12--

DDoorruu AAddrriiaann PPnneessccuu

KKnnoowwlleeddggee RReepprreesseennttaattiioonn

aanndd RReeaassoonniinngg

Course Lectures

EDITURA CONSPRESS

2013

2

Copyright 2013, Editura Conspress

EDITURA CONSPRESS este recunoscut de

Consiliul Naional al Cercetrii tiinifice din nvmntul Superior

Lucrare elaborat n cadrul proiectului: "Reea naional de centre pentru

dezvoltarea programelor de studii cu rute flexibile i a unor instrumente

didactice la specializarea de licen i masterat, din domeniul Ingineria

Sistemelor"

Colecia Carte universitar

CONSPRESS B-dul Lacul Tei nr. 124, sector 2

cod 020396, Bucureti Tel: (021) 242 2719 / 300; Fax: (021) 242 0781

Descrierea CIP a Bibliotecii Naionale a Romniei Pnescu, Doru Adrian

Suport de curs pentru Reprezentarea cunoaterii i raionament

Doru Adrian Pnescu Bucureti, Editura Conspress 2013ISBN

3

CCoonntteennttss

BBiibblliiooggrraapphhyy ....55

CCoouurrssee oovveerrvviieeww ....55

CChhaapptteerr II IInnttrroodduuccttiioonn ttoo KKnnoowwlleeddggee RReepprreesseennttaattiioonn aanndd

RReeaassoonniinngg ((KKRRRR)) ....66

BBaassiicc nnoottiioonnss iinn KKRRRR ....77

KKRRRR:: RReepprreesseennttaattiioonn aanndd RReeaassoonniinngg 1177

KKnnoowwlleeddggee BBaasseedd SSyysstteemmss ((KKBBSSss)) ......1199

KKBBSSss IImmppoorrttaannccee ooff rreeaassoonniinngg 2233

KKBBSSss TThhee rroollee ooff LLooggiicc ..2266

HHoommeewwoorrkk ............2266

CChhaapptteerr IIII IImmppoorrttaanntt aassppeeccttss oonn tthhee LLaanngguuaaggee ooff

FFiirrsstt--OOrrddeerr LLooggiicc ((FFOOLL)) ....2277

IInnttrroodduuccttiioonn ..2288

LLaanngguuaaggee ooff FFiirrsstt--OOrrddeerr LLooggiicc SSyynnttaaxx 3300

LLaanngguuaaggee ooff FFiirrsstt--OOrrddeerr LLooggiicc SSeemmaannttiiccss ..3366

LLaanngguuaaggee ooff FFiirrsstt--OOrrddeerr LLooggiicc PPrraaggmmaattiiccss 4488

HHoommeewwoorrkk ............5588

CChhaapptteerr IIIIII KKnnoowwlleeddggee EEnnggiinneeeerriinngg ..6600

KKnnoowwlleeddggee eennggiinneeeerriinngg aanndd kknnoowwlleeddggee mmaannaaggeemmeenntt ......6611

4

OOnnttoollooggyy ..7711

VVooccaabbuullaarryy ..9911

TTyyppeess ooff ffaaccttss ..9944

RReeiiffiiccaattiioonn ..110022

EEnnttaaiillmmeenntt 110055

HHoommeewwoorrkk 111111

CChhaapptteerr IIVV RReessoolluuttiioonn ..111122

IInnttrroodduuccttiioonn ....111133

RReessoolluuttiioonn iinn PPrrooppoossiittiioonnaall CCaallccuulluuss ......111144

RReessoolluuttiioonn ddeerriivvaattiioonnss ......111177

AA RReessoolluuttiioonn bbaasseedd eennttaaiillmmeenntt pprroocceedduurree ..111199

RReessoolluuttiioonn iinn FFOOLL ......112233

AAnnsswweerr eexxttrraaccttiioonn ......113333

SSkkoolleemmiizzaattiioonn ....113366

TTrreeaattmmeenntt ooff eeqquuaalliittyy ..113399

TThhee ccoommppuuttaattiioonnaall ccoommpplleexxiittyy ooff rreessoolluuttiioonn ......114422


CChhaapptteerr VV PPrroodduuccttiioonn SSyysstteemmss ....114488

IInnttrroodduuccttiioonn DDiirreeccttiioonn ooff rreeaassoonniinngg ..114499

BBaassiicc ooppeerraattiioonn ....115511

WWoorrkkiinngg mmeemmoorryy ....115533

PPrroodduuccttiioonn rruulleess ......115544

CCoonnfflliicctt rreessoolluuttiioonn ..116611

EEffffiicciieennccyy iinn PPrroodduuccttiioonn SSyysstteemmss ..116655


5

BibliographyThis course notes follow to a large extent the following book:

Ron Brachman, Hector Levesque, Knowledge representation and reasoning, Elsevier, Amsterdam, 2004

Other sources that were used are: Frank van Harmelen, Vladimir Lifschitz, Bruce Porter, Handbook

of Knowledge Representation, Elsevier, Amsterdam, 2008 Natasha Noy, Ontology Development 101, Stanford University,

2008 Irma Becerra-Fernandez, Avelino Gonzalez, Rajiv Sabherwal,

Knowledge Management, Prentice Hall, 2004 Amrit Tiwana, The Knowledge Management Toolkit, Prentice

Hall, 2000 Joseph Giarratano, Gary Riley, Expert Systems: Principles and

Programming, PWS-KENT, Boston, 1989

Course overview

Knowledge Representation (KR) Logic based KR

Reasoning Expert Systems (ES)

6

Chapter 1 Introduction to KRR

Basic notions in Knowledge Representation and Reasoning (KRR)

Intelligence Knowledge Representation Reasoning The role of Logic


Objectives for Chapter 1

Understanding the relation between intelligence knowledge representation

Establishing the importance of reasoning Setting up the borders for a knowledge

based system

7

Basic notions in KRR

Intelligence complex and mysterious phenomenon

Intelligence is dependent on knowledge

Thinking (reasoning) bring out what is relevant from what one knows to solve a problem


KRR is a chapter of Artificial Intelligence (AI)

AI - the study of intelligent behavior achieved through computational means

KRR is that part of AI that is concerned with how an agent (software program) uses what it knows in deciding what to do

8


KRR is considered to be the sub-field of AI concerned with the representation of information in computers in ways that allow computers to draw reasonable conclusions from them.


Reasoner+

KnowledgeWorld

Sensoryinformation

Actions/Decisions

9


Approaches of AI

Symbolic approach KRR (Exp. Syst., KBSyst)

Connectionist approach neural networks

Evolutionary approach genetic algorithms

Basic notions in KRR: Knowledge

Knowledge definition:

Information and skills acquired through experience or education; the theoretical or practical understanding of a subject.

What is known in a particular field; facts and information

10


How we refer to knowledge?

Ann knows that

Ann knows that Tom will go to school

John knows that Chemistry is easy to learn.


Knowledge is a relation between a knower, like John, and a proposition, that is, the idea expressed by a simple declarative sentence

Propositions are abstract entities that can be true or false, right or wrong.

John knows that p John knows that it is true that p

11


John knows something John has formed a judgment of some sort, and has come to realize that the world is one way and not another

Propositions are used to classify the two cases


Propositions can be of different kinds

John hopes that Mary will come to the party.

Verbs: know, regret, fear, doubt denote propositional attitudes (relations between agents and propositions)

12

There are cases involving knowledge that do not explicitly mention propositions: John knows who Mary is taking to the party

John knows how to fix the robot

John knows how to make a good first impression

John knows Mary well


A further concept used in AI belief BDI (Belief Desire Intention) software

agents architecture

John knows that p. Johns believes that p.

The second approach is used when one does not want to claim that the judgment about the world is necessarily accurate.


13

There is a full range of propositional attitudes:

John is absolutely certain that p

John is confident that p

John is of the opinion that p

John suspects that p


Representation is a relationship between two domains, where the first is meant to stand for or take the place of the second.

Usually, the first domain, the representor, is more concrete, immediate, simple or accessible in some way.

The first domain models the second one.

Basic notions in KRR: Representation

14

In AI we use symbols to construct representations

A group of symbols (a word) stand for (represent) a certain entity; e.g. John, love, truth, robot

A more meaningful representation is got by a set of words; e.g. John enjoys the lecture


Knowledge representation - the field of AI concerned with using formal symbols to represent a collection of propositions believed by an agent.

