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Symbolic Representation and Reasoning an Overview

Symbolic Representation and Reasoning an Overview

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Symbolic Representation and Reasoning an Overview

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  1. SymbolicRepresentation and Reasoningan 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. Introduction • Knowledge Representation • Reasoning • Symbols • Logics S. C. Shapiro

  3. 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. S. C. Shapiro

  4. 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 S. C. Shapiro

  5. 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. S. C. Shapiro

  6. 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] S. C. Shapiro

  7. 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. S. C. Shapiro

  8. Parts of Specifying a Logic • Syntax • Semantics • Proof Theory S. C. Shapiro

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

  10. 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 S. C. Shapiro

  11. 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. S. C. Shapiro

  12. Semantic Values • Terms could denote • Objects • Categories of objects • Properties… • Wffs could denote • Propositions • Truth values S. C. Shapiro

  13. 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) S. C. Shapiro

  14. 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. S. C. Shapiro

  15. 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 S. C. Shapiro

  16. 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: , , ,  S. C. Shapiro

  17. 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 S. C. Shapiro

  18. Extensional (Denotational) Semantics 5 of 26 = 64 possible situations S. C. Shapiro

  19. Properties of WffsSatisfiableT in some situation S. C. Shapiro

  20. Properties of WffsContingentT in some, F in some S. C. Shapiro

  21. Properties of WffsValidT in all situations S. C. Shapiro

  22. Properties of WffsContradictoryT in no situation S. C. Shapiro

  23. Logical Implication P1, …, Pn logically imply Q P1, …, Pn |= Q In every situation that P1, …, Pn are True, so is Q. S. C. Shapiro

  24. Example: CarPool World KB Let KBCPW = Bd  Bp Td  Tp BdT  Bd  Tp TdB  Td  Bp TdB  BdT S. C. Shapiro

  25. Extensional (Denotational) Semantics Only 2 of the 64 situations where KBCPW are T So, e.g., KBCPW, BdT |= Bd  Bp This is how a KB constrains a model to the domain we want. S. C. Shapiro

  26. Proof TheorySome Rules of Inference P P  Q P  Q P Q Q Modus Ponens or  Elimination Elimination P  Q Q P P P  Q Elimination Introduction S. C. Shapiro

  27. Derivation from Assumptions Q is derivable from P1, …, Pn P1, …, Pn |- Q Starting from the collection P1, …, Pn, one can repeatedly apply rules of inference, and eventually get Q. S. C. Shapiro

  28. Example: CarPool World Proof BdT  Bd  Tp Bd  Bp BdT Bd  Tp Bd Bp Bd  Bp So, KBCPW, BdT |- Bd  Bp S. C. Shapiro

  29. Theoremhood If Q is derivable from no assumptions, |- Q We say that Q is provable, and that Q is a theorem. S. C. Shapiro

  30. Deduction Theorem P1, …, Pn |= Q iff |= (P1  · · ·  Pn )  Q P1, …, Pn |- Q iff |- (P1  · · ·  Pn )  Q So theorem-proving is relevant to reasoning. S. C. Shapiro

  31. 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.) S. C. Shapiro

  32. Soundness vs. Completeness • Soundness is the essence of correct reasoning • Completeness is less important because it doesn’t indicate how long it might take. S. C. Shapiro

  33. Commutativity DiagramforSound and Complete Logics |= (P1  · · ·  Pn )  Q P1, …, Pn |= Q completeness soundness soundness completeness |- (P1  · · ·  Pn )  Q P1, …, Pn |- Q So, whenever you want one, you can do another. S. C. Shapiro

  34. Use of Commutativity Diagram Refutation proof techniques, such as resolution refutation or semantic tableaux, prove that there can be no situation in which P1, …, and Pnare True and Q is False. These are semantic proof techniques. S. C. Shapiro

  35. Decision Procedure • A procedure that is guaranteed • to terminate • and tell whether or not P is provable. S. C. Shapiro

  36. Semidecision Procedure • A procedure that, if P is a theorem • is guaranteed • to terminate • and say so. • Otherwise, it may not terminate. S. C. Shapiro

  37. A Tour ofSome Classes of Logics • Propositional Logics • Elementary Predicate Logics • Full First-Order Logics S. C. Shapiro

  38. Propositional Logics • Smallest Unit: Proposition/Sentence •  propositional logics that are • Sound • Complete • Have decision procedures S. C. Shapiro

  39. What You Can Dowith Propositional Logic • BettyDrivesTom  TomDrivesBetty • BettyDrivesTom  NearTomBetty • TomDrivesBetty  NearTomBetty •  NearTomBetty Can derive conclusions even though the “facts” aren’t entirely known. S. C. Shapiro

  40. Elementary Predicate Logics • Propositions plus • Predicate (Relation) symbols, • Individual terms, variables, quantifiers •  elementary predicate logics that are • Sound • Complete • Have decision procedures S. C. Shapiro

  41. What You Can Say withElementary 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. S. C. Shapiro

  42. 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 S. C. Shapiro

  43. What You Can Say withFull First-Order Logic p[HasProp(0, p) x[HasProp(x, p)  HasProp(x+1, p)]  x HasProp(x, p)] Principle of induction. S. C. Shapiro

  44. 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. S. C. Shapiro

  45. Unsound ReasoningInduction 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)] S. C. Shapiro

  46. Unsound ReasoningAbduction From x[Person(x)  Injured(x)  Bandaged(x)] Person(Tom) Bandaged(Tom) To Injured(Tom) S. C. Shapiro

  47. What’s “First-Order” aboutFirst-Order Logics • Can’t quantify over • Function symbols • Predicate symbols • Propositions S. C. Shapiro

  48. Examples ofSNePS Reasoning Using a Logic Designed for KRR S. C. Shapiro

  49. 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 S. C. Shapiro

  50. “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) S. C. Shapiro