Natural Language Pro cessing Using a Prop ositional Semantic

Network with Structured Variables

Syed S Ali and Stuart C Shapiro

Department of Computer Science

State University of New York at Bualo

Bell Hall

Bualo NY

fsyali shapirogcsbualoedu

Abstract

We describ e a knowledge representation and inference formalism based on an intensional

prop ositional semantic network in which variables are structured terms consisting of quanti

er type and other information This has three imp ortant consequences for natural language

pro cessing First this leads to an extended more natural formalism whose use and rep

resentations are consistent with the use of variables in natural language in two ways the

structure of representations mirrors the structure of the language and allows reuse phenom

ena such as pronouns and ellipsis Second the formalism allows the sp ecication of description

subsumption as a partial ordering on related concepts variable no des in a semantic network

that relates more general concepts to more sp ecic instances of that concept as is done in

language Finally this structured variable representation simplies the resolution of some rep

resentational diculties with certain classes of natural language sentences namely donkey

sentences and sentences involving branching quantiers The implementation of this formal

ism is called ANALOG A NAtural Logic and its utility for natural language pro cessing tasks

is illustrated

Introduction

This work is based on the assumption that the kind of knowledge to b e represented and its

asso ciated goals have a profound eect on the design of a knowledge representation and reasoning

KRR system In particular we present a KRR system for the representation of knowledge

asso ciated with natural language dialog We will argue that this may b e done with minimal loss

of inferential p ower and will result in an enriched representation language capable of supp orting

complex natural language descriptions some discourse phenomena standard rstorder inference



This pap er will app ear in the journal Minds and Machines in a sp ecial issue on Knowledge Representation

for Natural Language Pro cessing It is also available as technical rep ort from the Department of Computer

Science SUNY at Bualo A p ostscript version is available for anonymous ftp from ftpcsbuffaloedu in the le

pubtechreportsps

inheritance and terminological subsumption We characterize our goals for a more natural logic

and its computational implementation in a knowledge representation and reasoning system b elow

The mapping from natural language sentences into the representation language should b e as

direct as p ossible The representation should reect the structure of the natural language NL

sentence it purp orts to represent This is particularly evident in rule sentences such as small

dogs bite harder than big dogs where the representation takes the form of an implication

x y smallx dogx largey dogy bitesharder x y

This is in contrast with the predicateargument structure of the original sentence By comparison

the representation of Fido bites harder than Rover is more consistent with the structure of the

original sentence

bitesharderFido Rover

This is so despite the intuitive observation that the two sentences have nearly identical syntactic

structure and similar meaning

The subunits of the representation should b e what we term conceptually complete By this we

mean that any comp onent of the representation of a sentence should have a meaningful interpre

tation indep endent of the entire sentence For example for the representation of the sentence as

in ab ove we might ask what the meaning of x or y is Presumably some thing in the world

Note that the original sentence mentions only dogs We suggest that a b etter translation might

b e

bitesharder all x such that smalldogx all y such that largedogy

where the variables x and y would have their own internal structure that reects their conceptual

ization Note that we are suggesting something stronger than just restricted quantication simple

type constraints can certainly b e felicitously represented using restricted quantiers Complex

internalized constraints that is other than simple type and internalized quantier structures

characterize this approach to the representation of variables Thus representation of the sentence

Every smal l dog that is owned by a badtempered person bites harder than a large dog should reect

the structure of the representation of

A high degree of structure sharing should b e p ossible In language multisentence connected

discourse often uses reduced forms of previously used terms in subsequent reference to those terms

This can b e reected in the representation language by structure sharing and corresp onds to the

use of pronouns and some forms of ellipsis in discourse An example of this phenomenon is the

representation of intersentential pronominal reference to scop ed terms e g

Every apartment had a dishwasher In some of them it had just b een installed

Every chess set comes with a spare pawn It is tap ed to the top of the b ox

examples from The structures that are b eing shared in these sentences are the variables

corresp onding to the italicized noun phrases Logical representations can only mo del this sharing

by combining multiple sentences of natural language into one sentence of logic This metho d is

unnatural for at least two reasons First when several sentences must b e combined into one

sentence the resulting logical sentence as a conjunction of several p otentially disparate sentences

is overly complex Second this approach is counter intuitive in that a language user can re

articulate the original sentences that heshe represents This argues for some form of separate

representations of the original sentences The problem with logic in this task is that logic requires

the complete sp ecication of a variable corresp onding to a noun phrase and its constraints in the

scop e of some quantier This diculty is not restricted to noun phrases indeed it is frequently

the case that entire sub clauses of sentences are referred to using reduced forms such as to o e g