A knowledge representation model is not supposed to stand for all the propositions believed by the agent.


15

There may very well be an infinite number of propositions believed, only a finite number of which are ever represented.

It will be the role of reasoning to bridge the gap between what is represented and what is believed.


It is the formal manipulation of those symbols representing a collection of believed propositions to produce representations of new ones.

Computer based reasoning benefits from the fact that symbols are more accessible than the propositions they represent.

Basic notions in KRR: Reasoning

16

Reasoning manipulates the groups of symbols (move them around, take them apart, copy them, string them together) in such a way as to construct representations of new propositions.

An example (two hypotheses and a conclusion): H1: John is friend with Tom H2: Tom attends the Robotics lecture

C: Someone being friend with John attends theRobotics lecture


Such a form of reasoning is named logical inference because the final sentence represents a logical conclusion of the propositions represented by the hypotheses.

Reasoning is a form of calculation over symbols (first noticed by Gottfried Leibniz in the seventeenth century).


17

Knowledge representation is important for AI since it provides means to describe the behavior of complex systems.

Knowledge representation and reasoning can deal with terms like beliefs, desires, goals,intentions, hopes

KRR: Representation and Reasoning

An example: an AI program for chess playing

Variant1: It moved this way because it believed its king was vulnerable, but still wanted to move forward the pawn

Variant2: It moved this way because theevaluation procedure P using a staticevaluation function F returned a value of +10,after a depth search of 4 steps


18

The AI approach: to consider an intentional stance toward the chess-playing system.

The intentional mechanism (used by the BDI software architecture) is not always appropriate.

An example: The thermostat knew the room is cold and wanted to warm it up.


Knowledge representation a stance towards representing a complex system.

A knowledge representation must possess two important properties: we (from the outside) can understand the symbolic

representation as standing for propositions

the system can correctly operate due to the symbolic representation


19

When these properties apply, one can say that the Knowledge Representation Hypothesis takes place.

Knowledge Representation Hypothesis construction of systems using an intentional stance based on a symbolic representation.

Such a system: Knowledge Based Systems (KBS)

The symbolic representation involved: Knowledge Base (KB)


Two forms of a program V1

(defrule R=>

(printout t snow has color white crlfgrass has color green crlf) )

Knowledge Based Systems

20

V2

(deffacts information(entity snow color white)(entity vegetation color green)

(class vegetation tree grass plant))


V2(defrule R1

(entity ?x color ?y) =>(printout t ?x has color ?y crlf))

(defrule R2(entity ?x color ?y) (class ?x $? ?z $?) =>(printout t ?z has color ?y crlf))


21

Both programs print the color of snow and grass.

Only the second program is designed according to the Knowledge Representation Hypothesis.

KBS the presence of a KB: a collection of symbolic structures representing what the system believes and reasons with during its operation.

There may be a KBS lacking the symbolic representation (e.g. neural networks).


There are both advantages and disadvantages for KBSs

A procedural approach can determine a faster solution in comparison with a KBS

There is a paradox of Expert Systems (Hubert Dryfus)

ESs are superior as being KB: they reason over explicitly represented knowledge


22

The paradox: novices are the ones who think and reason; experts recognize and to react

KBSs are superior in dealing with a set of tasks that is open-ended

KBSs can receive new tasks and easily make them depend on previous knowledge

KBSs can debug faulty behavior by locating the erroneous beliefs of the system


KBSs can concisely explain and justify the behavior of the system

KBS - by design, it has the ability to be told facts about its world and adjust its behavior correspondingly

KBSs possess cognitive penetrability


23

KBSs action to depend on what the system believes about the world, as opposed to just what the system has explicitly represented

A KB will involve quite general facts, which will then need to be applied to particular situations

An example (two facts) A part transfer operation solves task x. (1)

Any part transfer operation can be solved by a robot. (2)

KBSs Importance of reasoning

Problem to solve: it is appropriate to involve a robot in order to resolve task x

A KBS can derive the solution to the above problem, though this does not result from the presented facts.

A KBS is beyond a data base system (the behavior of a data base system is conditioned only on the facts it is able to retrieve).


24

The KBS should provide the answer p, if it believes it.

The system should believe p if, according to the beliefs it has represented, the world it is imagining is one where p is true.

Logical entailment: if the world is so that propositions (1) and (2) are true, then it is also true: A robot can solve task x


A KBS should believe all and only the entailments of what it has explicitly represented.

The job of reasoning, then, according to this account, is to compute the entailments of a KB; this is a simplistic view!

It can be too difficult computationally to decide which sentences are entailed by the kind of KB we will want to use.

The decision problem in Logics is undecidable.


25

Any procedure that always gives us answers in a reasonable amount of time will occasionally either:

miss some entailments (it is logically incomplete);return some incorrect answers (it is logically unsound).

There are also conceptual reasons why a KBS might consider unsound or incomplete reasoning.


Examples: something not entailed by the KB, but that is a reasonable answer

a KBS might come to believe a collection of facts from various sources that, taken together, cannot all be true (the KB is not sound)

While it would be a mistake to identify reasoning in a KBS with logically sound and complete inference, it is the right place to begin.


26

Importance of Logic for KBS:

It studies:Entailment relationsTruth conditions

Rules of inference

AI (the symbolic approach) is based on the calculus of First Order Predicates (First Order Logic)

KBSs The role of Logic

Homework

Consider a manufacturing control system; assess the system intelligence; propose some enhancements in order to obtain a KBS.

Explain the role of a knowledge representation language and a reasoning mechanism in KBSs.

Give examples of two different knowledge representation languages and reasoning mechanisms.

Problems 1 and 4, Brachman and Levesque, pp. 13-14.


27

Chapter 2 Important aspects on the

Language of First-Order Logic

Chapter Overview

Syntax Semantics Pragmatics Explicit and implicit belief




To understand how first-order logic (FOL) can provide a modelling tool for KBSs

To undestand the way FOL can establish a sound reasoning mechanism

To discover the limitations of FOL

28

Introduction Basic Aspects

First point for an AI based system: to be able to express (formulate) the ideas to be used

There is the need of a language

Language of first-order logic

Introduction Components of a language

Syntax: specifies which groups of symbols, arranged in what way, are to be considered properly formed

This student has sound knowledge on Robtics well formed English sentence

29


Semantics: specifies what the well-formed expressions are supposed to mean

Without semantics we cannot consider beliefs

Pragmatics: specifies how the meaningful expressions in the language are to be used

There is someone right behind you could be used as a warning


For knowledge representation, this involves how we use the meaningful sentences of a representation language as part of a KB from which inferences will be drawn.

These three aspects apply mainly to declarative languages, the sort we use to represent knowledge.

30

Language of First-Order Logic Syntax

An alphabet the set of admissible symbols

In FOL, there are two sorts of symbols: the logical ones

the nonlogical ones.

Intuitively, the logical symbols are those that have a fixed meaning or use in the language.


There are three sorts of logical symbols:

punctuation: (, ), and ..

connectives: 7, , , , , , and =. variables: an infinite supply of symbols, which we will denote here using x, y, and z, sometimes with subscripts and superscripts (V is the set of variables).

31


The nonlogical symbols are those that have an application-dependent meaning or use.

In FOL, there are two sorts of nonlogical symbols:1. function symbols: an infinite supply of symbols, which we will write in uncapitalized mixed case, e.g., bestFriend, and which we will denote more generally using a, b, c, f , g, and h, with subscripts and superscripts.


2. predicate symbols: an infinite supply of symbols, which we will write in capitalized mixed case, e.g., OlderThan, and which we will denote more generally using P, Q, and R, with subscripts and superscripts.

One distinguishing feature of nonlogical symbols is that each one is assumed to have an arity, that is, a nonnegative integer indicating how many arguments it takes.

32


By convention, a, b, and c are only used for function symbols of arity 0, which are called constants g and h are only used for function symbols of nonzero arity. Predicate symbols of arity 0 are sometimes called propositional symbols.


Note that = is not treated as a predicate symbol, but as a special logical symbol.

There are two types of legal syntactic expressions in FOL: terms and formulas. Intuitively, a term will be used to refer to something in the world, and a formula will be used to express a proposition.

