A Logic of Arbitrary and Indefinite Objects

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

3. 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

4. Outline of Paper • Introduction and Motivations • Introduction to Arbitrary Objects • Informal Introduction to LA • Formal Syntax of LA • Translations Between and LA Standard FOL • Semantics of LA • Proof Theory of A • Soundness & Completeness Proofs • Subsumption Reasoning in LA • MRS and LA • Implementation Status S. C. Shapiro

5. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with examples S. C. Shapiro

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

7. Motivations • See paper for other logics that each satisfy some of these motivations S. C. Shapiro

8. Motivation 1Uniform Syntax • Standard FOL: White(Dolly) x(Sheep(x)  White(x)) x(Sheep(x)  White(x)) • LA: White(Dolly) White(any x Sheep(x)) White(some x ( ) Sheep(x)) S. C. Shapiro

9. Motivation 2Locality of Phrases • Every elephant has a trunk. • Standard FOL x(Elephant(x)  y(Trunk(y)  Has(x,y)) • LA: Has(any x Elephant(x), some y (x) Trunk(y)) S. C. Shapiro

10. 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

11. 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

12. 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

13. Outline of Talk • Introduction and Motivations • Informal Introduction to LA with examples S. C. Shapiro

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

15. 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

16. 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

17. 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

18. 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

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

20. 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

21. 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

22. 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

23. 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

24. 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

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

26. 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

27. 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

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