SNePS: A Logic for Natural Language Understanding and Commonsense Reasoning

1
SNePS A Logic for Natural Language Understanding
and Commonsense Reasoning

• Stuart C. Shapiro
• Department of Computer Science and Engineering
• and Center for Cognitive Science
• State University of New York at Buffalo
• shapiro_at_cse.buffalo.edu

2
Based on

• Stuart C. Shapiro, SNePS A Logic for Natural
  Language Understanding and Commonsense
  Reasoning,
• in Lucja Iwanska and Stuart C. Shapiro, Eds.,
  Natural Language Processing and Knowledge
  Representation Language for Knowledge and
  Knowledge for Language,
• AAAI Press/The MIT Press, 2000.

3
Abstract

• Design a better logic
• Than FOPL
• For NLU
• and CSR
• SNePS

4
Presentation Approach

• Problem
• Difficulties of FOPL
• Solution in SNePS
• Use SNePSLOG.

5
Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

6
Twenty Questions

• Everything is an animal, a vegetable, or a
  mineral.
• Squash is a vegetable
• Is squash an animal?
• a mineral?
• Marble is neither an animal nor a vegetable.
• Is marble a mineral?

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Twenty Questions in FOPL?

• ?xAnimal(x) ? Vegetable(x) ? Mineral(x)
• but dont want inclusive or
• ?xAnimal(x) Vegetable(x) Mineral(x)

T
T
T
F
T
So dont want exclusive or either
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andor

• andor(i, j)Pi, ..., Pn
• True iff at least i, and at most j of the Pi are
  True

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Twenty Questions in SNePSLOG

• all(x)(andor(1,1)animal(x),
• vegetable(x), mineral(x)).
• vegetable(squash)!
• VEGETABLE(SQUASH)
• ANIMAL(SQUASH)
• MINERAL(SQUASH)

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Twenty Questions in SNePSLOG II

• andor(0,0)animal(marble),
• vegetable(marble)!
• ANIMAL(MARBLE)
• VEGETABLE(MARBLE)
• MINERAL(MARBLE)

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Equivalent Statements

• For every object, the following statements are
  equivalent
• It is human. It is a featherless biped. It
  is a rational animal.
• Socrates is human.
• Is Socrates a featherless biped? A rational
  animal?
• Snoopy is not a featherless biped.
• Is Snoopy a rational animal? A human?

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Equivalent Statements in FOPL?

• ?xHuman(x) ? Featherless-Biped(x)
• ?Rational-Animal(x)
• wrong
• F ? F ? T

T
T
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thresh

• thresh(i, j)Pi, ..., Pn
• True iff either fewer than i,
• or more than j
• of the Pi are True
• Note thresh(i, j)? andor(i, j)

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thresh Abbreviation

• thresh(i)Pi, ..., Pn
• for
• thresh(i, n-1)Pi, ..., Pn

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Equivalent Statementsin SNePSLOG 1

• all(x)(thresh(1)human(x),
• featherless-biped(x),
• rational-animal(x)).
• human(Socrates)!
• HUMAN(SOCRATES)
• FEATHERLESS-BIPED(SOCRATES)
• RATIONAL-ANIMAL(SOCRATES)

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Equivalent Statementsin SNePSLOG 2

• featherless-biped(Snoopy)!
• RATIONAL-ANIMAL(SNOOPY)
• FEATHERLESS-BIPED(SNOOPY)
• HUMAN(SNOOPY)

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Putative Inclusive or

• If Hilda is in Boston or Kathy is in Las Vegas,
  then Eve is in Providence.
• What if Hilda is in Boston and Kathy is in Las
  Vegas?
• Rips 1983 ___ ? ___ ? ___

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or-entailment

• Pi, ..., Pn vgt Qi, ..., Qn
• True iff for all i, j Pi ? Qj

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Hilda and Kathy in SNePSLOG

• in(Hilda, Boston),
• in(Kathy, Las_Vegas)
• vgt in(Eve, Providence).
• in(Hilda, Boston)!
• Since IN(HILDA,BOSTON),IN(KATHY,LAS_VEGAS) vgt
  IN(EVE,PROVIDENCE)
• and IN(HILDA,BOSTON)
• I infer IN(EVE,PROVIDENCE)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Disappointed Voter Problem

• If someone votes for X and someone votes for Y,
  one of them will be disappointed
• all(u,v,x,y)(
• votesfor(u,x), votesfor(v,y)
• gt
• andor(1,1)disappointed(u),
• disappointed(v)).
• all(u,x)(votesfor(u,x),wins(x)
• gt disappointedu).

