); (8; 12); (9; 6); (9; 11); (10; 6); (10; 12); (13; 11); (13; 17); (14; 11); (14; 18);(15; 12); (15; 17); (16; 12); (16; 18); (19; 17); (19; 23); (20; 17); (20; 24); (21; 18);82

(21; 23); (22; 18); (22; 24)gDCS = ;Dom(P ) = f1; 2; 3; 4; 7; 8; 9; 10; 13; 14; 15; 16; 19; 20; 21; 22gRan(P ) = f5; 6; 11; 12; 17; 18; 23; 24gThe concurrency structure of �g. 6:X = f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16gco = l f(11; 1); (11; 2); (11; 3); (12; 3); (12; 4); (12; 5); (12; 11); (13; 5); (13; 6);(13; 7); (13; 12); (14; 7); (14; 8); (14; 9); (14; 13); (15; 1); (15; 9); (15; 10);(15; 11); (15; 14); (16; 1); (16; 7); (16; 8); (16; 9); (16; 10); (16; 11); (16; 13);(16; 14); (16; 15)gli = l f(2; 1); (3; 1); (3; 2); (4; 1); (4; 2); (4; 3); (5; 1); (5; 2); (5; 3); (5; 4); (6; 1); (6; 2);(6; 3); (6; 4); (6; 5); (7; 1); (7; 2); (7; 3); (7; 4); (7; 5); (7; 6); (8; 1); (8; 2); (8; 3);(8; 4); (8; 5); (8; 6); (8; 7); (9; 1); (9; 2); (9; 3); (9; 4); (9; 5); (9; 6); (9; 7); (9; 8);(10; 1); (10; 2); (10; 3); (10; 4); (10; 5); (10; 6); (10; 7); (10; 8); (10; 9); (11; 4);(11; 5); (11; 6); (11; 7); (11; 8); (11; 9); (11; 10); (12; 1); (12; 2); (12; 6); (12; 7);(12; 8); (12; 9); (12; 10); (13; 1); (13; 2); (13; 3); (13; 4); (13; 8); (13; 9); (13; 10);(13; 11); (14; 1); (14; 2); (14; 3); (14; 4); (14; 5); (14; 6); (14; 10); (14; 11); (14; 12);(15; 2); (15; 3); (15; 4); (15; 5); (15; 6); (15; 7); (15; 8); (15; 12); (15; 13); (16; 2);(16; 3); (16; 4); (16; 5); (16; 6); (16; 12)gCuts(CS) = ff1; 11; 15; 16g; f2; 11g; f3; 11; 12g; f4; 12g; f5; 12; 13g; f6; 13g;f7; 13; 14; 16g; f8; 14; 16g; f9; 14; 15; 16g; f10; 15; 16ggLines(CS) = ff1; 2; 3; 4; 5; 6; 7; 8; 9; 10g; f1; 2; 3; 4; 5; 6; 10; 14g; f1; 2; 3; 4; 8; 9; 10; 13g;f1; 2; 6; 7; 8; 9; 10; 12g; f1; 2; 6; 10; 12; 14g; f2; 3; 4; 5; 6; 7; 8; 15g;f2; 3; 4; 5; 6; 16g; f2; 3; 4; 8; 13; 15g; f2; 6; 7; 8; 12; 15g; f2; 6; 12; 16g;f4; 5; 6; 7; 8; 9; 10; 11g; f4; 5; 6; 10; 11; 14g; f4; 8; 9; 10; 11; 13ggPCS = f(1; 2); (1; 10); (3; 2); (3; 4); (5; 4); (5; 6); (7; 6); (7; 8); (9; 8); (9; 10); (11; 4);(11; 10); (12; 2); (12; 6); (13; 4); (13; 8); (14; 6); (14; 10); (15; 2); (15; 8); (16; 2); (16; 6)gDCS = f(1; 11); (1; 15); (1; 16); (2; 11); (3; 11); (3; 12); (4; 12); (5; 12); (5; 13); (6; 13);(7; 13); (7; 14); (7; 16); (8; 14); (8; 16); (9; 14); (9; 15); (9; 16); (10; 15); (10; 16);(14; 16); (15; 16)gDom(P ) = f1; 3; 5; 7; 9; 11; 12; 13; 14; 15; 16gRan(P ) = f2; 4; 6; 8; 10gDom(D) = f1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 14; 15gRan(D) = f11; 12; 13; 14; 15; 16g 83