22.4.2008 Formal Methods of Systems Specification Logical Specification of Hard- and Software Prof. Dr. Holger Schlingloff Institut für Informatik der

22.4.2008

Formal Methods of Systems SpecificationLogical Specification of Hard- and Software

Prof. Dr. Holger SchlingloffInstitut für Informatik der Humboldt Universität

and

Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

http://www.hu-berlin.de/index.html

22.4.2008 Slide 2H. Schlingloff, Logical Specification

A first example

A new video camcorder (“DCR-PC330”) owner's manual almost incomprehensible can be found in the internet typical for such devices

off

memorytape play

dn

dn dn

dn

up up up


• Such models can help in the development of complex systems ("model-driven design")

• The more concrete the formalism, the closer it is to an implementation executable code may be generated from state

diagrams We might add additional information such as

timing, communication, variables and such.

• Specification as opposed to modeling describes properties of the targeted system not aiming at a complete description of the system not aiming at the generation of executable code


Screen menu

• The power-switch by itself is not a "complex system“ (Even I didn't need long to understand it).

• Let's look at the screen menu.


Screen menu (contd.)

greyed out

invisible


• There are menus, items and settings menus: Camera Set,... items: Volume, LCD

Brightness, ... settings: on/off, 0-100%, ...

• Items may be nested in two levels

• Setting screen allows to choose the value of a particular variable only the relevant variables

may be accessed


Modelling as a tree

Menu-off

MemorySet Pict.Appli. StandardSet CameraSet...

Volume LCD/VFSet RemoteCtrl

LCDBright LCD Color

... ...

...

Menu


Modelling as a tree

Menu-off

MemorySet Pict.Appli. StandardSet CameraSet...

Volume LCD/VFSet RemoteCtrl

LCDBright LCD Color

... ...

...

Menu


• Menus are mode-dependent As a consequence, the up- and

down-relations in the graph aremode-dependent

Since the first line is not uniform,also the menu-relation is mode-dependent

• Formalization shows weaknessin the design (usability) what is hard to formalize is hard to

understand and likely to contain orcause errors

• How to describe such a structure? homework

(consider cases that an item disappears and that it is greyed out)

Camera /Tape

Camera /Memory

Play /Edit

Camera Set + + -Memory Set - + +Pict.Appl. + + +Edit/Play + - +Standard Set + + +Time/Langu + + +


Propositional Logic

• A formal specification method consists of three parts syntax, i.e., what are well-formed specifications semantics, i.e., what is the meaning of a specification calculus, i.e., what are transformations or deductions

of a specification

• Propositional logic: probably the first and most widely used specification method dates back to Aristotle, Chrysippus, Boole, Frege, … base of most modern logics fundamental for computer science


Syntax of Propositional Logic

•Let Ρ be a finite set {p1,…,pn} of propositionsand assume that , and (, ) are not in

•SyntaxPL ::= Ρ | | (PL PL)

every p is a wff is a wff („falsum“) if and are wffs, then () is a wff nothing else is a wff


Remarks

• Ρ may be empty still a meaningful logic!

• Minimalistic approach infix-operator necessitates parentheses other connectives can be defined as usual

¬ ≙ ( ) (linear blowup!)Τ ≙ ¬() ≙ (¬)() ≙ ¬(¬¬) ≙ ¬(¬)() ≙ (()()) (exponential blowup!)

operator precedence as usual literal = a proposition or a negated proposition


Semantics of Propositional Logic

• Propositional Model Truth value universe U: {true, false} Interpretation I: assignment Ρ ↦ U Model M: (U,I)

• Validation relation ⊨ between model M and formula M ⊨ p if I(p)=true M ⊭ M ⊨ () if M ⊨ implies M ⊨

• M validates or satisfies iff M ⊨ is valid (⊨) iff every model M validates is satisfiable (SAT()) iff some model M satisfies


Propositional Calculus

• Various calculi have been proposed boolean satisfiability (SAT) algorithms tableau systems, natural deduction, enumeration of valid formulæ

• Hilbert-style axiom system⊢ (()) (weakening)

⊢ ((()) (()())) (distribution)

⊢ (¬¬) (excluded middle)

, () ⊢ (modus ponens)

• Derivability All substitution instances of axioms are derivable If all antecedents of a rule are derivable, so is the

consequent


An Example Derivation

Show ⊢ (pp)

(1)⊢(p((pp)p))((p(pp))(pp)) (dis)

(2)⊢(p((pp)p)) (wea)

(3)⊢((p(pp))(pp)) (1,2,mp)

(4)⊢(p(pp)) (wea)

(5)⊢(pp) (3,4,mp)


Correctness and Completeness

•Correctness: ⊢ ⊨Only valid formulæ can be derived Induction on the length of the derivation Show that all axiom instances are valid, and

thatthe consequent of (mp) is valid if both antecedents are

•Completeness: ⊨ ⊢All valid formulæ can be derived Show that consistent formulæ are satisfiable

~⊢¬ ~⊨¬


Consistency and Satisfiability

• A finite set Φ of formulæ is consistent, if ~⊢¬ΛΦ• Extension lemma: If Φ is a finite consistent set of formulæ

and is any formula, then Φ{} or Φ{¬} is consistent Assume ⊢¬(Φ) and ⊢¬(Φ¬). Then ⊢(Φ¬) and ⊢(Φ¬¬).

Therefore ⊢¬Φ, acontradiction.

• Let SF() be the set of all subformulæ of • For any consistent , let #

be a maximal consistent extension of (i.e., # and for every SF(), either #or #. (Existence guaranteed by extension lemma)


Canonical models

• For a maximal consistent set #, the canonical model CM(#) is defined by I(p)=true iff p#.

• Truth lemma: For any SF(), I()=true iff #

Case =p: by construction Case =: Φ{} cannot be consistent Case =(12): by induction hypothesis and derivation

• Therefore, if is consistent, then for any maximal consistent set #, CM(#)⊨ any consistent formula is satisfiable any unsatisfiable formula is inconsistent any valid formula is derivable


Example: Combinational Circuits

•Multiplexer

S selects whether I0 or I1 is output to Y

Y = if S then I1 else I0 end

(Y((SI1)(¬SI0)))

Pictures taken from: http://www.scs.ryerson.ca/~aabhari/cps213Chapter4.ppt

I0 I1 S Y

0 0 0 0

1 0 0 1

0 1 0 0


Boolean Specifications

•Evaluator (output is 1 if input matches a certain binary value)

•Encoder (output i is set if binary number i is on input lines)

•Majority function (output is 1 if half or more of the inputs are 1)

•Comparator (output is 1 if input0 > input1)

•Half-Adder, Full-Adder, …


Software Example

•Code generator optimization if (p and q) then if (r) then x else y else if (q

or r) then y else if (p and not r) then x else y

•Loop optimization


Verification of Boolean Functions

• Latch-Up: can a certain line go up? does (¬L0) hold? is (L0) satisfiable?

• Given , ; does () hold? usually reduced to SAT:

is ((¬)(¬)) satisfiable? efficient SAT-solver exist (annual competition) partitioning techniques

• any output depends only on some inputs find which ones generate test patterns (BIST: built-in-self-test)


Optimizing Boolean Functions

•Given ; find such that () holds and is „optimal“ much harder question optimal wrt. speed / size / power /… translation to normal form (e.g., OBDD)

Documents

22.4.2008 Formal Methods of Systems Specification Logical Specification of Hard- and Software Prof. Dr. Holger Schlingloff Institut für Informatik der