Knowledge Representation and Reasoning  Representação do Conhecimento e Raciocínio

1. Knowledge Representation and ReasoningRepresentação do Conhecimento e Raciocínio José Júlio Alferes

2. Part 1: Introduction

3. What is it ? • What data does an intelligent “agent” deal with? - Not just facts or tuples. • How does an “agent” knows what surrounds it? What are the rules of the game? • One must represent that “knowledge”. • And what to do afterwards with that knowledge? How to draw conclusions from it? How to reason? • Knowledge Representation and Reasoning  AI Algorithms and Data Structures  Computation

4. What is it good for ? • Fundamental topic in Artificial Intelligence • Planning • Legal Knowledge • Model-Based Diagnosis • Expert Systems • Semantic Web (http://www.w3.org) • Reasoning on the Web (http://www.rewerse.com) • Ontologies and data-modeling

5. What is this course about? • Logic approaches to knowledge representation • Issues in knowledge representation • semantics, expressivity, complexity • Representation formalisms • Forms of reasoning • Methodologies • Applications

6. Bibliography • Will be pointed out as we go along (articles, surveys) in the summaries at the web page • For the first part of the syllabus: • Reasoning with Logic Programming J. J. Alferes and L. M. Pereira Springer LNAI, 1996 • Nonmonotonic Reasoning G. Antoniou MIT Press, 1996.

7. What prior knowledge? • Computational Logic • Introduction to Artificial Intelligence • Logic Programming

8. Logic for KRR • Logic is a language conceived for representing knowledge • It was developed for representing mathematical knowledge • What is appropriate for mathematical knowledge might not be so for representing common sense • What is appropriate for mathematical knowledge might be too complex for modeling data.

9. Mathematical knowledge vs common sense • Complete vs incomplete knowledge • " x : x Î N → x Î R • go_Work → use_car • Solid inferences vs default ones • In the face incomplete knowledge • In emergency situations • In taxonomies • In legal reasoning • ...

10. Monotonicity of Logic • Classical Logic is monotonic T |= F → T U T’ |= F • This is a basic property which makes sense for mathematical knowledge • But is not desirable for knowledge representation in general!

11. Non-monotonic logics • Do not obey that property • Appropriate for Common Sense Knowledge • Default Logic • Introduces default rules • Autoepistemic Logic • Introduces (modal) operators which speak about knowledge and beliefs • Logic Programming

12. Logics for Modeling • Mathematical 1st order logics can be used for modeling data and concepts. E.g. • Define ontologies • Define (ER) models for databases • Here monotonicity is not a problem • Knowledge is (assumed) complete • But undecidability, complexity, and even notation might be a problem

13. Description Logics • Can be seen as subsets of 1st order logics • Less expressive • Enough (and tailored for) describing concepts/ontologies • Decidable inference procedures • (arguably) more convenient notation • Quite useful in data modeling • New applications to Semantic Web • Languages for the Semantic Web are in fact Description Logics!

14. In this course (revisited) • Non-Monotonic Logics • Languages • Tools • Methodologies • Applications • Description Logics • Idem…

15. Part 2: Default and Autoepistemic Logics

16. Default Logic • Proposed by Ray Reiter (1980) go_Work → use_car • Does not admit exceptions! • Default rules go_Work : use_car use_car

17. More examples anniversary(X)  friend(X) : give_gift(X) give_gift(X) friend(X,Y)  friend(Y,Z) : friend (X,Z) friend(X,Z) accused(X) : innocent(X) innocent(X)

18. Default Logic Syntax • A theory is a pair (W,D), where: • W is a set of 1st order formulas • D is a set of default rules of the form: j : Y1, … ,Yn g • j (pre-requisites), Yi (justifications) and g (conclusion) are 1st order formulas

19. The issue of semantics • If j is true (where?) and all Yi are consistent (with what?) then g becomes true (becomes? Wasn’t it before?) • Conclusions must: • be a closed set • contain W • apply the rules of D maximally, without becoming unsupported

20. Default extensions • G(S) is the smallest set such that: • W G(S) • Th(G(S)) = G(S) • A:Bi/C  D, A G(S) and Bi  S → C G(S) • E is an extension of (W,D) iff E = G(E)

21. Quasi-inductive definition • E is an extension iff E = Ui Ei for: • E0 = W • Ei+1 = Th(Ei) U {C: A:Bj/C  D, A  Ei, Bj E}

