410 likes | 510 Views
A Logic of Arbitrary and 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
E N D
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/
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
Collaborators • Jean-Pierre Koenig • David R. Pierce • William J. Rapaport • The SNePS Research Group S. C. Shapiro
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
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
Basic Idea • Arbitrary Terms (any x R(x)) • Indefinite Terms (some x (y1 … yn) R(x)) S. C. Shapiro
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
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
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
Motivation 2Locality of Phrases Every elephant has a trunk. • Standard FOL x(Elephant(x) y(Trunk(y) Has(x,y)) • LR: xElephantyTrunk Has(x,y)) S. C. Shapiro
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
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
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
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
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
Quantified Terms • Arbitrary terms: (any x [R(x)]) • Indefinite terms: (some x ([y1 … yn]) [R(x)]) S. C. Shapiro
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
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
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
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
Closure x … contains the scope of x Compatibility and capture rules only apply within closures. S. C. Shapiro
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
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
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
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
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
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
Proof Theory:anyE (abbreviated) From B(any x A(x)) and A(a) conclude B(a) S. C. Shapiro
Proof Theory:anyI (abbreviated) From A(a) as Hyp and derive B(a) Conclude B(any x A(x)) S. C. Shapiro
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
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)) anyE3,67 • Busy(some u (v) childOf(u, any v Woman(v) Doctor(any w sonOf(w,v)))) anyI,45—8 QED S. C. Shapiro
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
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
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
Current Implementation Status Partially implemented as the logic of SNePS 3 S. C. Shapiro
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
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
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
Closure Contains wide-scoping of quantified terms S. C. Shapiro
Implementation Status Partially implemented as the logic of SNePS 3 S. C. Shapiro
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