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A Logic of Arbitrary and Indefinite Objects

This paper presents a logic, based on arbitrary and indefinite objects, that supports natural language understanding, generation, and commonsense reasoning in knowledge representation and reasoning (KRR) systems.

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A Logic of Arbitrary and Indefinite Objects

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  1. A Logic of Arbitraryand Indefinite Objects Stuart C. Shapiro Department of Computer Science and Engineering, 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. Based On Stuart C. Shapiro, A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004, 565-575. S. C. Shapiro

  3. Collaborators • Jean-Pierre Koenig • David R. Pierce • William J. Rapaport • The SNePS Research Group S. C. Shapiro

  4. What Is It? • A logic • For KRR systems • Supporting NL understanding & generation • And commonsense reasoning • LA • Sound & complete (via translation to Standard FOL) • Based on Arbitrary Objects, Fine (’83, ’85a, ’85b) • And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93) S. C. Shapiro

  5. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with Examples • Examples of Proof Theory • Implementation as Logic of SNePS 3 S. C. Shapiro

  6. Basic Idea • Arbitrary Terms (any x R(x)) • Indefinite Terms (some x (y1 … yn) R(x)) S. C. Shapiro

  7. Motivation 1Uniform Syntax • Standard FOL (Ls ): • Dolly is white. White(Dolly) • Every sheep is white. x(Sheep(x)  White(x)) • Some sheep is white. x(Sheep(x)  White(x)) S. C. Shapiro

  8. Motivation 1Uniform Syntax • FOL with Restricted Quantifiers (LR ): • Dolly is white. White(Dolly) • Every sheep is white. xSheep White(x) • Some sheep is white. xSheep White(x) S. C. Shapiro

  9. Motivation 1Uniform Syntax • LA : • Dolly is white. White(Dolly) • Every sheep is white. White(any x Sheep(x)) • Some sheep is white. White(some x ( ) Sheep(x)) S. C. Shapiro

  10. Motivation 2Locality of Phrases Every elephant has a trunk. • Standard FOL x(Elephant(x)  y(Trunk(y)  Has(x,y)) • LR: xElephantyTrunk Has(x,y)) S. C. Shapiro

  11. Motivation 2Locality of Phrases Every elephant has a trunk. • Logical Form, or FOL with “complex terms” (LC): Has(<x Elephant(x)>, <yTrunk(y)>) • LA: Has(any x Elephant(x), some y (x) Trunk(y)) S. C. Shapiro

  12. Motivation 3Prospects for Generalized Quantifiers • Most elephants have two tusks. • Standard FOL ?? • LA: Has(most x Elephant(x), two y Tusk(y)) (Currently, just notation.) S. C. Shapiro

  13. Motivation 4Structure Sharing • Every elephant has a trunk. It’s flexible. • Quantified terms are “conceptually complete”. • Fixed semantics (forthcoming). Has( , ) Flexible( ) some y ( ) Trunk(y) any x Elephant(x) S. C. Shapiro

  14. Motivation 5Term Subsumption Hairy(any x Mammal(x)) Mammal(any y Elephant(y)) • Hairy(any y Elephant(y)) Pet(some w () Mammal(w))  Hairy(some z () Pet(z)) Hairy Mammal Pet Elephant S. C. Shapiro

  15. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with Examples • Examples of Proof Theory • Implementation as Logic of SNePS 3 S. C. Shapiro

  16. Quantified Terms • Arbitrary terms: (any x [R(x)]) • Indefinite terms: (some x ([y1 … yn]) [R(x)]) S. C. Shapiro

  17. Compatible Quantified Terms (Q v ([a1 … an]) [R(v)]) (Q u ([a1 … an]) [R(u)]) (Q v ([a1 … an]) [R(v)]) (Q v ([a1 … an]) [R(v)]) different or same All quantified terms in an expression must be compatible. S. C. Shapiro

  18. Quantified Terms in an Expression Must be Compatible • Illegal: White(any x Sheep(x))  Black(any x Raven(x)) • Legal White(any x Sheep(x))  Black(any y Raven(y)) White(any x Sheep(x))  Black(any x Sheep(x)) S. C. Shapiro

  19. Capture free bound White(any x Sheep(x)) Black(x) White(any x Sheep(x))  Black(x) same Quantifiers take wide scope! S. C. Shapiro

  20. Examples of Dependency Has(any x Elephant(x), some(y (x) Trunk(y)) Every elephant has (its own) trunk. (any x Number(x)) < (some y (x) Number(y)) Every number has some number bigger than it. (any x Number(x)) < (some y ( ) Number(y)) There’s a number bigger than every number. S. C. Shapiro