22. Some properties • (W,D) has an inconsistent extension iff W is inconsistent • If an inconsistent extension exists, it is unique • If W  Just  Conc is inconsistent , then there is only a single extension • If E is an extension of (W,D), then it is also an extension of (W  E’,D) for any E’  E

23. Operational semantics • The computation of an extension can be reduced to finding a rule application order (without repetitions). • P = (d1,d2,...) and P[k] is the initial segment of P with k elements • In(P) = Th(W  {conc(d) | dP}) • The conclusions after rules in P are applied • Out(P) = {Y | Y just(d) and dP } • The formulas which may not become true, after application of rules in P

24. Operational semantics (cont’d) • d is applicable in P iff pre(d)  In(P) and Y In(P) • P is a process iff dkP, dk is applicable in P[k-1] • A process P is: • successful iff In(P) ∩ Out(P) = {}. • Otherwise it is failed. • closed iff d D applicable in P→dP • Theorem: E is an extension iff there exists P, successful and closed, such that In(P) = E

25. Computing extensions (Antoniou page 39) extension(W,D,E) :- process(D,[],W,[],_,E,_). process(D,Pcur,InCur,OutCur,P,In,Out) :- getNewDefault(default(A,B,C),D,Pcur), prove(InCur,[A]), not prove(InCur,[~B]), process(D,[default(A,B,C)|Pcur],[C|InCur],[~B|OutCur],P,In,Out). process(D,P,In,Out,P,In,Out) :- closed(D,P,In), successful(In,Out). closed(D,P,In) :- not (getNewDefault(default(A,B,C),D,P), prove(In,[A]), not prove(In,[~B]) ). successful(In,Out) :- not ( member(B,Out), member(B,In) ). getNewDefault(Def,D,P) :- member(Def,D), not member(Def,P).

26. Normal theories • Every rule has its justification identical to its conclusion • Normal theories always have extensions • If D grows, then the extensions grow (semi-monotonicity) • They are not good for everything: • John is a recent graduate • Normally recent graduates are adult • Normally adults, not recently graduated, have a job (this cannot be coded with a normal rule!)

27. Problems • No guarantee of extension existence • Deficiencies in reasoning by cases • D = {italian:wine/wine french:wine/wine} • W ={italian v french} • No guarantee of consistency among justifications. • D = {:usable(X),  broken(X)/usable(X)} • W ={broken(right) v broken(left)} • Non cummulativity • D = {:p/p, pvq:p/p} • derives p v q, but after adding p v q no longer does so

28. Auto-Epistemic Logic • Proposed by Moore (1985) • Contemplates reflection on self knowledge (auto-epistemic) • Allows for representing knowledge not just about the external world, but also about the knowledge I have of it

29. Syntax of AEL • 1st Order Logic, plus the operator L (applied to formulas) • Lj means “I know j” • Examples: MScOnSW →L MScSW (or  L MScOnSW → MScOnSW) young (X) Lstudies (X) → studies (X)

30. Meaning of AEL • What do I know? • What I can derive (in all models) • And what do I not know? • What I cannot derive • But what can be derived depends on what I know • Add knowledge, then test

31. Semantics of AEL • T* is an expansion of theory T iff T* = Th(T{Lj : T* |= j}  {Lj : T* |≠j}) • Assuming the inference rule j/Lj : T* = CnAEL(T  {Lj : T* |≠j}) • An AEL theory is always two-valued in L, that is, for every expansion: j | Lj T* Lj T*

32. Knowledge vs. Belief • Belief is a weaker concept • For every formula, I know it or know it not • There may be formulas I do not believe in, neither their contrary • The Auto-Epistemic Logic of knowledge and belief (AELB), introduces also operator B j – I believe in j

33. AELB Example • I rent a film if I believe I’m neither going to baseball nor football games Bbaseball Bfootball → rent_filme • I don’t buy tickets if I don’t know I’m going to baseball nor know I’m going to football  Lbaseball  Lfootball → buy_tickets • I’m going to football or baseball baseball  football • I should not conclude that I rent a film, but do conclude I should not buy tickets

34. Axioms about beliefs • Consistency Axiom B • Normality Axiom B(F → G) → (B F →B G) • Necessitation rule F B F

35. Minimal models • In what do I believe? • In that which belongs to all preferred models • Which are the preferred models? • Those that, for one same set of beliefs, have a minimal number of true things • A model M is minimal iff there does not exist a smaller model N, coincident with M on Bj e Lj atoms • When j is true in all minimal models of T, we write T |=minj