John went to the party Mary did too

A languagemotivated knowledge representation formalism should mo del this sort of reference

minimally by structure sharing

Finally collections of logical formulas do not seem to capture the intuitive use of concepts

by p eople This representation for knowledge is unstructured and disorganized What is missing

is that rstorder predicate logic do es not provide any sp ecial assistance in the problem of what

Brachman called knowledge structuring that is the sp ecication of the internal structure of

concepts in terms of roles and interrelations b etween them and the inheritance relationships b e

tween concepts Any computational theory must incorp orate knowledgestructuring mechanisms

such as subsumption and inheritance of the sort supp orted in framebased and semanticnetwork

based systems For example a taxonomy provides links that relate more general concepts to

more sp ecic concepts This allows information ab out more sp ecic concepts to b e asso ciated with

their most general concept so information can lter down to more sp ecic concepts in the taxon

omy via inheritance More general concepts in such a taxonomy subsume more sp ecic concepts

with the subsumee inheriting information from its subsumers For atomic concepts subsumption

relations b etween concepts are sp ecied by the links of the taxonomy A clear example of sub

sumption in natural language is the use of descriptions such as person that has children subsuming

person that has a son If one were told People that have children are happy then it follows that

People that have a son are happy The intuitive idea is that more general descriptions should

subsume more sp ecic descriptions of the same sort which in turn inherit attributes from their

more general subsumers

We have presented some general arguments for considering the use of a more natural with

resp ect to language logic for the representation of natural language sentences We have also

presented some characteristics of natural language that a knowledge representation and reasoning

system should supp ort In the remainder of this exp osition we will clarify the motivations for

this work with sp ecic examples present an alternative representation for simple unstructured

variables and reify some asp ects of the logic of these variables

Structured Variables

We are attempting to represent variables as a bundle of constraints and a binding structure

quantier We term these bundles structured variables b ecause variables in this scheme

are nonatomic terms The implemented language of representation is a semantic network rep

resentation system called ANALOG A NAtural LOGic which is a descendant of SNePS

Figure gives a case frame prop osed for the representation of variables in ANALOG The

shaded no de lab elled V is the structured variable The restrictions on the variable are expressed

by no des R R Scoping of existential structured variables with resp ect to universal structured

1 k

SYNTAX

If V V V n are distinct variable no des and if R R k are prop o

1 n 1 k

a

sition no des and all the R dominate V then i

Vn

Depends

Quantifier V R 1 Depends V 1

Quantifier

Rk

is a network actually a class of networks and V is a structured variable no de

SEMANTICS

b

V is an arbitrary quantierconstrained or constrained individual dep endent

V V such that all the restriction prop ositions R R hold for on

1 n 1 k

that arbitrary individual

a

One no de dominates another if there is a path of directed arcs from the rst no de to the second

no de

b

V is the intensional individual denoted by V

Figure Caseframe for Structured Variables

variables is expressed by the depends arcs to universal structured variable no des V V An

1 n

example of a structured variable the no de lab elled V is given in Figure describ ed b elow

We note that the semantics of structured variables is left largely unsp ecied here due to space

limitations However it is an augmented by the addition of arbitrary individuals semantic theory

based on and describ ed in The representations of sentences shown as examples

in this pap er use the case frames sp ecied by

Advantages of Structured Variables

Natural Form

We suggest that the representation of numerous types of quantifying expressions using structured

variables is more natural than typical logics b ecause the mapping of natural language sen

tences is direct We give an example in Figure and defer the formal sp ecication of no des to

Section No de M corresp onds to the asserted prop osition that al l men are mortal No de V

is the structured variable corresp onding to al l men Finally no de M is the restriction prop osition

that corresp onds to the arbitrary man b eing a member of the class of men The memberclass

case frame is the representation for the prop osition that an ob ject is a member of a class This

representation is more natural in that the toplevel prop osition is one of class membership rather

than a rulelike ifthen prop osition Note that any memberclass prop osition of the form X is Y

would b e represented as a similar network structure for example the representation of Al l rich

young men that own a car are mortal is given in Figure

M1!

