1. cse@buffalo Symbolic Representation and Reasoning an Overview Stuart C. Shapiro Department of Computer Science and Engineering, Center for Multisource Information Fusion, and Center for Cognitive Science University at Buffalo, The State University of New York 201 Bell Hall, Buffalo, NY 14260-2000 shapiro@cse.buffalo.edu http://www.cse.buffalo.edu/~shapiro/

2. cse@buffalo September, 2004S. C. Shapiro2 Introduction Knowledge Representation Reasoning Symbols Logics

3. cse@buffalo September, 2004S. C. Shapiro3 Knowledge Representation A subarea of Artificial Intelligence Concerned with understanding, designing, and implementing ways of representing information in computers So that programs can use this information to derive information that is implied by it, to converse with people in natural languages, to plan future activities, to solve problems in areas that normally require human expertise.

4. cse@buffalo September, 2004S. C. Shapiro4 Reasoning Deriving information that is implied by the information already present is a form of reasoning. Knowledge representation schemes are useless without the ability to reason with them. So, Knowledge Representation and Reasoning

5. cse@buffalo September, 2004S. C. Shapiro5 Knowledge vs. Belief Knowledge: Justified True Belief KR systems operate the same whether or not the information stored is justified or true. So, Belief Representation and Reasoning would be better. But we’ll stick with KR.

6. cse@buffalo September, 2004S. C. Shapiro6 What Is a Symbol? “A symbol token is a pattern that can be compared to some other symbol token and judged equal with it or different from it… Symbols may be formed into symbol structures by means of a set of relations… The `objects’ that symbols designate may include … objects in an external environment of sensible (readable) stimuli.” [Newell & Simon, Concise Encyclopedia of CS, 2004]

7. cse@buffalo September, 2004S. C. Shapiro7 What Is Logic? The study of correct reasoning. Not a particular KR language. There are many systems of logic. With slight abuse, we call a system of logic a logic. KR research may be seen as the search for the correct logic(s) to use in intelligent systems.

8. cse@buffalo September, 2004S. C. Shapiro8 Parts of Specifying a Logic Syntax Semantics Proof Theory

9. cse@buffalo September, 2004S. C. Shapiro9 Syntax The specification of a set of atomic symbols, and the grammatical rules for combining them into well-formed expressions (symbol-structures).

10. cse@buffalo September, 2004S. C. Shapiro10 Syntactic Expressions Atomic symbols Individual constants: Tom, Betty, white Variables: x, y, z Function symbols: motherOf Predicate symbols: Person, Elephant, Color Propositions: P, Q, BdT Terms Individual constants: Tom, Betty, white Variables : x, y, z Functional terms: motherOf(Fred) Well-formed formulas (wffs) Propositions (Proposition symbols) : P, Q, BdT Atomic formulas: Color(x, white), Duck(motherOf(Fred)) Non-atomic formulas: TdB  Td  Bp

11. cse@buffalo September, 2004S. C. Shapiro11 Semantics The specification of the meaning (designation) of the atomic symbols, and the rules for determining the meanings of the well-formed expressions from the meanings of their parts.

12. cse@buffalo September, 2004S. C. Shapiro12 Semantic Values Terms could denote Objects Categories of objects Properties… Wffs could denote Propositions Truth values

13. cse@buffalo September, 2004S. C. Shapiro13 Truth Values Could be 2, 3, 4, …, ∞ different truth values. Some truth values are “distinguished” Needn’t have anything to do with truth in the real world. By default, we’ll assume 2 truth values. Call distinguished one True (T) Call other False (F)

14. cse@buffalo September, 2004S. C. Shapiro14 Proof Theory The specification of a set of rules, which, given an initial collection of well-formed expressions, specify what other well-formed expressions can be added to the collection.

15. cse@buffalo September, 2004S. C. Shapiro15 Proof / Knowledge Base The collection could be A proof A knowledge base The initial collection could be Axioms Hypotheses Assumptions Domain facts & rules The added expressions could be Theorems Derived facts & rules

16. cse@buffalo September, 2004S. C. Shapiro16 Example Logic: Standard Propositional Logic Domain: CarPool World Atomic Proposition Symbols: BdT, TdB, Bd, Td, Bp, Tp Unary wff-forming connective:  Binary wff-forming connectives: , , , 

17. cse@buffalo September, 2004S. C. Shapiro17 Intended Interpretation (Intensional Semantics) BdT: Betty drives Tom TdB: Tom drives Betty Bd: Betty is the driver Td: Tom is the driver Bp: Betty is the passenger Tp: Tom is the passenger