36. AELB expansions • T* is a static expansion of T iff T* = CnAELB(T  {Lj : T* |≠j}  {Bj : T* |=minj}) where CnAELB denotes closure using the axioms of AELB plus necessitation for L

37. The special case of AEB • Because of its properties, the case of theories without the knowledge operator is especially interesting • Then, the definition of expansion becomes: T* = YT(T*) where YT(T*) = CnAEB(T  {Bj : T* |=minj}) and CnAEB denotes closure using the axioms of AEB

38. Least expansion • Theorem: Operator Y is monotonic, i.e. T  T1 T2→YT(T1) YT(T2) • Hence, there always exists a minimal expansion of T, obtainable by transfinite induction: • T0 = CnAEB(T) • Ti+1 = YT(Ti) • Tb = Ua < b Ta (for limit ordinals b)

39. Consequences • Every AEB theory has at least one expansion • If a theory is affirmative (i.e. all clauses have at least a positive literal) then it has at least a consistent expansion • There is a procedure to compute the semantics

40. Part 3: Logic Programming for Knowledge representation 3.1 Semantics of Normal Logic Programs

41. LP forKnowledge Representation • Due to its declarative nature, LP has become a prime candidate for Knowledge Representation and Reasoning • This has been more noticeable since its relations to other NMR formalisms were established • For this usage of LP, a precise declarative semantics was in order

42. Language • A Normal Logic Programs P is a set of rules: H ¬A1, …, An, not B1, … not Bm (n,m ³ 0) where H, Ai and Bj are atoms • Literal not Bj are called default literals • When no rule in P has default literal, P is called definite • The Herbrand base HP is the set of all instantiated atoms from program P. • We will consider programs as possibly infinite sets of instantiated rules.

43. Declarative Programming • A logic program can be an executable specification of a problem member(X,[X|Y]). member(X,[Y|L])¬ member(X,L). • Easier to program, compact code • Adequate for building prototypes • Given efficient implementations, why not use it to “program” directly?

44. flight from to flight ( lisbon , adam ). Lisbon Adam Þ flight ( lisbon , london ) Lisbon London M M M ¬ connection ( A , B ) flight ( A , B ). ¬ connection ( A , B ) flight ( A , C ), connection ( C , B ). ¬ chooseAnot her ( A , B ) not connection ( A , B ). LP and Deductive Databases • In a database, tables are viewed as sets of facts: • Other relations are represented with rules:

45. ¬ connection ( A , B ) flight ( A , B ). ¬ connection ( A , B ) flight ( A , C ), connection ( C , B ). ¬ chooseAnot her ( A , B ) not connection ( A , B ). LP and Deductive DBs (cont) • LP allows to store, besides relations, rules for deducing other relations • Note that default negation cannot be classical negation in: • A form of Closed World Assumption (CWA) is needed for inferring non-availability of connections

46. ¬ flies ( A ) bird ( A ), not abnormal ( A ) . ¬ bird ( P ) penguin ( P ). ¬ abnormal ( P ) penguin ( P ). bird ( a ). penguin ( p ). Default Rules • The representation of default rules, such as “All birds fly” can be done via the non-monotonic operator not

47. The need for a semantics • In all the previous examples, classical logic is not an appropriate semantics • In the 1st, it does not derive not member(3,[1,2]) • In the 2nd, it never concludes choosing another company • In the 3rd, all abnormalities must be expressed • The precise definition of a declarative semantics for LPs is recognized as an important issue for its use in KRR.

48. 2-valued Interpretations • A 2-valued interpretation I of P is a subset of HP • A is true in I (ie. I(A) = 1) iff AÎ I • Otherwise, A is false in I (ie. I(A) = 0) • Interpretations can be viewed as representing possible states of knowledge. • If knowledge is incomplete, there might be in some states atoms that are neither true nor false

49. 3-valued Interpretations • A 3-valued interpretation I of P is a set I = T U not F where T and F are disjoint subsets of HP • A is true in I iff A Î T • A is false in I iff AÎ F • Otherwise, A is undefined (I(A) = 1/2) • 2-valued interpretations are a special case, where: HP = T U F

50. Models • Models can be defined via an evaluation function Î: • For an atom A, Î(A) = I(A) • For a formula F, Î(not F) = 1 - Î(F) • For formulas F and G: • Î((F,G)) = min(Î(F), Î(G)) • Î(F ¬ G)= 1 if Î(F) £ Î(G), and = 0 otherwise • I is a model of P iff, for all rule H ¬ B of P: Î(H ¬ B) = 1