33


The set of terms of FOL is the set satisfying these conditions:

every variable is a term;

if t1, ... , tn are terms, and f is a function symbol of arity n, then f (t1, ... , tn) is a term.


The set of formulas of FOL is the set satisfying these constraints:

if t1, ... , tn are terms, and P is a predicate symbol of arity n, then P(t1, ... , tn) is a formula;

if t1 and t2 are terms, then t1 = t2 is a formula;

if and are formulas, and x is a variable, then 7,( ), ( ), x. , and x. are formulas.

34

Language of First-Order Logic SyntaxFormulas of the first two types (containing no other simpler formulas) are called atomic formulas or atoms. Notational abbreviations and conventions:

we will add or omit matched parentheses and periods freely we use square and curly brackets to improve readability in the case of predicates or function symbols of arity 0, we usually omit the parentheses since there are no arguments to enclose we reduce the number of parentheses by assuming that 7 has higher precedence than , and has higher precedence than


By the propositional subset of FOL, we mean the language with no terms, no quantifiers, and where only propositional symbols are used.

An example of propositional formula:

(P 7(Q R)), where P, Q, R are propositional symbols

35


We also use the following abbreviations (equivalences):

( ) (7 ); ( ) (( ) ( ))

A variable occurrence in a formula can be free or bound. A variable occurrence is bound in a formula if it lies within the scope of a quantifier, and free otherwise.


An example: y. P(x) x[P(y) Q(x)]y (P(x) x(P(y) Q(x)))

If x is a variable, a formula and t a term, then (x|t) means the substitution of all the free occurrences of x in by t.

A sentence of FOL is any formula without free variables.

The sentences of FOL are what we use to represent knowledge.

36

Language of First-Order Logic Semantics

It is important to understand what claim a sentence of FOL makes about the worldOnly thus, the beliefs of the KBS can be derived One cannot realistically expect to specify once and for all what a sentence of FOL meansThe nonlogical symbols are used in an application-dependent way


An example:One cannot agree what the sentence Happy(john) claims about the world This can be decided only if one knows the interpretation of the nonlogical symbols involvedConclusion: we need a clear specification of the meaning of sentences as a function of the interpretation of the predicate and function symbols.

37


Main points of semantics specification:There are objects (entities) in the world

For any predicate P of arity 1, some of the objects will satisfy P and some will not. An interpretation of P decides for each object whether it has or does not have the property specified by P. Predicates of other arities are handled similarly.


A function symbol is interpreted as a mapping from tuples of objects to an object

No other aspects of the world matter

To notice that the above statements are all one needs to say regarding the meaning of the nonlogical symbols

38


Examples:

DemocraticCountry a predicate symbol of arity 1

The meaning of DemocraticCountry in some interpretation will be no more and no less than those objects that are countries that we consider to be democratic.


Examples:

child_of a function symbol of arity 2

The meaning of the above function symbol is a mapping:

People x People People

child_of(john, mary) dan

39

Semantics - Interpretations

Meanings are typically captured by specific interpretations

An interpretation I in FOL is a pair , where D is any nonempty set of objects, called the domain of the interpretation (the universe of discourse - UoD), and I is a mapping, called the interpretation mapping, from the nonlogical symbols to functions and relations over D.

UoD - the set of entities over which the variables may range (UoD is part of the world considered by the current problem)


It is important to stress that an interpretation need not involve only mathematical objects D can be any set, including people, garages, numbers, sentences, fairness, unicorns, chunks of peanut butter, situations, and the universe, among other things.

The interpretation mapping I is devoted toassigning the meaning of the predicate and function symbols

40


To every predicate symbol P of arity n, I[P] is an n-ary relation over D:

I[P] D D (D is taken n times)

Examples: Robot a unary predicate symbol

I[Robot] would be some subset of D (presumably the set of robots in that interpretation)


Examples: SmarterThan a binary predicate symbol

I[SmarterThan] would be some subset of D x D (presumably the set of pairs of objects in D where the first element of the pair is smarter than the second)

To every function symbol f of arity n, I[f] is an n-ary function over D:

I[f] D D D

41


Examples: bestServant a unary function symbol

I[bestServant] would be a function D D (presumably the function that maps a person to his/her best servant)

johnSmith a 0-ary function I[johnSmith] would be an element of D (presumably somebody called John Smith)


It is useful to think of the interpretation of predicates in terms of their characteristic functions.

The interpretation of an n-ary predicate becomes an n-ary function to {0, 1}:

I[P] D D {0, 1}

42


Two distinct interpretations of a predicate:

as a relation as a function to {0, 1}

The relationship between the two specifications is that a tuple of objects is considered to be in the relation over D if and only if the characteristic function over those objects has value 1.


There is an advantage of using the second type of interpretation.

It allows us to see how predicates of arity 0 (i.e., the propositional symbols) are handled:

If P is 0-arity predicate, then I[P] is either 0, or 1.

43


It is normal to think as 0 represents the truth value False, while 1 stands for True.

Thus, for the propositional subset of FOL, one can ignore D completely.

For propositional calculus, an interpretation is simply a mapping, I, from the propositional symbols to either 0 or 1.

Semantics - Denotation

Denotation clarifies the interpretation of terms that contain variables.

Given an interpretation I = we can specify which elements of D are denoted by any variable-free term of FOL.

An example:bestServant(johnSmith); I[bestServant](I[johnSmith]) D

44

Semantics - Denotation

To deal with terms including variables, one needs a variable assignment over D, that is, a mapping from the variables of FOL to the elements of D. - variable assignment:

: V D, [x] D

By using both an interpretation I and a variable assignment the denotation of every term t can be calculated.

Semantics Denotation

If x is a variable, then ||x||I, = [x]

If t1, ... , tn are terms, and f is a function symbol of arity n, then

||f(t1, , tn)||I, = F(d1, , dn)F = I[f], di = ||ti||I,

The above rules are recursive According to these recursive rules, ||ti||I, is always an element of D

45

Semantics Satisfaction and Models

By the means of an interpretation and a variable assignment one can specify which sentences of FOL are true and which are false Example:Robot(bestServant(johnSmith)) is true in I if and only if the following holds: If we use I to get hold of the subset of D denoted by Robot and the object denoted by bestServant(johnSmith), then that object is in the set.


To deal with formulas containing free variables, we should use a variable assignment Formally, given an interpretation I and variable assignment , we say that the formula is satisfied in I (is true in I), written:

I, |= ,

according to the following rules:

Assume that t1, ... , tn are terms, P is a predicate of arity n, and are formulas, and x is a variable.

46


1. I, |= P(t1, ... , tn) iff P, where P = I[P], and di = ||ti||I, 2. I, |= t1 = t2 iff ||t1||I, and ||t2||I, are the same element of D;

3. I, |= 7 iff it is not the case that I, |= ;

4. I, |= ( ) iff I, |= and I, |= ;


5. I, |= ( ) iff I, |= or I, |= (or both);

6. I, |= x. iff I, |= , for some variable assignment that differs from on at most x;this means there exists an object a in D that substituted for x determines I, |= (a, aD, and I, |= (x|a) )

7. I, |= x. iff I,|= , for every variable assignment that differs from on at most x (a, aD, I, |= (x|a) )

47


When the formula is a sentence, it is easy to see that satisfaction does not depend on the given variable assignment (recall that sentences do not have free variables) In this case, we write I |= and say that is true in the interpretation I, or that is false otherwise In the case of the propositional subset of FOL, it is sometimes convenient to write I[]= 1 or I[]= 0 according to whether I |= or not.


We will also use the notation I |= S, where S is a set of sentences, to mean that all of the sentences in S are true in I.

We say in this case that I is a logical model of S.

48

Language of First-Order Logic Pragmatics

The semantic rules of interpretation tell us how to understand precisely the meaning of any term or formula of FOL in terms of a domain and an interpretation for the nonlogical symbols over that domain.