18. cse@buffalo September, 2004S. C. Shapiro18 Extensional (Denotational) Semantics BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF 5 of 2 6 = 64 possible situations

19. cse@buffalo September, 2004S. C. Shapiro19 Properties of Wffs Satisfiable T in some situation BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

20. cse@buffalo September, 2004S. C. Shapiro20 Properties of Wffs Contingent T in some, F in some BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

21. cse@buffalo September, 2004S. C. Shapiro21 Properties of Wffs Valid T in all situations BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

22. cse@buffalo September, 2004S. C. Shapiro22 Properties of Wffs Contradictory T in no situation BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

23. cse@buffalo September, 2004S. C. Shapiro23 Logical Implication P 1, …, P n logically imply Q P 1, …, P n |= Q In every situation that P 1, …, P n are True, so is Q.

24. cse@buffalo September, 2004S. C. Shapiro24 Example: CarPool World KB Let KB CPW = Bd   Bp Td   Tp BdT  Bd  Tp TdB  Td  Bp TdB  BdT

25. cse@buffalo September, 2004S. C. Shapiro25 Extensional (Denotational) Semantics BdTTF TdBFT BdTF TdFT BpFT TpTF Only 2 of the 64 situations where KB CPW are T So, e.g., KB CPW, BdT |= Bd   Bp This is how a KB constrains a model to the domain we want.

26. cse@buffalo September, 2004S. C. Shapiro26 Proof Theory Some Rules of Inference P Q P  Q P Q P  Q P Q P P  Q Modus Ponens or  Elimination  Elimination  Elimination  Introduction

27. cse@buffalo September, 2004S. C. Shapiro27 Derivation from Assumptions Q is derivable from P 1, …, P n P 1, …, P n |- Q Starting from the collection P 1, …, P n, one can repeatedly apply rules of inference, and eventually get Q.

28. cse@buffalo September, 2004S. C. Shapiro28 Example: CarPool World Proof BdT  Bd  Tp BdT Bd  Tp Bd Bd   Bp  Bp So, KB CPW, BdT |- Bd   Bp Bd   Bp

29. cse@buffalo September, 2004S. C. Shapiro29 Theoremhood If Q is derivable from no assumptions, |- Q We say that Q is provable, and that Q is a theorem.

30. cse@buffalo September, 2004S. C. Shapiro30 Deduction Theorem P 1, …, P n |= Q iff |= (P 1  · · ·  P n )  Q P 1, …, P n |- Q iff |- (P 1  · · ·  P n )  Q So theorem-proving is relevant to reasoning.

31. cse@buffalo September, 2004S. C. Shapiro31 Properties of Logics Soundness If |- P then |= P (If P is a provable, then P is valid.) Completeness If |= P then |- P (If P is valid, then P is a provable.)

32. cse@buffalo September, 2004S. C. Shapiro32 Soundness vs. Completeness Soundness is the essence of correct reasoning Completeness is less important because it doesn’t indicate how long it might take.

33. cse@buffalo September, 2004S. C. Shapiro33 Commutativity Diagram for Sound and Complete Logics P1, …, Pn |= Q |= (P1  · · ·  Pn )  Q P1, …, Pn |- Q |- (P1  · · ·  Pn )  Q completeness soundness So, whenever you want one, you can do another.

34. cse@buffalo September, 2004S. C. Shapiro34 Use of Commutativity Diagram Refutation proof techniques, such as resolution refutation or semantic tableaux, prove that there can be no situation in which P 1, …, and P n are True and Q is False. These are semantic proof techniques.

35. cse@buffalo September, 2004S. C. Shapiro35 Decision Procedure A procedure that is guaranteed to terminate and tell whether or not P is provable.

36. cse@buffalo September, 2004S. C. Shapiro36 Semidecision Procedure A procedure that, if P is a theorem is guaranteed to terminate and say so. Otherwise, it may not terminate.

37. cse@buffalo September, 2004S. C. Shapiro37 A Tour of Some Classes of Logics Propositional Logics Elementary Predicate Logics Full First-Order Logics

38. cse@buffalo September, 2004S. C. Shapiro38 Propositional Logics Smallest Unit: Proposition/Sentence  propositional logics that are Sound Complete Have decision procedures

39. cse@buffalo September, 2004S. C. Shapiro39 What You Can Do with Propositional Logic BettyDrivesTom  TomDrivesBetty BettyDrivesTom  NearTomBetty TomDrivesBetty  NearTomBetty  NearTomBetty Can derive conclusions even though the “facts” aren’t entirely known.