Member Class

Member Class men M2 V1 mortal

All

Figure Structured Variable Representation of Al l men are mortal

The representation of structured variables suggested here can represent most rstorder quan

tifying expressions directly Also we can represent general quantifying expressions directly al

though their semantics needs to b e detailed In general there is a direct mapping from natural

language quantifying expressions into structured variable representations as structured variables

corresp ond directly to noun phrases with restrictive relative clause complements Figure shows

the fragment of the generalized augmented transition network GATN grammar that pro cesses

noun phrases corresp onding to structured variables Note that in this NP subnetwork all

restrictions on a noun phrase corresp onding to a structured variable can b e lo cally collected and

the corresp onding structured variable built This includes restrictive relative clauses whuch are

parsed in the RREL subnetwork

Conceptual Completeness

In typical logics terms in one sentence are not referenced in other sentences In general reusing

a term involves rewriting the term in the new formula Ideally we would like to reuse exactly adjective

NP1 RREL article noun

noun pop

NP NP2

Figure GATN NP subnetwork for noun phrases corresponding to structured vari

ables

the same terms in dierent sentences in much the same way that language reuses noun phrases

and we would want this reuse to result in closed sentences This requires that variables typically

corresp onding to noun phrases or anaphora in such terms b e meaningful indep endent of the

sentences that use or dene them We call such variables conceptually complete and they may

b e shared by multiple sentences This is an issue in the representation of multisentential dialog

where intersentential reference to sententially scop ed ob jects frequently o ccurs For example Al l

cars come with a spare tire It is not a ful lsized tire can only b e represented in standard logics by

rewriting the terms corresp onding to the shared variables in all representations of sentences that

use them or combining several sentences into one representation To see the dicultly consider a

representation of the example

xcarx y sparetire y iny x

w z inw z car z sparetire w fullsizedw

In these cases there is a clear advantage to a conceptually complete closed variable representation

as well as the structure sharing asso ciated with a semantic network representation We would like

the representation of the example to b e

There is a spare tire y in every car x

y is not fullsized

using two distinct sentences however in the second sentence y is a free variable With structured

variables that contain their own binding structures quantiers no op en sentences are p ossible

Thus in the example the variable y would not b e free Further with structure sharing distinct

sentences may share constituents which would have b een op en sentences in a logical representation

Quantier Scoping

Any representation must account for the expression of quantier scoping at least as simply as the

rstorder predicate logic FOPL linear notation The linear notation implicitly expresses a partial

ordering of quantiers and terms by the order and bracketing of the quantiers With structured

variables quantier scoping is expressed explicitly by the presence or absence of dep endency arcs

However the nonlinearity of the representation introduces a new problem in sp ecifying limited

quantier scopings normally expressed by bracketing For example the dierences in the sentences

x y P x y Qx and x y P x y Qx are asso ciated with bracketing and cannot

b e expressed directly in the nonlinear representation suggested here The rst sentence can

b e represented using structured variables the second requires augmenting the representation as

its antecedent states a prop erty of the collection of all y s This should not b e seen as a ma jor

shortcoming of this representation as several solutions to the problem are p ossible Partitioned

semantic networks or a representation for collections would allow the representation

of all p ossible quantier scopings

Branching Quantiers

There is a class of natural language sentences that are not expressible in any linear notation

The standard example is Some relative of each vil lager and some relative of each

townsman hate each other A linear notation as in standard logics requires that one existentially

quantied relative of the townsman or villager scop e inside b oth universally quantied townsman

and villager This forces a dep endency that should not b e there since the relative of the villager

dep ends only on the particular villager and the relative of the townsman dep ends only on the

particular townsman Examples of these types of quantiers are called branching quantiers

b ecause expression of their scoping required a treelike notation the Henkin prex For

example the branched quantier sentence could b e expressed as in Figure

Since the dep endency arcs asso ciated with structured variables can sp ecify any partial order

of the variables in a sentence we may express any sort of dep endency including those of the type

asso ciated with branching quantiers Figure illustrates how this is done for the relativevillager

sentence No des V and V represent the existential relatives that are scop e dep endent on no des