Important point: The FOL language can be used to represent knowledge

Language of First-Order Logic Pragmatics

From the point of view of Pragmatics, one should understand that:

A KBS is supposed to reason about concepts like DemocraticCountry or Robot by the means of a given interpretation

A KBS has to be endowed with an interpretation, which could involve (perhaps infinite) sets of objects

49

Pragmatics Logical consequence

Although the semantic rules of interpretation depend on the interpretation of the nonlogical symbols, there are formulas that do not depend on the meaning of those symbols. Example: I is an interpretation so that formula is true; Let be the formula ; then is also true under the interpretation I (this does not depend on the nonlogical symbols in or ) is a logical consequence of


DefinitionLet S be a set of sentences (formulas), and any sentence. is a logical consequence of S, or that S logically entails , which we write:

S |=

if and only if, for every interpretation I, if I |= S then I |=

50


The definition of logical consequence has some equivalent formulations:

S |= iff every model of S is a model of

S |= iff there is no interpretation I so that I |= S U {7}

We say about the set S U {7} that is unsatisfiable, when there is no I to be its logical model


DefinitionA sentence is logically valid (tautology), which we write:

|=

when it is a logical consequence of the empty set.

Equivalent formulations: is valid if and only if, for every interpretation I, it is the case that I |=

is valid if and only if, the set {7} is unsatisfiable

51


Entailment is a special case of validity:

If S ={1, ... , n}, then S |= iff the formula

[(1 n) ] is logically valid.

The connection between KBS and logical entailment is at the heart of the knowledge representation enterprise.


KBSs means system that can reason

Given something like the fact that irb is a robot, it should be able to conclude that irb is also anindustrial equipment, an automatic device, and so on.

The KBS is told or can learn a sentence like Robot(irb) that is true in some user-intended interpretation, and that can then come to believe other sentences true in that interpretation.

52

Pragmatics Logical consequence A KBS will not and cannot have access to the interpretation of the nonlogical symbols itself.

A KBS cannot be given the set of sentences true in that interpretation as beliefs, because, among other things, there will be an infinite number of such sentences.

If a KBS is given a restricted set of sentences being true in the user intended interpretation, it can use logical entailment to derive new valid sentences.


An example Robot(irb) is true in the intended interpretation ()

A KBS can safely conclude: 7 7 Robot(irb)

Robot(irb) Happy(john)

This kind of conclusions are not very useful

53


To get an efficient KBS we need to include within the set of sentences S of the KB, statements connecting the nonlogical symbols involved. For the above example, such a sentence is:

x.Robot(x) Industrial_equipment(x) ()

If S = {, } then S |= Industrial_equipment (irb)


By including formula as one of the premises in S, we rule out interpretations like the one where the set of robots is not a subset of the set of industrial equipment.

One can conclude with the fundamental tenet of knowledge representation.

1. Reasoning based on only logical consequence allows safe, logically guaranteed conclusions to be drawn.

54

Pragmatics 2. The KBS starts with a collection of sentences as given premises (KB)

3. The KB includes not only facts about particulars of the intended application, but also those expressing connections among the nonlogical symbols involved

4. Calculating the entailments thus becomes a core part of a KBS, as it is like the form of reasoning we would expect of someone who understood the meaning of the terms involved (KB becomes a richer set)

Pragmatics

IMPORTANT CONCLUSION

This is all existing on knowledge representation and reasoning; the rest is just details.

55

Pragmatics Explicit and implicit belief

The KB can be viewed with the two components: working memory (WM) rule base (RB)

The WM - beliefs of the system that are explicitlygiven (introduced, sensed, or learned)

The RB - connections among the nonlogical symbols involved

Besides these, there are entailments of the KB -beliefs that are only implicitly given


It is often nontrivial to move from explicit to implicit beliefs.

An example from the blocks world

green

not green C

B

A

Fig. 2.1. Blocks world example

56


The problem can be formalized

a, b, c the names of the blocks (constants)

G, O predicate symbols, standing for greenand on

The initial KB:

S = {O(a, b), O(b, c), G(a), 7G(c)}


S |= (?) (*)

where

= x y. G(x) 7G(y) O(x, y)

A sketch of the demonstration: Two cases to be considered: I |= G(b); I |= 7G(b);

In both cases (*) takes place

57


Calculating what is implicit in a given collection of facts will sometimes involve subtle forms of reasoning

For FOL the problem of determining whether one sentence is a logical consequence of others is in general unsolvable

The process of calculating the entailments of a KB deductive inference


A reasoning process is logically sound if whenever it produces , then is guaranteed to be a logical consequence.

A reasoning process is logically complete if it is guaranteed to produce whenever is entailed.

The computational difficulty of FOL is one of the factors that will lead us to consider various other options of reasoning

58

Homework

Prove that the following two hypotheses do not entail the following conclusionHypotheses: Some students are hard-working. Some students are intelligent.

Conclusion: Some students are hard-working and intelligent.



Homework

Express the following sentences in FOL:Some students were enrolled in college and did not graduate.

All students that were enrolled attended the laboratories.

Some students attended the laboratories and did not graduate.



59

Homework Problems 1 and 4, Brachman and Levesque,

pp. 28-30. Express the following sentences in FOL:

Metallic objects are electrically conductive.

The wires are metallic objects. Metallic objects are not electrically conductive and insulating.

The wires are not insulating.



Homework

Consider the following set of sentences*:H1 Andrew is the father of Bob.H2 Bob is the father of Chris.H3 Every grandfather is someone's father.C Andrew is a grandfather of Chris.Translate these sentences into FOL. Show semantically

(by reasoning about interpretations) that H1 H3 do not logically entail C.

* This problem is part of an examination paper at University of Nottingham, School of Computer Science



60

Chapter 3 Knowledge Engineering

Chapter Overview

Knowledge engineering & Knowledge management

Ontology Vocabulary Types of facts Reification Entailment



Understanding the role and importance of knowledge engineering

Understanding the way an ontology can be a designing tool for KBSs

Studying some issues regarding the knowledge base development

61

Knowledge engineering and knowledge management

Knowledge management (KM) and knowledge engineering (KE) are topical terms

They are invoked when talking about: Knowledge-based society

Knowledge-based economy

Knowledge-based company

Knowledge-based marketing

Knowledge-based development


The terms knowledge management -knowledge engineering seem to be used as interchangeably, as the terms data and information used to be.

The difference stems from the distinctness between manager and engineer Manager - exercise executive, administrative and

supervisory direction

Engineer - lay out, construct, contrive or plan out, usually with a specific skill

62


The knowledge manager establishes the direction a process should take

The knowledge engineer develops the means to accomplish that direction

Though they have distinct tasks, they should cooperate


Tasks for knowledge managers:

To establish the knowledge needs of the enterprise (what knowledge is necessary to make what decision and enable what actions)

To ascertain the enterprise knowledge management policies

63


Tasks for knowledge engineers:

Knowledge representation and encoding

Manage data repositories, work flows, groupware technologies

Set up the processes by which knowledge requests are examined, information assembled, and knowledge returned to the requestors


Knowledge manager - can be the Chief Information Officer (CIO) or the person in charge of the Information Resource Management (IRM).

Knowledge engineer - most likely a computer scientist specialized in the development of artificial intelligence knowledge bases.

64


Knowledge management (KM) comprises a range of strategies and practices used in an organization to identify, create, represent, distribute, and enable adoption of insightsand experiences.

Such insights and experiences comprise knowledge, either embodied in individuals or embedded in organizations as processes or practices.


Knowledge engineering (KE) - an engineering discipline that involves integrating knowledge into computer systems in order to solve complex problems normally requiring a high level of human expertise

KM & KE ensure the creation, communication and application of knowledge of all types to meet enterprise goals

Companies - physical assets versus intangible assets

65


Adaptability of companies is related to KM

Creating new elements of knowledge

Fast & efficient dissemination of knowledge

Incorporation of knowledge into new products and services


Nine reasons to assume KM:

1. Companies are "knowledge intensive" rather than "capital intensive"

2. Unstable markets require a new organization for companies

3. KM allows companies to control the change (lets you lead change, so that change does

not lead you)

66



4. Only knowledge survives

5. Avoiding loss determined by tacit knowledge

6. Knowledge should be disseminated



7. Complexity of production situations can be mastered by KM

8. KM provides decision support systems

9. Use of the opportunities resulting from globalization

67


KM is important for two types of companies:

1. Companies that must keep the place in dispute with competitors (maintenance and upgrading the kernel of knowledge)

2. Companies that rely on products that require innovation


KM influences the main organizational processes of an enterprise

The impact of KM can be considered on three important features:

Optimality (best decisions/ production mechanisms)

Efficiency (minimizing time and costs)

Adaptability (making various processes in a creative way, complying to requests)

68


Knowledge Management

People Processes Products Performance

Dimensions of impacts of KM


KM impacts on people

Knowledge Employee Learning

Employee Adaptability

Employee Job

Satisfaction

KM Syst

69


KM case studies

1. Ford and Firestone Companies have the information necessary on the incompatibility of the Ford Explorer and Firestone tires Information was not properly integrated and communicated between companies Results: accidents, lawsuits, loss of customers


KM case studies

2. Xerox

KM is conducted as a scientific community (Community of Practice)

Provides visibility of advanced solutions

Ability to solve unstructured problems

Knowing the latest technologies

70


KM case studies

2. Toyota

It uses an advanced KM System Motivate employees to participate in the dissemination of knowledge Eliminate "free riders Reduces costs associated with finding and accessing valuable knowledge elements


Investment in KM is a capital investment that produces long term benefits.