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Hillary and Elizabeth Vote

• votesfor(Hillary, Bill).
• votesfor(Elizabeth, Bob).
• wins(Bill).
• disappointed(?x)?
• DISAPPOINTED(ELIZABETH)
• DISAPPOINTED(HILLARY)

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FOPL Disappointment

• votesfor(Hillary, Bill),
• votesfor(Hillary, Bill)
• gt
• andor(1,1)
• disappointed(Hillary),
• disappointed(Hillary)
• ? disappointed(Hillary)

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Unique Variable Binding Rule (UVBR)

• Two variables in one wff cannot be replaced by
  the same term.
• or
• Two terms in an mgu cannot be equal.

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Sisters

• Mary, Sue, and Sally are sisters.

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Sisters in FOPL

• Sisters(Mary, Sue) ? Sisters(Sue, Sally)
• ?(x,y)Sisters(x,y) ? Sisters(y,x)
• ?(x,z)x ? z
• ? (?(y)sisters(x,y) ? sisters(y,z)
• ? Sisters(x,z))

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Reduction Inference
P(s1,...,si,?,si1,...,sm), ?? ?
P(s1,...,si,?,si1,...,sm)
P(s1,...,si,t1,,tn,si1,...,sm)
P(s1,...,si,ti,si1,...,sm)
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Sisters in SNePSLOG

• sisters(Mary, Sue, Sally).
• all(x,y)(sisters(x,y) gt likes(x,y),
  likes(y,x)).
• likes(?x,?y)?
• LIKES(SUE,MARY)
• LIKES(MARY,SUE)
• LIKES(SALLY,SUE)
• LIKES(SUE,SALLY)
• LIKES(MARY,SALLY)
• LIKES(SALLY,MARY)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Inadequacy of FOPL 1

• If R is a transitive relation
• and R(x, y) and R(y, z)
• then R(x, z).

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Term Logic

• Every expression in SNePS is a term.

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SNePSLOG Transitivity Rule

• all(R)(Transitive(R) gt
• all(x,y,z)(R(x,y), R(y,z)
• gt R(x,z))).
• Transitive(bigger).
• bigger(elephant,lion).
• bigger(lion,mouse).
• bigger(elephant,mouse)?
• BIGGER(ELEPHANT,MOUSE)

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Inadequacy of FOPL 2

• Everything Bob believes is true.
• Bob believes everything Bill believes.
• Bill believes Kevins favorite proposition.
• Kevins favorite proposition is that John is
  taller than Mary.
• Is John taller than Mary?

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Quantifying Over Propositions

• all(p)(Believes(Bob, p) gt p).
• all(p)(Believes(Bill, p) gt
  Believes(Bob, p)).
• all(p)
• (Favorite-proposition(Kevin, p) gt
  Believes(Bill, p)).
• Favorite-proposition(Kevin, Taller(John,
  Mary)).
• Taller(John, Mary)?
• TALLER(JOHN,MARY)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Opaque Contexts

• George IV wished to know whether Scott was the
  author of Waverly
• Russell 1906

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Intensional Representation

• Every SNePS term represents (denotes) an
  intensional (mental) entity.

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Uniqueness Principle

• No two SNePS terms denote the same entity.

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McCarthys Telephone Number Problem

• all(R)(Transparent(R)
• gt all(a,x,y)(R(a,x), (x,y)
• gt R(a,y))).
• Transparent(Dial).
• (Telephone(Mike), Telephone(Mary)).
• Know(Pat, Telephone(Mike)).
• Dial(Pat, Telephone(Mike)).

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Correct Answer to Telephone Number Problem

• ?what(Pat, ?which)?
• DIAL(PAT,TELEPHONE(MARY))
• KNOW(PAT,TELEPHONE(MIKE))
• DIAL(PAT,TELEPHONE(MIKE))

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Reasoning by Elimination

• No one has more than one mother.
• Jane is John's mother.
• Is Mary John's mother?
• The committee members are Chris, Leslie, Pat, and
  Stevie. At least two of them are women.
• Leslie and Stevie are men.
• Is Pat a man or a woman?