V each villager and V each townsman resp ectively

x y

n

villagerxtownsmanz relativex y relativez w Hatesy w

z w

Figure Branching quantier representation for Some relative of each vil lager and

some relative of each townsman hate each other

Donkey Sentences

Another class of sentences that are dicult for rstorder logics are the socalled donkey sentences

These are sentences that use pronouns to refer to quantied variables in closed sub clauses

outside of the scop e of the sub clauses In the example sentence Every farmer who owns a donkey M1!

Object1 Object2

Some Rel Some Rel Rel related-to V1 V3 related-to hate Object1 Object1 Object2 Depends Depends Object2

All All Class Class villager V2 V4 townsman

Member Member

Figure Representation for the Branched Quantier Sentence Some relative of

each vil lager and some relative of each townsman hate each other

beats it the noun phrase a donkey is a variable inside the scop e of a universally quantied variable

every farmer and is referred to pronominally outside the scop e of the existentially quantied

donkey Consider some attempts to represent the ab ove sentence in FOPL

a x farmerx y donkeyy ownsx y b eatsx y

b x farmerx y donkeyy ownsx y b eatsx y

c x y farmerx donkeyy ownsx y b eatsx y

Representation a says that every farmer owns a donkey that he b eats which is clearly more

than the original sentence intends Representation b is a b etter attempt since it captures the

notion that we are considering only farmers who own donkeys however it contains a free variable

Representation c fails to capture the sense of the original sentence in that it quanties over al l

farmers and donkeys rather than just farmers that own donkeys To see this consider the case of

the farmer that owns two donkeys and b eats only one of them Clearly the donkey sentence can

apply to this case but interpretation c do es not

Since there are no op en subformulas it is p ossible for formulas to use constituent variables at

any level including at a level which would result in an unscop ed variable in an FOPL representa

tion Figure shows the representation for Every farmer who owns a donkey beats it Note that

M denotes the prop osition Every farmer who owns a donkey beats it and V and V are every

farmer that beats a donkey he owns and a beaten donkey that is owned by any farmer resp ectively

Related Work

This pap er suggests a hybrid approach to knowledge representation and reasoning for natural

language pro cessing by combining semantic network representations and structured ob ject rep M1!

Agent Act Action beat

Object

Class All Some Depends Class farmer V1 V2 donkey Member Member Object All Agent

Act

Action

own

Figure Representation for the Donkey Sentence Every farmer that owns a don

key beats it

resentations corresp onding to structured variables to pro duce a KR formalism that addresses

the previously outlined goals of knowledge representation for natural language pro cessing There

is a large b o dy of work that addresses similar issues For comparative purp oses it is useful to

categorize this related work into three groups structured variable representations atomic variable

representations and hybrid variable representations

There is a large b o dy of work in structured ob ject representation that characterizes very com

plex structured variables as frames scripts and so on Framebased systems such as KLONE

and KRL use highly structured concept representations to express the meaning of concepts

These concept representations are constructed using structural primitives These highly

structured ob jects which typically consist of slots and llers with other mechanisms such as

defaults can b e viewed as complex structured variables that bind ob jects with the appropriate

internal structure although they are not typically so viewed Their use of structural primitives

allows the sp ecication of a subsumption mechanism b etween concepts A diculty with these

representations is that structured representations corresp ond directly to predicates in the under

lying logic Thus constituents of a structured concept are not available as terms in the logic In

the donkey sentence the donkey in farmer that owns a donkey cannot b e used as a term

An alternative to the framebased highly structured ob ject representations is that of a log

ical form representation The general motivation for a logical form is the need for a mediating

representation b etween syntactic and meaning representations usually in the context of determin

ing quantier scoping Representative examples of logicalformbased approaches include

