Knowledge Representation and Reasoning

1. Knowledge Representation and Reasoning José Júlio Alferes Luís Moniz Pereira

2. What is it ? • What data does an intelligent “agent” deal with? - Not just facts or tuples. • How does an “agent” know what surrounds it? 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

3. What is it good for ? • Basic subject matter for Artificial Intelligence • Planning • Legal Knowledge • Model-Based Diagnosis • Expert Systems • Semantic Web (http://www.w3.org) • Web of Knowledge (http://www.rewerse.com)

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

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

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. 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

8. 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 • ...

9. 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 !

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

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

12. 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)

13. Default Logic Syntaxe • 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

14. 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

15. 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)

16. 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}

17. 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

18. 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

19. 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

20. 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).

21. 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!)

22. 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