40. cse@buffalo September, 2004S. C. Shapiro40 Elementary Predicate Logics Propositions plus Predicate (Relation) symbols, Individual terms, variables, quantifiers  elementary predicate logics that are Sound Complete Have decision procedures

41. cse@buffalo September, 2004S. C. Shapiro41 What You Can Say with Elementary Predicate Logic  x[Elephant(x)  HasA(x, trunk)] Can state generalities before all individuals are known.  x[Elephant(x)  Color(x, white)] Can describe individuals Even when they are not specifically known.

42. cse@buffalo September, 2004S. C. Shapiro42 Full First-Order Logics Elementary predicate logic plus Function symbols/ functional terms  full first-order logics that are Sound None are Complete Have decision procedures

43. cse@buffalo September, 2004S. C. Shapiro43 What You Can Say with Full First-Order Logic  p[HasProp(0, p)  x[HasProp(x, p)  HasProp(x+1, p)]   x HasProp(x, p)] Principle of induction.

44. cse@buffalo September, 2004S. C. Shapiro44 Example of Undecidability Large KB about ducks, etc.  x[  y (Duck(y)  WalksLike(x,y))   y (Duck(y)  TalksLike(x,y))  Duck(x)]  x Duck(motherOf(x))  Duck(x) Duck(Fred)? If Fred is not a duck, possible ∞ loop.

45. cse@buffalo September, 2004S. C. Shapiro45 Unsound Reasoning Induction From Raven(a)  Black(a) Raven(b)  Black(b)  Raven(c)  Black(c)  Raven(d)   Black(d) … Raven(n)  Black(n) To  x[Raven(x)  Black(x)]

46. cse@buffalo September, 2004S. C. Shapiro46 Unsound Reasoning Abduction From  x[Person(x)  Injured(x)  Bandaged(x)] Person(Tom) Bandaged(Tom) To Injured(Tom)

47. cse@buffalo September, 2004S. C. Shapiro47 What’s “First-Order” about First-Order Logics Can’t quantify over Function symbols Predicate symbols Propositions

48. cse@buffalo September, 2004S. C. Shapiro48 Examples of SNePS Reasoning Using a Logic Designed for KRR

49. cse@buffalo September, 2004S. C. Shapiro49 SNePS, A “Higher-Order” Logic : all(R)(Transitive(R) => (all(x,y,z)(R(x,y) and R(y,z) => R(x,z)))). : Bigger(elephants, lions). : Bigger(lions, mice). : Transitive(Bigger). : Bigger(elephants, mice)? Bigger(elephants,mice) Really a higher-order language for a first-order logic

50. cse@buffalo September, 2004S. C. Shapiro50 “Higher-Order” Example 2 : all(source)(Trusted(source) => all(p)(Says(source, p) => p)). : Trusted(Agent007). : Says(Agent007, Dangerous(Dr_No)). : Dangerous(Dr_No)? Dangerous(Dr_No)

51. cse@buffalo September, 2004S. C. Shapiro51 Designing New Connectives : andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}. : OnFloor(G1). : OnFloor(?where)? ~OnFloor(G2) ~OnFloor(1) ~OnFloor(2) OnFloor(G1)

52. cse@buffalo September, 2004S. C. Shapiro52 Belief Change : andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}. : {OnFloor(G2), OnFloor(G1)} => {Location(belowGround)}. : {OnFloor(1), OnFloor(2)} => {Location(aboveGround)}. : perform believe(OnFloor(G2)) : Location(?where)? Location(belowGround) : perform believe(OnFloor(2)) : Location(?where)? Location(aboveGround)

53. cse@buffalo September, 2004S. C. Shapiro53 Summary 1 Symbolic KRR uses logic. There are many logics. The question is which to use.

54. cse@buffalo September, 2004S. C. Shapiro54 Summary 2 A logic has a Syntax Semantics Proof Theory Logics may Be sound Be complete Have a decision procedure

55. cse@buffalo September, 2004S. C. Shapiro55 Summary 3 Logics provide non-atomic wffs That can describe situations Without knowing all specifics

56. cse@buffalo September, 2004S. C. Shapiro56 Summary 4 One can design and build Useful new logics And reasoning systems using them.