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Numerical Quantifiers

• nexists(i,j,k)(x)
• (P1(x),..., Pn(x) Q(x))
• There are k individuals that satisfy
• P1(x) ?...? Pn(x)
• and, of them, at least i and at most j also
  satisfy
• Q(x)

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Numerical QuantifierRule of Inference 1

• If j individuals are known that satisfy
• P1(x) ? ? Pn(x) ? Q(x)
• then every other individual that satisfies
• P1(x) ?...? Pn(x)
• also satisfies
• Q(x)

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Numerical QuantifierRule of Inference 2

• If k-i individuals are known that satisfy
• P1(x) ? ? Pn(x) ? Q(x)
• then every other individual that satisfies
• P1(x) ? ? Pn(x)
• also satisfies
• Q(x)

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Numerical Quantifier Forms

• nexists(i,j,k)
• nexists(_,j,_)
• nexists(i,_,k)

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Reasoning by Eliminationin SNePSLOG 1

• all(x)(Person(x) gt nexists(_,1,_)(y)(Person(y)
  Mother(y,x))).
• Person(John, Jane, Mary).
• Mother(Jane, John).
• Mother(Mary, John)?
• MOTHER(MARY,JOHN)

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Reasoning by Eliminationin SNePSLOG 2

• Member(Chris, Leslie, Pat, Stevie).
• nexists(2,_,4)(x) (Member(x) Woman(x)).
• all(x)(Member(x) gt andor(1,1)Man(x),
  Woman(x)).

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Committee Problem Answer

• Man(Leslie, Stevie).
• ?What(Pat)?
• WOMAN(PAT)
• MEMBER(PAT)
• MAN(PAT)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Contexts

• Pegasus is a winged horse in mythology
• But a normal horse in the real world.
• So Bellerophon travels by air in mythology
• but by ground in the real world.

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Pegasus in the Real and Mythological Worlds

• set-context real-world ()
• set-default-context real-world
• all(x)(Bird(x), Beast(x) vgt
  Animal(x)).
• all(x)(Animal(x) gt andor(1,1)Bird(x),
  Beast(x)).
• all(x)(Horse(x) gt Beast(x)).
• Horse(Pegasus).

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Bellerophon in the Real and Mythological Worlds

• all(x,y)(Rides(x,y) gt andor(1,1)Travelsby(x
  , air), Travelsby(x, ground)).
• all(x,y)(Rides(x,y) gt thresh(1,1)Winged(y),
  Travelsby(x, air)).
• Rides(Bellerophon, Pegasus).

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Initialize Mythology

• describe-context ((ASSERTIONS (WFF1 WFF2
  WFF3 WFF4 WFF5 WFF6 WFF7)) (RESTRICTION
  NIL) (NAMED (REAL-WORLD)))
• set-context mythology (WFF1 WFF2 WFF3 WFF4
  WFF5 WFF6 WFF7)

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Riding in the Real World

• all(x)(Winged(x) ltgt Bird(x)).
• Travelsby(Bellerophon,?what)?
• TRAVELSBY(BELLEROPHON,GROUND)
• TRAVELSBY(BELLEROPHON,AIR)

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Riding in Mythology

• set-default-context mythology
• Winged(Pegasus).
• Travelsby(Bellerophon,?what)?
• TRAVELSBY(BELLEROPHON,GROUND)
• TRAVELSBY(BELLEROPHON,AIR)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Relevance LogicA Paraconsistent Logic

• A contradiction should not imply anything
  whatsoever.

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Opus Flies and Doesnt

• all(x)(Flies(x)gtFeathered(x)).
• all(x)(Flies(x) gt Swims(x)).
• Flies(Opus).
• Flies(Opus).
• A contradiction was detected within context
  DEFAULT-DEFAULTCT.
• ...
• FLIES(OPUS)

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Opus is Feathered and Swims, but the Earth isnt
Flat

• Feathered(Opus)?
• FEATHERED(OPUS)
• Swims(Opus)?
• SWIMS(OPUS)
• Flat(Earth)?

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The Inconsistent Belief Space

• list-asserted-wffs
• all(X)(FLIES(X) gt FEATHERED(X))
• all(X)((FLIES(X)) gt SWIMS(X))
• FLIES(OPUS)
• FLIES(OPUS)
• FEATHERED(OPUS)
• SWIMS(OPUS)
• but the earth isnt flat.

