A Logic of Arbitrary and Indefinite Objects

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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_at_cse.buffalo.edu
• http//www.cse.buffalo.edu/shapiro/

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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.

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Collaborators

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

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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)

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Outline of Talk

• Introduction and Motivations
• Informal Introduction to LA
• with Examples
• Examples of Proof Theory
• Implementation as Logic of SNePS 3

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Basic Idea

• Arbitrary Terms
• (any x R(x))
• Indefinite Terms
• (some x (y1 yn) R(x))

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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))

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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)

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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))

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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))

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Motivation 2Locality of Phrases

• Every elephant has a trunk.
• Logical Form,
• or FOL with complex terms (LC)
• Has(lt?x Elephant(x)gt, lt?yTrunk(y)gt)
• LA
• Has(any x Elephant(x), some y (x) Trunk(y))

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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.)

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Motivation 4Structure Sharing

• Every elephant has a trunk. Its flexible.
• Quantified terms are conceptually complete.
• Fixed semantics (forthcoming).

Has( , )
Flexible( )
some y ( ) Trunk(y)
any x Elephant(x)
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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
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Outline of Talk

• Introduction and Motivations
• Informal Introduction to LA
• with Examples
• Examples of Proof Theory
• Implementation as Logic of SNePS 3

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Quantified Terms

• Arbitrary terms
• (any x R(x))
• Indefinite terms
• (some x (y1 yn) R(x))

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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.
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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))

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Capture
free
bound

• White(any x Sheep(x))
  Black(x)
• White(any x Sheep(x)) ? Black(x)

same
Quantifiers take wide scope!
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Examples of Dependency

• Has(any x Elephant(x), some(y (x) Trunk(y))
• Every elephant has (its own) trunk.
• (any x Number(x)) lt (some y (x) Number(y))
• Every number has some number bigger than it.
• (any x Number(x)) lt (some y ( ) Number(y))
• Theres a number bigger than every number.

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Closure

• ?x ? contains the scope of x
• Compatibility and capture rules
• only apply within closures.

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Closure and Negation

• ?White(any x Sheep(x))
• Every sheep is not white.
• ? ?x White(any x Sheep(x)) ?
• It is not the case that every sheep is white.
• White(some x () Sheep(x))
• Some sheep is not white.
• ?x White(some x () Sheep(x)) ?
• No sheep is white.

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Closure and Capture

• Odd(any x Number(x)) ? Even(x)
• Every number is odd or even.
• ?x Odd(any x Number(x)) ?
• ? ?x Even(any x Number(x)) ?
• Every number is odd or every number is even.

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Tricky SentencesDonkey Sentences

• Every farmer who owns a donkey beats it.
• Beats(any x Farmer(x)
• ? Owns(x, some y (x)
  Donkey(y)),
• y)

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Tricky SentencesBranching 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))

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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 isnt a spy.
• Believes(Mike, ?Spy(some x ( ) Person(x))
• Mike believes that no one is a spy.
• Believes(Mike, ? ?xSpy(some x ( ) Person(x)?)

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Outline of Talk

• Introduction and Motivations
• Informal Introduction to LA
• with Examples
• Examples of Proof Theory
• Implementation as Logic of SNePS 3

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Proof TheoryanyE (abbreviated)

• From B(any x A(x))
• and A(a)
• conclude B(a)

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Proof TheoryanyI (abbreviated)

• From A(a) as Hyp
• and derive B(a)
• Conclude B(any x A(x))

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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
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Example Proof

1. Person(any x Woman(x))
2. Professional(any y Doctor(y))
3. Busy(some u (v) childOf(u, any v
   Person(v) ? Professional(any w
   sonOf(w,v))))
4. Woman(a) Hyp
5. Doctor(any z sonOf(z,a)) Hyp
6. Person(a) anyE,1,4
7. Professional(any z sonOf(z,a)) anyE,2,6
8. Busy(some u ( ) childOf(u,a)) anyE3,6?7
9. Busy(some u (v) childOf(u, any v Woman(v)
   ? Doctor(any w sonOf(w,v)))) anyI,4?
   58 QED

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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))

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Syllogistic Reasoningas Subsumption(Derived
Rules of Inference)

• Darii
• From A(any x B(x))
• and C(some y f B (y))
• conclude A(some y f C(y))

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Outline of Talk

• Introduction and Motivations
• Informal Introduction to LA
• with Examples
• Examples of Proof Theory
• Implementation as Logic of SNePS 3

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Current Implementation Status

• Partially implemented as the logic of SNePS 3

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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)
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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.

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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.

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Closure

• Contains wide-scoping of quantified terms

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Implementation Status

• Partially implemented as the logic of SNePS 3

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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.h
  tml