Ideas are capital. The rest is just money.

Deutsche Bank, Wall Street Journal

71

Ontology Introduction; Basic Aspects

The same way as for any software system, for a KBS a principled architecture is to be designed first

Some main points should be decided before implementation

Reasons and times that inference will be necessary in our systems behavior

Ontology a key notion for KBS

An ontology must be staked out as a starting point

Ontology defines the set of representational primitives with which to model a UoD

Ontology a description of the concepts and relations existing in a UoD, which are established to support a software system

72


The definitions of concepts/relations include information about their meaning and constraints on their logically consistent application

The purpose of an ontology can be viewed as setting up the rules so that a group of entities should give the same interpretation for a part of the world


An ontology is an explicit description of a domain:

concepts

properties and attributes of concepts

constraints on properties and attributes

individuals

73


An ontology defines a common vocabulary

a shared understanding (among people and/or software entities)

An ontology should enable reuse of domain knowledge to avoid re-inventing the wheel

to introduce standards


To make domain assumptions explicit easier to change domain assumptions easier to understand and update legacy data

To separate domain knowledge from the operational knowledge re-use domain and operational knowledge

separately

74

Ontology Development

Defining terms in the domain and relations among them Defining concepts in the domain (classes) Arranging the concepts in a hierarchy (subclass-

superclass hierarchy)

Defining which attributes/properties (slots) classes can have and constraints on their values

Defining individuals and filling in slot values (instances)


An example of ontology

75

Ontology Development Ontology development versus Object-oriented modeling

- It reflects the structure of the world - It reflects the structure of data and code

- It is often about structure of concepts - It is usually about behavior (methods)

- Actual physical representation is not - It describes the an issue physical representation

of data (long int, char, etc.)

An ontology An OO Structure


What is the domain that the ontology will cover?

For what we are going to use the ontology? For what types of questions the information in

the ontology should provide answers?

Who will use and maintain the ontology?

determinescope

determinescope

considerreuse

enumerateterms

defineclasses

defineproperties

defineconstraints

createinstances

76


Example: questions for the wine ontology

Which wine characteristics should I consider when choosing a wine?

Is Bordeaux a red or white wine? Does Cabernet Sauvignon go well with seafood? What is the best choice of wine for grilled meat? Which characteristics of a wine affect its

appropriateness for a dish? Does a bouquet or body of a specific wine change

with vintage year?


Why reuse other ontologies? to save the effort

to interact with the tools that use other ontologies to use ontologies that have been validated

through use in applications

determinescope

considerreuse

considerreuse

enumerateterms

defineclasses

defineproperties

defineconstraints

createinstances

77

Ontology DevelopmentExamples of existing ontologies

Ontology libraries Protg ontology library (protege.stanford.edu) Ontolingua ontology library

(www.ksl.stanford.edu/software/ontolingua/)

Upper ontologies IEEE Standard Upper Ontology (suo.ieee.org) Cyc (www.cyc.com)

Domain-specific ontologies UMLS Semantic Net GO (Gene Ontology) (www.geneontology.org) OBO (Open Biological Ontologies) (obo.sourceforge.net)


What are the terms we need to talk about?

What are the properties of these terms?

What do we want to say about the terms?

determinescope

considerreuse

enumerateterms

enumerateterms

defineclasses

defineproperties

defineconstraints

createinstances

78


Examples of terms for wine ontology

wine, grape, winery, location

wine color, wine body, wine flavor, sugar content

white wine, red wine, Bordeaux wine

food, seafood, fish, meat, vegetables, cheese


A class is a concept in the domain a class of wines

a class of wineries

a class of red wines

A class is a collection of elements with similar properties

Instances of classes a California wine youll have for lunch

determinescope

considerreuse

defineclasses

defineclasses

defineproperties

defineconstraints

createinstances

enumerateterms

79


Inheritance is a common mechanism in a class hierarchy

Classes usually constitute a taxonomic hierarchy (a subclass-superclass hierarchy)

A class hierarchy is usually an IS-A hierarchy: an instance of a subclass is an instance of

a superclass If you think of a class as a set of elements, a

subclass is a subset


Multiple inheritance can create problemsA class can have more than one superclassThe subclass inherits properties and restrictions from all the parents

Different systems resolve conflicts differently

Industrial Equipment

Programmable System

Robot

80


Properties in a class definition describe attributes of instances of the class

each wine will have color, sugar content, producer, etc.

determinescope

considerreuse

defineproperties

defineproperties

defineconstraints

createinstances

enumerateterms

defineclasses


A taxonomy of properties Types of properties

intrinsic properties: flavor and color of wine

extrinsic properties: name and price of wine

parts: ingredients in a dish

relations to other objects: producer of wine (winery)

Simple and complex properties simple properties (attributes): contain primitive values

(strings, numbers)

complex properties: contain other objects (e.g., a winery instance)

81


Properties A subclass inherits all the properties from the

superclass If a wine has a name and flavor, a red wine also

has a name and flavor

If a class has multiple superclasses, it inherits properties from all of them Port is both a dessert wine and a red wine. It

inherits sugar content: high from the former and color:red from the latter


Property constraints (restrictions) describe or limit the set of possible values for a property

the name of a wine is a string

the wine producer is an instance of Winery a winery has exactly one location

determinescope

considerreuse

defineconstraints

defineconstraints

enumerateterms

defineclasses

defineproperties

createinstances

82


Property constraints Global property constraints apply to the

property throughout the ontology range for hasColor is always an instance of

WineColor

Local property constraints (restrictions) valid only for instances of the class and its subclasses hasColor for RedWine has the value Red


Property constraints Cardinality the number of values a property can or must

have

Minimum cardinality Minimum cardinality 1 means that the property must have a

value (required)

Minimum cardinality 0 means that the property value is optional

Maximum cardinality Maximum cardinality 1 means that the property can have at

most one value (functional property)

Maximum cardinality greater than 1 means that the property can have more than one value (multiple-valued property)

83


Create an instance of a class

The class becomes a direct type of the instance

Any superclass of the direct type is a type of the instance

Assign property values for instances

Property values should conform to the constraints

Knowledge-acquisition tools often make the check

determinescope

considerreuse

enumerateterms

defineclasses

defineproperties

createinstancescreate

instancesdefine

constraints


All the siblings in the class hierarchy must be at the same level of generality

If a class has only one child, there may be a modeling problem

If a class has more than a dozen children, additional subcategories may be necessary

84


Danger of multiple inheritance: cycles in the class hierarchy

Classes A, B, and C have equivalent sets of instances

A

B

C

subclass-of

subclass-of

subclass-of


Different modes of the development

top-down - define the most general concepts first and then specialize them

bottom-up - define the most specific concepts and then organize them in more general classes

combination

85


Classes represent concepts in the domain, not their names

The class name can change, but it will still refer to the same concept

Synonym names for the same concept are not different classes

Many systems allow listing synonyms as part of the class definition


A wine is not a kind-of wines

A wine is an instance of the class Wines

Class names should be either

all singular all plural

86


Subclasses of a class usually have

Additional/new properties

Additional/new restrictions

Participate in different relationships


It is to decide either for a new class or a property value

Wine Wine White wine Colour: white, rose, red

Rose wine

Red wine

OR

87


How important is the distinction for the domain?