Logical form representations resemble this work in that typically quantiers are bundled

with typed variables that are complete in the manner describ ed here Additionally the utility

of such representations lies in the relative ease of mapping natural language into a representation

which is also clearly a goal of this work However logical form is not a meaning representation

unlike the other representational work considered here In general logical form representations

provide a weakest ambiguous interpretation that is sub ject to further computation b efore its

meaning is apparent It is p ossible to view this work as an improved logical form that has

the advantage of having a natural mapping from language to representation We improve on

logical form however by including clear sp ecication of some types of dicult quantier scoping

incorp orated into a prop ositional semantic network system allowing structure sharing of no des

and terms and cyclic structures of the sort seen in English which are not easily represented in

a linear notation Further we provide a subsumption mechanism not typically present in logical

form work

Previous work using atomic that is unstructured variable representation has b een primarily

based on FOPL In the work of Schubert et al which uses a semanticnetworkbased

formalism variables are atomic no des in the network Type and other restrictions are sp ecied by

links to the variable no des There are no explicit universal or existential quantiers Free variables

are implicitly universally quantied Skolem arcs sp ecify existentially quantied variable no des

Because this is a nonlinear notation sentences with branching quantiers can b e represented

However the separation of variables from their constraints causes the representation to b e not

natural relative to the original natural language Moreover since restrictions on p ossible llers

for variables app ear to b e simple type restrictions there is no representation for noun phrases with

restrictive relative clause complements and consequently no representation for donkey sentences

Fahlmans representation of variables is more general p otentially variables have complex

structure but has similar shortcomings

An alternative atomic variable representation theory is that of Discourse Representation The

ory DRT is a semantic theory that among other things accords indenite noun phrases

the status of referential terms rather than the standard quantied variables and denite noun

phrases the status of anaphoric terms These terms are scop ed by discourse representation struc

tures DRSs and the theory provides rules to expand these DRSs based on the discourse b eing

represented as well as rules for interpreting the DRSs DRT was directly motivated by the dif

culties in the representation of donkey sentences and deals with them by making the scop e of

terms variables b e the DRS rather than the sentence prop osition DRSs themselves may scop e

inside other DRSs creating a hierarchy of DRSs and scop ed terms The approach is similar to

that of Hendrixs partitioned semantic networks As with all the atomic variable represen

tations there is a separation of constraints from variables and the form of DRSs is not natural

in the same sense that a prop osition that represented a sentence would b e Further the rules of

construction of DRS explicitly prohibit the representation of intersentential pronominal reference

to scop ed terms Variables are still scop ed by the DRS and not conceptually complete although

all their constraints are asso ciated with the DRS in whose scop e they lie

An imp ortant philosophically motivated attempt to represent the semantics of natural language

is Montague grammar Montague grammar is a theory of natural language pro cessing in

that it is a complete formal sp ecication of the syntax semantics and knowledge representation for

natural language understanding Montague grammar mimics the syntactic structure of the surface

sentence in sp ecifying the mapping to logic and interpretation In that it resembles this work but

its coverage is far more ambitious than and exceeds the scop e of the work in this pap er However

as a comp ositional semantic theory based on a higherorder intensional logic it provides no inherent

facility for the description of discourse relations and anaphoric connections Further it suers

from the same previously describ ed problems that all logicbased variable representations do

A related b o dy of work is that of Barwise and Co op er on generalized quantiers in natural

language They attempt to represent and provide semantics for more general types of quantied

natural language sentences e g many most and sp ecify a translation of a fragment of English

using phrase structure rules Their discussion of semantic issues related to these generalized

quantiers which are typically manifested as noun phrases forms a pro ductive basis for any

attempt to sp ecify the semantics of quantied noun phrases

Hybrid variable representations accord variables p otentially complex internal structure A

representative system is the work of Brachman with KRYPTON KRYPTON is a KR

system that supp orts and separates two kinds of knowledge terminological in the TBox and

assertional in the ABox Since KRYPTON can represent complex descriptions in the TBox in

principle general structured variables with arbitrary restrictions are p ossible in the TBox with

logicbased assertional representations in the ABox Descriptions in the TBox are used in the

ABox syntactically as unary predicates or as binary relations The form of the representation

is more natural than FOPL since restrictions on ob jects which can take the form of complex

terminological constraints in the TBox are simple predicates in the ABox on those ob jects