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Information Provided by Real People

• is circular
• and left- and right-recursive

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Circular Definitions

• all(x,y)(thresh(1,1) North-of(x,y),
  South-of(y,x)).
• North-of(Seattle, Portland).
• South-of(San_Francisco, Portland).
• North-of(San_Francisco, Los_Angeles).
• South-of(San_Diego, Los_Angeles).

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Using a Circular Definition

• North-of(?x, ?y)?
• NORTH-OF(SEATTLE,PORTLAND)
• NORTH-OF (SAN_FRANCISCO,LOS_ANGELES)
• NORTH-OF(LOS_ANGELES,SAN_DIEGO)
• NORTH-OF(PORTLAND,SAN_FRANCISCO)

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Recursive Rules

• all(x,y)(parent(x,y) gt ancestor(x,y)).
• all(x,y,z)(ancestor(x,y), ancestor(y,z)
  gt ancestor(x,z)).
• parent(John, Mary).
• ancestor(Mary, George).
• ancestor(George, Sally).
• parent(Sally, Jimmy).

68
Using a Recursive Rule

• ancestor(John, ?y)?
• ANCESTOR(JOHN,JIMMY)
• ANCESTOR(JOHN,MARY)
• ANCESTOR(JOHN,GEORGE)
• ANCESTOR(JOHN,SALLY)

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In the Other Direction

• ancestor(?x, Jimmy)?
• ANCESTOR(SALLY,JIMMY)
• ANCESTOR(JOHN,JIMMY)
• ANCESTOR(GEORGE,JIMMY)
• ANCESTOR(MARY,JIMMY)

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Outline

• Set-Oriented Logical Connectives
• The Unique Variable Binding Rule
• Set Arguments
• Higher-Order Logic
• Intensional Representation
• The Numerical Quantifiers
• Contexts
• Relevance Logic
• Circular and Recursive Rules
• Practical Donkey Sentences

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Practical Donkey Sentences

• Donkey Sentence
• Every farmer who owns a donkey beats it.
• Practical Donkey Sentence
• If you have a quarter in your pocket, put it in
  the meter.
• Lenhart Schubert

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FOPL Practical Sentence

• ?(x)Quarter(x) ? In(x, pocket) ?
  PutIn(x, meter)
• puts all quarters in pocket in the meter
• ?(x) )Quarter(x) ? In(x, pocket) ?
  PutIn(x, meter)
• last x unbound
• ?(x) )Quarter(x) ? In(x, pocket)
  ??(x) )Quarter(x) ? In(x, pocket) ?
  PutIn(x, meter)
• Finds one, may put in another

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Withsome

• withsome(var, suchthat, do,else)
• Term denoting an act
• if there is some a satisfying suchthata/x
• perform doa/x
• else perform else.

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Practical Sentence in SNePSLOG

• (define-primaction Putin (object
  place) (format t "I am putting
  A in the A. object place))

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Practical Sentence in SNePSLOG 2

• Nickel(N1).
• Nickel(N2).
• Quarter(Q1).
• Quarter(Q2).
• Quarter(Q3).
• In(N1, pocket).
• In(Q1, pocket).
• In(Q2, pocket).

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Practical Sentence in SNePSLOG 3

• perform
• withsome(?x,
• Quarter(?x) and In(?x, pocket),
• Putin(?x, meter))
• I am putting Q1 in the METER.

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Summary

• SNePS Logic has been designed specifically to
  support Natural Language Understanding and
  CommonSense Reasoning.
• The aspects of that design fall under several
  categories

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Basic SNePS Principles

• Term Logic
• Every expression in SNePS is a term.
• Intensional Representation
• Every SNePS term represents (denotes) an
  intensional (mental) entity.
• Uniqueness Principle
• No two SNePS terms denote the same entity.

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Specialized Syntax

• Term Logic
• Set Arguments
• Set-Oriented Logical Connectives
• Numerical Quantifiers
• Higher-Order Logic
• Acts

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Specialized Inference Rules

• The Unique Variable Binding Rule
• Relevance Logic

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Specialized Semantics

• Intensional Representation
• Uniqueness Principle

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Specialized Inference Mechanism

• Contexts
• Belief Revision
• Use of circular and recursive rules
• Integrated reasoning and acting

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For More Information

• Personnel
• Manual
• Tutorial
• Bibliography
• ftpable SNePS source code
• etc.
• http//www.cse.buffalo.edu/sneps/