A class of an instance should not change often

Individual instances are the most specific objects in an ontology

If concepts form a natural hierarchy, represent them as classes


Domain of a property the class (or classes) that have the property More precisely: class (or classes) of

instances which can have the property Range of a property the class (or classes)

to which property values belong

88


When defining a domain or range for a slot, find the most general class or classes

Consider the produces slot for a Winery:

Range: Red wine, White wine, Ros wine

Range: Wine

Consider the flavor slot

Domain: Red wine, White wine, Ros wine

Domain: Wine

Ontology Development Inverse properties (e.g., maker produces)

contain redundant information, but Allow acquisition of the information in either direction

Enable additional verification

Allow presentation of information in both directions

The actual implementation differs from system to system Are both values stored?

When are the inverse values filled in?

What happens if we change the link to an inverse property?

89


Default value a value the slot gets when an instance is created and a specific value is not provided

A default value can be changed

The default value is a common value for the slot, but is not a required value

Ontology DevelopmentNaming convention Define a naming convention for classes and

properties and adhere to it

Features of an ontology tool to be considered:

Can classes and properties have the same names?

Is the system case-sensitive?

What delimiters are allowed?

90

Concluding Remarks about Ontologies

There is no single correct class hierarchy, but there are some guidelines

An ontology should not contain all the possible information about the domain No need to specialize or generalize more than the

application requires No need to include all possible properties of a

class Only the most salient properties Only the properties that the applications require

Concluding Remarks about Ontologies

Ontology of wine, food, and their pairings probably will not include Bottle size Label color My favorite food and wine

An ontology of biological experiments will contain Biological organism Experimenter

Is the class Experimenter a subclass of Biological organism?

91

VocabularyA main point for an ontology is to agree on the use of a certain vocabulary

The nonlogical symbols are those that have an application-dependent meaning or use

Vocabulary establishing means the setting up of the nonlogical symbols

Considered example: a soap opera world

VocabularyConstants:

Named individuals: maryJones, johnSmith, tomJones

One needs to allow multiple identifiers that could used to refer to the same individual (e.g. tom,johnSmith)

Other entities: corporations, restaurants, places, objects (e.g. astraInsuranceCompany, iasiTownCouncil, tomsHouse, studentgroup1)

92

Vocabulary After capturing the set of individuals that will be central to the agents world, it is next essential to circumscribe the basic types of entities that those individuals are.

The unary predicates should be used Person(x) Other Examples:

Man, Woman, Place, Company, Jewelry, Pen Restaurant, Bar, House, University

Vocabulary The unary predicates can also describe the set of attributes that the entities can have A vocabulary of properties that can hold for individuals are to be established Examples:

Rich, Beautiful, Unscrupulous, Bankrupt, ClosedForRepairs

Important remark: The syntax of FOL is limited in that it does not allow us to distinguish between the predicates for properties and those used to describe the types of entities

93

Vocabulary The n-ary predicates can express relationships

Examples of predicates of arity 2: MarriedTo, DaughterOf, LivesAt

Examples of predicates of higher arity: ConspiresWith,OccursInTimeInterval

Vocabulary The n-ary functions are also part of the vocabulary

Both unary and higher arity functions can be included

Examples of functions fatherOf, bestFriendOf (unary functions)

childOf, gradeOf (binary functions; e.g. gradeOf(John, KRR) = 10) )

94

Vocabulary Important remark: there may be cases when both a function or a predicate can be involved

Some criteria to decide: Functions are taken to be total in FOL; if

exceptions are possible, then use predicates (e.g. bestFriendOf versus BestFriendOf)

Functions are needed as a means to keep using the FOL (e.g. Happy(bestFriendOf(x)) is an FOL formula, while Happy(BestFriendOf(x)) is beyond FOL)

Types of facts - Simple, core facts of the UoD

Simple facts

Atomic formula Negations of atomic formula

Examples:Man(john), Company(faultyInsuranceCompany),

Knife(butcherknife1)

These define the basic ontology of UoD Suggestive names are not a form of knowledge

representation, as they do not support logical inference

95

Properties of the entities These are important, since we most often want to see what properties (and relationships) are implied by a set of facts

Examples:Rich(john), 7HappilyMarried(jim), WorksFor(jim, ann), bestFriendOf(jim) = john,fic = faultyInsuranceCompany

Types of facts - Simple, core facts of the UoD

Complex facts are expressed by non-atomic formula Whenever we use quantifiers the domain of the variables must be specified (this is usually the UoD)

Examples:

All the rich men love Jane x[Rich(x) Man(x) Loves(x, jane)]

Types of facts - Complex facts

96

All women, with the possible exception of Jane, love John

x[Woman(x) 7(x = jane) Loves(x, john)]

Universals are also useful for expressing very general facts, not even involving any known individuals

x y [Loves(x, y) 7Blackmails(x, y)]

Universal quantification helps us to express statements which otherwise would mean to enumerate all the individuals of the UoD (a kind of abbreviation)


To express incomplete knowledge about our world we also need complex facts

Examples Friends(jane, john) Friends(jane, jim) x [Adult(x) Friends(x, ann)]

The use of the existential quantification is not necessarily an abbreviation, but determined by the lack of knowledge


97

Complex facts are used to express closure sentences the limits of the UoD

Examples x [Lawyer(x) x = jane x = john ]

For a KBS, it may be necessary to make explicit all the distinct individuals of the UoD

x [x = jane x = fic x = tomsHouse ] jane john; fic tomsHouse;


We need to provide a set of facts about the terminology we are using

Terminological facts are supporting the inferential component of the KBS

Terminological facts capture the relationships between statements expressed by predicates and functions

Types of facts - Terminological facts

98

Examples of terminological facts:

Disjointness two predicates that are disjoint x [Man(x) 7 Woman(x)]

Subtypes - predicates that imply a form of specialization, wherein one type is subsumed by another

x [Surgeon(x) Doctor(x)]



Exhaustiveness two or more subtypes completely account for a supertype (reverse of subtype assertion)

x [Adult(x) Man(x) Woman(x)]

Symmetry used for symmetric relationships x y [MarriedTo(x, y) MarriedTo(y, x)]


99


Inverse to make evident that some relationships are the opposite of others

x y[ChildOf(x, y) ParentOf(y, x)]

Type restrictions the arguments of some predicates must be of certain types

x y [MarriedTo(x, y) Person(x) Person(y)]



Full definitions to create compound predicates that are completely defined by a logical combination of other predicates (the use of biconditional/equivalence)

x [RichMan(x) Man(x) Rich(x)]

Conclusion: terminological facts are typically captured in FOL as universally quantified conditionals


100

In certain domains, there are some other types of facts that we may want to capture. Each of these is problematical for a straightforward application of FOL. Extensions of FOL or other knowledge representation languages allow their handling. The choice of the language to use in a KBS will ultimately depend on what types of facts/ conclusions are most important for the application.

Types of facts Other sorts of facts

Statistical and probabilistic facts include those facts that involve portions of the sets of individuals satisfying a predicate, in some cases exact subsets and in other cases less exactly quantifiable. Examples:

Half of the companies are located on the East Side.

Most of the employees are restless. Almost none of the employees are completely

trustworthy.


101

Default and prototypical facts regard characteristics that are usually true, or reasonable to assume true unless told otherwise Examples:

Company presidents typically have secretaries intercepting their phone calls.

Cars have four wheels.

Companies generally do not allow employees that work together to be married.

Birds fly.


Intentional facts express peoples mental attitudes and intentions, that is, they can reflect the reality of peoples beliefs but not necessarily the real world itself

Examples: John believes that Henry is trying to blackmail him. Jane does not want Jim to know that she loves

him.

Ultimately, a KBS should be able to express & reason with anything that can be expressed by natural language, anything that we can imagine as being either true or false.


102

The FOL language gives us the basic tools for representing facts with a great deal of flexibility There is also considerable flexibility in what we consider to be the individuals in the domain Sometimes is useful to introduce new abstract individuals that might not have been considered in a first analysis Reification the idea of making up new individuals as needed for the most appropriate detail level

Reification

Example1: the event of Johns buying a bike

Purchases(john, bike) Purchases(john, bike, feb15)

Purchases(john, bike, feb15, $200)

The problem here is that the arity of the Purchases predicate depends on how much detail we will want to express

Reification

103

We may not be able to predict in advance the level of detail; we need another approach, more flexible. To introduce an abstract individual, together with a unary predicate and some functions

p23 a constant representing the event of purchasing

Purchase(p23) agent(p23) = john object(p23) = bike time(p23) = feb15 cost(p23) = $200

Reification

We gain flexibility more/less details by adjusting the formula

Advantage - the arity of the predicate and function symbols involved can be determined in advance

Example2: a set of predicates to represent the marriage relationships: MarriedTo(x, y); PreviouslyMarriedTo(x, y); ReMarriedTo(x, y)

Reification

104

We can reify the marriage events as abstract individuals (entities)

We can determine anyones current marital status and complete marital history directly from these individuals

Marriage(m20) husband(m20) = john wife(m20) = ann date(m23) =

Reification

In representing commonsense information we need individuals for numbers, dates, times, addresses, and so on.