However variables are still atomic in the ABox and since sentences of the ABox are FOPLbased

KRYPTON cannot represent branched quantiers or donkey sentences Additionally constituents

of complex terms concepts in the TBox are not available in the ABox so donkey sentences

cannot b e represented

The Ontic system of McAllester also provides the ability to dene structured variables

using a combination of type expressions and functions that reify these types into sets Ontic is a

system for verifying mathematical arguments and as such the selection of type expressions and

functions is limited to the mathematical domain Additionally Ontic is rstorder and settheoretic

with quantication over terms which may b e variables of complex type In principle one could

represent natural language in Ontic however the type system would have to b e enriched and it

would still suer from the disadvantages outlined for KRYPTON In later work McAllester has

addressed natural language issues similar this work particularly the natural form issue

The Knowledge Representation Formalism

Syntax and Semantics of the Formalism

In this section we provide a syntax and semantics of a logic whose variables are not atomic

and have structure We call these variables structured variables The syntax of the logic is

sp ecied by a complete denition of a prop ositional semantic network representation formalism

an augmentation of By a prop ositional semantic network we mean that all information

including prop ositions facts etc is represented by no des The implemented system ANALOG

is used here for convenience to refer to the logical system

Semantics

As a prop ositional semantic network formalism any theory of semantics that ascrib es prop ositional

meaning to no des can b e the semantics used in ANALOG In this pap er examples and representa

tions are used that follow the case frame semantics of which provide a collection of prop osi

tional case frames and their asso ciated semantics based on an extended rstorder predicate logic

We augment that logic further with arbitrary individuals for the semantics of structured variables

in a manner similar to the semantic theory of We will provide semantics for no des

in this pap er as necessary For a complete sp ecication of the semantics of ANALOG see

The Domain of Interpretation

ANALOG no des are terms of a formal language The interpretation of a no de is an ob ject in the

domain of interpretation called an entity Every ANALOG no de denotes an entity and if n is

n denotes the entity represented by n It is useful for discussing the an ANALOG no de then

semantics of ANALOG networks to present them in terms of an agent Said agent has b eliefs

and p erforms actions and is actually a mo del of a cognitive agent

Metapredicates

To help formalize this description we introduce the metapredicates Conceive Believe and If

n n n are metavariables ranging over no des and p is a metavariable ranging over prop osition

1 2

no des the semantics of the metapredicates listed ab ove are

Conceiven Means that the no de is actually constructed in the network

n b eing known to b e true Conceiven may b e true without

or false

p Believep Means that the agent b elieves the prop osition

n n Means that n and n are the same identical no de

1 2 1 2

Belief implies conception as sp ecied in axiom one

Axiom Believep Conceivep

Denition of No des

Informally a no de consists of a set of lab eled by relations directed arcs to one or more no des

Additionally a no de may b e lab eled by a name e g BILL M V as a useful but extra

theoretic way to refer to the no de This naming of a rule or prop osition no de is of the form Mn

where n is some integer A is app ended to the name to show that the prop osition represented

by the no de is b elieved However the do es not aect the identity of the no de or the prop osition

it represents Similarly variable no des are lab eled Vn where n is some integer and base no des are

named Bn where n is some integer additionally base no des may b e named for the concept they

represent e g man More formally a no de is dened as follows

Denition There is a noneempty collection of lab elled atomic no des called base no des

Typically base no des are lab elled by the entity they denote Example bill is a base no de

Denition A wire is an ordered pair r n where r is a relation and n is a no de Metavari

ables w w w range over wires Example member john is a wire

1 2

Denition A no deset is a set of no des fn n g Metavariables ns ns ns range

1 k 1 2

over no desets Example fjohn billg is a no deset if john and bill are no des

Denition A cable is an ordered pair r ns where r is a relation and ns is a nonempty

no deset Metavariables c c c range over cables Example member fjohn billg is a

1 2

cable

Denition A cableset is a nonempty set of cables fr ns r ns g such that

1 1 k k

r r i j Metavariables cs cs cs range over cablesets Example fmember

i j 1 2

fjohn billg class fmangg is a cableset

Denition Every no de is either a base no de or a cableset Example bill is a base no de

fmember fjohn billg class fmangg is a cableset

Denition We overload the membership relation so that x s holds just under the

following conditions

If x is a no de and s is a no deset x s y y s Subsumey x

Example M fM M Mg

If x is a wire such that x r n and s is a cable such that s r ns then

1 2

x s r r n ns

1 2

Example member john member fjohn billg

If x is a wire and s is a cableset then x s cc s x c

Example member john fmember fjohn billg class fmangg

Because we need more denitions b efore Subsume can b e dened we defer its denition to Figure