  21. Closure x … contains the scope of x Compatibility and capture rules only apply within closures. S. C. Shapiro

  22. Closure and Negation White(any x Sheep(x)) Every sheep is not white.  xWhite(any x Sheep(x))  It is not the case that every sheep is white. • White(some x () Sheep(x)) Some sheep is not white. • xWhite(some x () Sheep(x))  No sheep is white. S. C. Shapiro

  23. Closure and Capture Odd(any x Number(x))  Even(x) Every number is odd or even. xOdd(any x Number(x))   xEven(any x Number(x))  Every number is odd or every number is even. S. C. Shapiro

  24. Tricky Sentences:Donkey Sentences Every farmer who owns a donkey beats it. Beats(any x Farmer(x)  Owns(x, some y (x) Donkey(y)), y) S. C. Shapiro

  25. Tricky Sentences:Branching Quantifiers Some relative of each villager and some relative of each townsman hate each other. Hates(some x (any v Villager(v)) Relative(x,v), some y (any u Townsman(u)) Relative(y,u)) S. C. Shapiro

  26. Closure & Nested Beliefs(Assumes Reified Propositions) There is someone whom Mike believes to be a spy. Believes(Mike, Spy(some x ( ) Person(x)) Mike believes that someone is a spy. Believes(Mike, xSpy(some x ( ) Person(x)) There is someone whom Mike believes isn’t a spy. Believes(Mike, Spy(some x ( ) Person(x)) Mike believes that no one is a spy. Believes(Mike,  xSpy(some x ( ) Person(x)) S. C. Shapiro

  27. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with Examples • Examples of Proof Theory • Implementation as Logic of SNePS 3 S. C. Shapiro

  28. Proof Theory:anyE (abbreviated) From B(any x A(x)) and A(a) conclude B(a) S. C. Shapiro

  29. Proof Theory:anyI (abbreviated) From A(a) as Hyp and derive B(a) Conclude B(any x A(x)) S. C. Shapiro

  30. Example Proof From Every woman is a person. Every doctor is a professional. Some child of every person all of whose sons are professionals is busy. Conclude Some child of every woman all of whose sons are doctors is busy. [Based on an example of W. A. Woods] S. C. Shapiro

  31. Example Proof • Person(any x Woman(x)) • Professional(any y Doctor(y)) • Busy(some u (v) childOf(u, any v Person(v)  Professional(any w sonOf(w,v)))) • Woman(a) Hyp • Doctor(any z sonOf(z,a)) Hyp • Person(a) anyE,1,4 • Professional(any z sonOf(z,a)) anyE,2,6 • Busy(some u ( ) childOf(u,a)) anyE3,67 • Busy(some u (v) childOf(u, any v Woman(v)  Doctor(any w sonOf(w,v)))) anyI,45—8 QED S. C. Shapiro

  32. Syllogistic Reasoningas Subsumption(Derived Rules of Inference) Barbara: From A(any x B(x)) and B(any y C(y)) conclude A(any y C(y)) S. C. Shapiro

  33. Syllogistic Reasoningas Subsumption(Derived Rules of Inference) Darii: From A(any x B(x)) and C(some y φB (y)) conclude A(some y φC(y)) S. C. Shapiro

  34. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with Examples • Examples of Proof Theory • Implementation as Logic of SNePS 3 S. C. Shapiro

  35. Current Implementation Status Partially implemented as the logic of SNePS 3 S. C. Shapiro

  36. SNePS 3 Example snepsul(25): #L#!(build object (any x (build member x class Mammal)) property hairy) Is((any Arb1 Isa(Arb1, Mammal)), hairy) snepsul(26): #L#!(build member (any y (build member y class Elephant)) class Mammal) Isa((any Arb2 Isa(Arb2, Elephant)), Mammal) snepsul(27): #L#?(build object (any y (build member y class Elephant)) property hairy) Is((any Arb2 Isa(Arb2, Elephant)), hairy) snepsul(28): #L#!(build member Clyde class Elephant) Isa(Clyde, Elephant) snepsul(29): #L#?(build object Clyde property hairy) Is(Clyde, hairy) S. C. Shapiro

  37. Summary • LA is • A logic • For KRR systems • Supporting NL understanding & generation • And commonsense reasoning • Uses arbitrary and indefinite terms • Instead of universally and existentially quantified variables. S. C. Shapiro

  38. Arbitrary & Indefinite Terms • Provide for uniform syntax • Promote locality of phrases • Provide prospects for generalized quantifiers • Are conceptually complete • Allow structure sharing • Support subsumption reasoning. S. C. Shapiro

  39. Closure Contains wide-scoping of quantified terms S. C. Shapiro

  40. Implementation Status Partially implemented as the logic of SNePS 3 S. C. Shapiro

  41. For More Information The SNePS Research Group web site: http://www.cse.buffalo.edu/sneps/ The SNePS 3 Project page: http://www.cse.buffalo.edu/sneps/Projects/sneps3.html S. C. Shapiro

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