Basically, any object about which we can ask a wh-question should have an individual standing for it in the KB so it can be returned as the result of a query.

Reification

105

We gain flexibility in representing quantitative knowledge, by using proper functions

Example: ageInYears(suzzy) = 12 ageInMonths(suzzy) = 144

years(age(suzzy)) = 12 months(x) = 12 years(x)

Reification

Entailment is part of the reasoning process Entailment allows deriving of implicit conclusions from statements explicitly represented in the KB

KB collection of simple and complex facts (considered example, soap opera world) If the only tasks are to answer simple questions (e.g. Is John rich man?), then entailments will not be needed

Entailment

106

Entailments are needed when complex statements are analyzed Example1: Is there a company whose CEO (chief executive officer) loves Jane?

: x [Company(x) Loves(ceoOf(x), jane)]?

KB |= (I, if I |= KB (*), then I |= )

Entailment

I |= {Rich(john), Man(john)} - from (*) (1)

I|=x[Rich(x)Man(x)Loves(x,jane)] - from (*) (2)

(1), (2) I |= Loves(john, jane) (3)

I |= {john = ceoOf(fic), Company(faultyInsuranceCompany), from (*) (4)fic = faultyInsuranceCompany}

Entailment

107

(3),(4)I |= Company(fic)Loves(ceoOf(fic), jane)

I |= x [Company(x) Loves(ceoOf(x), jane)]

KB |=

Entailment

Remarks:

The provided solution determined not only that there is a company whose CEO loves Jane, but also what that company is. We can be interested in finding out not only whether something is true or not, but also which individuals satisfy a property of interest. A KBS has to face not only to yesno questions, but to wh-questions as well (who? what? where? when? how? why?)

Entailment

108

Example2: If no man is blackmailing John, then is he being blackmailed by someone he loves?

: x [Man(x) 7 Blackmails(x, john)] y [Loves(john, y) Blackmails(y, john)] ?

KB |=

It is more easy to demonstrate the requested entailment by using:

KB |= 1 1 iff KB U {1} |= 1

Entailment

I |= KB (*) and

I |= x [Man(x) 7 Blackmails(x, john)] (**)

I |= y [Loves(john, y) Blackmails(y, john)] ?

I |= x [Adult(x) Blackmails(x, john)] from (*) (1)I |= x [Adult(x) Man(x)Woman(x)] from (*) (2)

Entailment

109

I |= x [Woman(x) Blackmails(x, john)] from (**), (1), (2) (3)

I |= Loves(john, jane) from previous example (4)

I |= x[Woman(x) 7(x = jane) Loves(x, john)] from (*) (5)

I |= x y [Loves(x, y) 7Blackmails(x, y)]from (*) (6)

Entailment

I |=x[Woman(x)7(x = jane)7Blackmails(x, john)] from (5, 6) (7)

I |= Blackmails(jane, john) from (7, 3) (8)

I |= Blackmails(jane, john) Loves(john, jane)from (8, 4) (9)

I |= y [Loves(john, y) Blackmails(y, john)] from (9)

Entailment

110

We have illustrated, in intuitive form, how a proof can be thought of as a sequence of FOL sentences, starting with those known to be true in the KB or surmised as part of the assumptions dictated by the query.

The proof proceeds logically using the facts in the KB and the rules of logic.

Entailment

In some cases the answer to a question may be a negative one.

Entailment can be involved in such a case, in a more complicated way.

We have to prove:KB |

We must produce a specific interpretation and argue that it satisfies every sentence in the KB as well as the negation of

Entailment

111

Homework

Consider the following universes of discourse and sketch an ontology for each of them: the manufacturing cell from the Robotics Laboratory

the laboratory of Programable Logic Controlers

Problems 1, 2 and 3, Brachman and Levesque, pp. 45-46.


Homework

Consider the rule-based solution (in CLIPS) for planning a robot activity in the world of blocks. Analyze the working memory and establish which facts are simple, complex and terminological.


112

Chapter 4 Resolution

Chapter Overview

Introduction Resolution in Propositional Calculus

Resolution Derivations A Resolution based Entailment Procedure Resolution in FOL Answer Extraction Skolemization Equality Dealing with the computational complexity of

resolution

Chapter 4 Resolution


Understading the resolution based reasoning mechanism

A procedure to apply the resolution rule for all the cases of FOL formulas

A procedure to extract an answer from a knowledge base

Understanding the resolution limitations

113

Resolution - Introduction

Resolution is the main method applied by automated theorem proving

It details how to automate a deductive reasoning procedure

Problems to solve: Given a knowledge base KB and a sentence ,

we would like a procedure that can determine whether or not KB |=

If [x1, ... , xn] is a formula with free variables among the xi, we want a procedure that can find terms ti, if they exist, such that KB |= [t1, ... , tn]


There is no procedure that can fully satisfy this specification

The possible result: a procedure that does deductive reasoning in as sound and complete a manner as possible, and in a language as close as possible to that of full FOL

114


If we take the KB to be a finite set of sentences {1, ... , n}, there are several equivalent ways of formulating the deductive reasoning task:

KB |= |= [(1 n) ] KB U {7} is not satisfiable

Resolution is simpler for propositional calculus

Resolution in Propositional Calculus

It is easy to show that for any formula there is an equivalent formula so that is in the conjunctive normal form.

Example: p, q, r, s predicate symbols of arity zero

( p 7q) (q r 7s p) (7r q)

Automated theorem proving uses only formulas in the conjunctive normal form (CNF)

115


The procedure to convert any propositional formula to CNF is as follows:

1. Eliminate the logical operations and by using only 7, and

2. Use the tautology: |= 77 3. Use De Morgans laws

4. Distribute over , using the following equivalences:

|= ( ( ))(( ) )(( )())


5. Collect terms, using the following equivalences:

|= ( ) |= ( )

The CNF is also named clausal normal form: a formula is a conjunction of clauses, and a clause is a disjunction of literals (p is a positive literal, 7p is a negative literal)

116


It is convenient to use a shorthand representation for CNF:

a clausal formula is a set of clauses

a clause is a set of literals

An example: (( p 7q r) q {[p, 7q, r], [q]}


A clause with a single literal unit clause

The empty clause - [ ] - is understood as a representation of 7 TRUE (something always FALSE)

The procedure applied by automated theorem proving to conclude

KB |=

is:

117


1. Put the formulas in KB and 7 into CNF;

2. If the resulting set of cluases is un-satisfiable, then KB |=

In other words, any question about entailment can be reduced to a question about the satisfiability of a set of clauses.

Resolution derivations

To discuss reasoning at the symbol level, it is common to posit what are called rules of inference: statements of what formulas can be inferred from other formulas.

Automated theorem proving uses the resolution rule of inference:

A B, 7B C |= A C Expert systems use the modus ponens rule of

inference: A, A B |= B

118

Resolution derivations

Resolution rule can be expressed for set representation of formulas:

Given a clause of the form C1 = c1 U {} containing some literal , and a clause of the form C2 = c2 U {7} containing the complement of , infer the clause c1 U c2 consisting of those literals in the first clause other than and those in the second other than 7.

We say in this case that c1 U c2 is a resolvent of the two input clauses with respect to (c1 U c2 = Res(C1, C2) )

Resolution derivations Examples:[w, p, 7q], [s, w, 7p] |= [w, 7q, s]

[p], [7p] |= [ ] (this means {[p], [7p]} is un-satisfiable)

A resolution derivation of a clause c from a set of clauses S is a sequence of clauses c1, ... , cn, where the last clause, cn, is c, and where each ciis either an element of S or a resolvent of two earlier clauses in the derivation. We write S |- c if there is a derivation of c from S.