Denition A variable no de is a cableset of the form fall nsg universal variable

or fsome ns depends ns g existential variable A variable no de is further restricted

1 2

in the form it may take in the following ways

If it has the form fall nsg then every n ns must dominate it

If it has the form fsome ns depends ns g then every n ns must dominate it and

1 2 1

every n ns must b e a universal variable no de

2

Nothing else is a variable no de

Example V fall ffmember fVg class fmangggg is the variable no de corre

sp onding to every man The variable lab el V is just a convenient extratheoretic

metho d of referring to the variable

We dene two selectors for variable no des



ns if v fal l nsg

restv

ns if v fsome ns depends ns g

1 1 2

dependsv ns if v fsome ns depends ns g

2 1 2

Informally restv is the set of restriction prop ositions on the types of things that may b e b ound

to the variable no de v Dependv is the set of universal variable no des on which an existential

variable no de v is scop edep endent

Denition A molecular no de is a cableset that is not a variable no de Example fmember

fjohn billg class fmangg is a molecular no de since it is a cableset but not a variable no de

Denition An nrnpath from the no de n to the no de n is a sequence

1 k +1

n r n r n

1 1 k k k +1

for k where the n are no des the r are relations and for each i r n is a wire in

i i i i+1

n Example If M fmember fjohn billg class fmangg then M member john and

i

M class man are some nrnpaths

Denition A no de n dominates a no de n just in case there is an nrnpath from n

1 2 1

to n The predicate dominaten n which is true if and only if n dominates n Example If

2 1 2 1 2

M fmember fjohn billg class fmangg then M dominates john bill and man

Denition A rule no de is a molecular no de that dominates a variable no de that do es

not in turn dominate it

Example V fall fMgg

M fmember fVg class fmangg

M fmember fVg class fmortalgg

M is a rule no de since M dominates V which do es not in turn dominate M M

is not a rule no de b ecause while it dominates V it is also dominated by V The

nonrule no des that dominate variable no des corresp ond to restrictions on binders of

those same variable no des

The ANALOG mo del

Denition An ANALOG mo del is a tuple A B M R U E where A is a set of

relations B is a set of base no des M is a set of nonrule molecular no des R is a set of rule no des

U is a set of universal variable no des and E is a set of existential variable no des and M R

B M R U and E are disjoint consists of b elieved prop ositions Note that the metapredicates

B el iev e and C onceiv e are by denition

Believen n

Conceiven n M R

Reduction

We follow in arguing for a form of reduction inference dened in axioms and b elow

as b eing useful This is a form of structural subsumption p eculiar to semantic network

formalismss which allows a prop osition to reduce to logically imply prop ositions whose wires

are a subset of the wires of the original prop osition Figure gives an example of a prop osition

expressing a brotherho o d relation among a group of men No de M represents the prop osition that

bill john ted and joe are brothers By reduction subsumption all prop osition no des such

as M and M involving fewer brothers follow

In the following mo del A B M R U E

A frelation argg

B fbill john ted joeg

M fM M Mg

fM M Mg

where

M frelation fbrothersg arg fbill john ted joegg

M frelation fbrothersg arg fjohn ted joegg

M frelation fbrothersg arg fbill johngg

Some reductions

reduceM M T

reduceM M T

Figure Example of Subsumption by Reduction for a Particular Mo del

However we must restrict the use of reduction inference to precisely those prop ositions and

rules which are reducible through the use of the I sReducibl e metapredicate

Axiom Reducecs cs w w cs w cs IsReducible cs

1 2 2 1 1

Note that the semantics of the metapredicate IsReducible will b e sp ecied in terms of the partic

