A Logic of Arbitrary and Indefinite Objects - PowerPoint PPT Presentation

A logic of arbitrary and indefinite objects l.jpg
Download
1 / 28

  • 199 Views
  • Updated On :
  • Presentation posted in: Pets / Animals

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

Related searches for A Logic of Arbitrary and Indefinite Objects

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Presentation

A Logic of Arbitrary and Indefinite Objects

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


A logic of arbitrary and indefinite objects l.jpg

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/


Collaborators l.jpg

Collaborators

  • Jean-Pierre Koenig

  • David R. Pierce

  • William J. Rapaport

  • The SNePS Research Group

S. C. Shapiro


What is it l.jpg

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 paper l.jpg

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


Outline of talk l.jpg

Outline of Talk

  • Introduction and Motivations

  • Informal Introduction to LA

    with examples

S. C. Shapiro


Basic idea l.jpg

Basic Idea

  • Arbitrary Terms

    (any x R(x))

  • Indefinite Terms

    (some x (y1 … yn) R(x))

S. C. Shapiro


Motivations l.jpg

Motivations

  • See paper for other logics

    that each satisfy some of these motivations

S. C. Shapiro


Motivation 1 uniform syntax l.jpg

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


Motivation 2 locality of phrases l.jpg

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


Motivation 3 prospects for generalized quantifiers l.jpg

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 4 structure sharing l.jpg

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 5 term subsumption l.jpg

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 talk13 l.jpg

Outline of Talk

  • Introduction and Motivations

  • Informal Introduction to LA

    with examples

S. C. Shapiro


Quantified terms l.jpg

Quantified Terms

  • Arbitrary terms:

    (any x [R(x)])

  • Indefinite terms:

    (some x ([y1 … yn]) [R(x)])

S. C. Shapiro


Compatible quantified terms l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

Closure

x … contains the scope of x

Compatibility and capture rules

only apply within closures.

S. C. Shapiro


Closure and negation l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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


Current implementation status l.jpg

Current Implementation Status

Partially implemented as the logic of SNePS 3

S. C. Shapiro


Summary l.jpg

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 l.jpg

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


Closure28 l.jpg

Closure

Contains wide-scoping of quantified terms

S. C. Shapiro


  • Login