119

Resolution derivations The importance of resolution for reasoning:

If S |- c, then S |= c (the converse does not hold)

Conclusion 1: In general, as a form of reasoning, Resolution is sound, but not complete.

Conclusion 2: For c = [ ], resolution is both sound and complete (S |- [ ] if and only if S |= [ ])

A Resolution based entailment procedure

The main steps for a symbol-level procedure able to determine if KB |=

1. Put both KB and {7} into CNF2. Check if S = KB U {7} is un-satisfiable by

applying resolution derivation3. If [ ] is got by resolution derivation, then

positive answer: KB |=

120


The resolution procedure:

Input: a finite set S of propositional clausesOutput: satisfiable or unsatisfiable1. check if [ ] S; if so, return unsatisfiable2. otherwise, check if there are two clauses in S

such that they resolve to produce another clause not already in S; if not, return satisfiable

3. otherwise, add the new resolvent clause to S, and go back to step 1


Remarks: The procedure can be made deterministic

quite simply: we need to settle on a strategy for choosing which pair of clauses to use when there is more than one pair that would produce a new resolvent.

One possibility is to use the first pair encountered; another is to use the pair that would produce the shortest resolvent. It might also be a good idea to keep track of which pairs have already been considered to avoid redundant checking.

121


Remarks: If we were interested in returning or printing

out a derivation, we would want to store with each resolvent pointers to its input clauses.

The procedure does not distinguish between clauses that come from the KB and those that come from the negation of , which we will call the query.

Observe that if we have a number of queries we want to ask for the same KB, we need only convert the KB to CNF once and then add clauses for the negation of each query.


Remarks: If we want to add a new fact to the KB, we

can do so by adding the clauses for to those already calculated for KB.

To use this type of entailment procedure, it makes good sense to keep KB in CNF, adding and removing clauses as necessary.

122


Example KB:

Toddler; Toddler Child; Child Male Boy;

Infant Child; Child Female Girl; Female

KB |= Girl ?


S = {Toddler, 7Toddler Child, 7Child 7Male Boy, 7Infant Child, 7Child 7Female Girl, Female, 7Girl}

S = {[Toddler] , [7Toddler, Child], [7Child, 7Male, Boy], [7Infant, Child], [7Child, 7Female, Girl], [Female], [7Girl]}

123


Derivation by resolution:C1 = [Toddler] , C2 = [7Toddler, Child], C3 = [7Child, 7Male, Boy], C4 = [7Infant, Child], C5 = [7Child, 7Female, Girl], C6 = [Female], C7 = [7Girl]}C8 = [Child] Res(C1, C2)C9 = [7Female, Girl] Res(C5, C8)C10 = [Girl] Res(C9, C6)C11 = [ ] Res(C10, C7)

Resolution in FOL

For the general case of FOL we have to handle terms and quantifiers

Again we have to convert the formulas in the CNF (clausal normal form)

First, we neglect existential quantifiers

124

Resolution in FOL

Procedure to obtain CNF:Important remark a formula in CNF is a sentence

which if containing variables, these are universally quantified

1. Eliminate the logical operations and by using only 7, and

2. Move 7 inward so that it appears only in front of an atom, using De Morgans laws and the following two:

|= 7 x. x. 7 |= 7 x. x. 7

Resolution in FOL

3. Standardize variables, that is, ensure that each quantifier is over a distinct variable by renaming them as necessary. This uses the following equivalences (provided that x does not occur free in ):

|= y. x. (y|x) |= y. x. (y|x)

125

Resolution in FOL

4. Eliminate all remaining existentials (discussed later)

5. Move universals outside the scope of and using the following equivalences (provided that x does not occur free in )

|= ( x. ) (x. ) x ( ) |= ( x. ) (x. ) x ( )

Resolution in FOL

6. Distribute over , as before7. Collect terms as before

The result of this procedure is a quantified version of CNF, a universally quantified conjunction of disjunctions of literals that is once again logically equivalent to the original formula (ignoring existentials).

126

Resolution in FOL

As for the propositional calculus, the clausal form allows us to use sets instead of formulas.

In the general case, a literal can be:

P(t1, , tn) P predicate symbol of arity n, ti term (variable or function symbol)

Resolution in FOL

Example: x y [(P(x) 7R(a, f(b, x))) Q(x, y)]

{[P(x) 7R(a, f(b, x))], [Q(x, y)]}

To apply resolution in FOL we use substitutions

= {x1|t1, , xn|tn}; xi variable; ti term (xidoes not appear in ti)

= (x1|t1, , xn|tn); is an instance of

127

Resolution in FOL

Example: = {x|a, y|g(a, b, z)}

= P(x, c, y)

= P(a, c, g(a, b, z))

A term, a literal or a formula (clause) is ground if it contains no variables.

Resolution in FOL

General rule of resolution for FOLSuppose we are given a clause of the form c1 U {1} containing some literal 1, and a clause of the form c2 U {72} containing the complement of a literal 2. Suppose we rename the variables in the two clauses so that each clause has distinct variables, and there is a substitution such that 1 = 2. Then, we can infer by resolution the clause (c1Uc2) consisting of those literals in the first clause other than 1 and those in the second other than 2, after applying .

128

Resolution in FOL

We say in this case that unifies 1 and 2, and that is a unifier of the two literals.

With this new general rule of Resolution, the definition of a derivation by resolution stays the same, and ignoring equality, it is the case that S |- [ ] if and only if S |= [ ], as for the propositional calculus.

Resolution in FOL

Example 1 KB

x(GradStudent(x) Student(x)) x(Student(x) HardWorker(x)) GradStudent(sue)

Question HardWorker(sue)

Problem KB |= HardWorker(sue)

129

Resolution in FOL

Derivation by resolution:C1 = [7GradStudent(x), Student(x)] , C2 = [7Student(y), HardWorker(y)],

C3 = [GradStudent(sue)], C4 = [7HardWorker(sue)], C5 = [7Student(sue)] - Res(C2, C4); = {y|sue}C6 = [7GradStudent(sue)] - Res(C5, C1); ={x|sue}C7 = [ ] Res(C3, C6)

Resolution in FOL

Example 2 KB

On(a, b) = C1 On(b, c) = C2 Green(a) = C3 7Green(c) = C4

Question Q = x y (On(x, y) Green(x) 7Green(y))

130

Resolution in FOL

Q = x y 7(On(x, y) Green(x) 7Green(y)) =x y (7On(x, y) 7Green(x) Green(y)) = C5

C6 = [7Green(a), Green(b)] - Res(C1, C5); ={x|a, y|b}

C7 = [Green(b)] - Res(C6, C3)

C8 = [7Green(b), Green(c)] - Res(C2, C5); ={x|b, y|c}

C9 = [7Green(b)] - Res(C8, C4)

C10 = [ ] - Res(C9, C10)

Resolution in FOL

Example 3

Using resolution derivation it is possible to get answers to queries that we might think of as requiring some computation, too.

To do arithmetic, for example, we can use the constant zero to stand for 0, and succ to stand for the successor function.

131

Resolution in FOL

Every natural number can then be written as a ground term using these two symbols.

4 = succ(succ(succ(succ(zero))))

The predicate Plus(x, y, z) stands for the relation: x + y = z

KB x Plus(zero, x, x) x y z(Plus(x, y, z) Plus(succ(x), y, succ(z)) )

Resolution in FOL

From this KB one can derive that: 2 + 3 = 5 (namely, Plus(2, 3, 5)).

The same KB and the query: u Plus(2, 3, u) = Q

KB C1 = [Plus(0, s, s)] C2 = [7Plus(x, y, z), Plus(succ(x), y, succ(z))]

C3 = 7Q = [7 Plus(2, 3, u)]

132

Resolution in FOL

[7Plus(x, y, z), Plus(succ(x), y, succ(z))] [7Plus(2, 3, u)]

[7Plus(1, 3, v)]

= {x|1, y|3, u|succ(v), z|v}

[7Plus(0, 3, w)]

= {x|0, y|3, v|succ(w), z|w}

[Plus(0, s, s)]

[ ]

= {s|3, w|3}

Resolution in FOL

The answer is obtained by examining the bindings of variables

u succ(v)

v succ(w) w 3

This form of computation, including locating the answers in a derivation of an existential, is

Documents

12--