ular case frames used in a representation Prop ositions like M are clearly reducible but not all

prop ositional case frames are reducible For example

xmanx richx happyx

should not allow the reduction involving fewer constraints on x

xmanx happyx

as the latter do es not follow from the former Note that reduction is appropriate when the

constraints in the antecedent of the rule are disjunctive

A prop osition that is a reducible reduction of a b elieved prop osition is also a b elieved prop o

sition Since no des are cablesets we state this as in axiom

Axiom Reducen n B el iev en B el iev en

1 2 1 2

Types of No des

We have dened four orthogonal types of no des base molecular rule and variable no des In

formally base no des corresp ond to individual constants in a standard predicate logic molecular

no des to sentences and functional terms rule no des to closed sentences with variables and variable

no des to variables Note that syntactically all are terms in ANALOG however

The Uniqueness Principle

No two nonvariable no des in the network represent the same individual prop osition or rule

Axiom n n n n

1 2 1 2

This is a consequence of the intensional semantics A b enet of this is a high degree of structure

sharing in large networks Additionally the network representation of some types of sentences

such as the donkey sentence can reect the reuse of natural language terms expressed by pro

nouns and other reduced forms

Subsumption

Semantic network formalisms provide links that relate more general concepts to more sp ecic

concepts this is called a taxonomy It allows information ab out concepts to b e asso ciated with

their most general concept and it allows information to lter down to more sp ecic concepts in the

taxonomy via inheritance More general concepts in such a taxonomy subsume more sp ecic con

cepts the subsumee inheriting information from its subsumers For atomic concepts subsumption

relations b etween concepts are sp ecied by the links of the taxonomy To sp ecify subsumption

some additional denitions are required

Denition A binding is a pair v u where either u and v are b oth structured variables

of the same type universal or existential or u is a universal SV and v is any no de

Examples VV

JOHNV

Denition A substitution is a p ossibly empty set of bindings ft v t v g

1 1 n n

Examples fVV JOHNVg

fBV MVg

Denition The result of applying a substitution ft v t v g to a no de n is

1 1 m m

the instance n of n obtained by simultaneously replacing each of the v dominated by n with t

i i

If fg then n n

Example If M fmember fVg class fMANgg then

MfJOHNVg fmember fJOHNg class fMANgg

Denition Let fs u s u g and ft v t v g b e substitutions Then

1 1 n n 1 1 m m

the comp osition of and is the substitution obtained from the set

fs u s u t v t v g

1 1 n n 1 1 m m

by deleting any binding s u for which u s

i i i i

Example fVV VVg

fVV JOHNVg

fJOHNV JOHNVg

Denition A substitution is consistent i neither of the following hold

u t stu su s t

u v ttu tv u v

A substitution that is not consistent is termed inconsistent The motivation for the second con

straint called the unique variable binding rule UVBR is that in natural language users seldom

want dierent variables in the same sentence to bind identical ob jects For example Every

elephant hates every elephant has a dierent interpretation from Every elephant hates himself

Typically the most acceptable interpretation of the former sentence requires that it not b e inter

preted as the latter UVBR requires that within an individual sentence that is a rule has b ound

variables any rule use binding of variables must involve dierent terms for each variable in the

rule to b e acceptable

Examples fJOHNV BILLVg is inconsistent

fJOHNV JOHNVg is inconsistent

fJOHNV BILLVg is consistent

Denition The predicate occursinx y where x is a variable is dened

occursinx y dominatey x

occursin enforces the standard o ccurs check of the unication algorithm and is just a more

p erspicacious naming of dominate

In ANALOG we sp ecify subsumption as a binary relation b etween arbitrary no des in the net

work We dene subsumption b etween two no des x and y in Figure This denition of subsump

tion includes subsumption mechanisms that Woo ds classies as structural recorded axiomatic

and deduced subsumption In Figure case corresp onds to identical no des a no de obvi

ously subsumes itself Case is the reduction inference case discussed in section Case

Subsumex y in a mo del A B M R U E if any one of

x y

Reducex y

For x U and y B M R if not occursinx y and there exists a

substitution S such that

r r r estx r fy xg S

Logical derivation is here denoted by Substitution and substitution

application with resp ect to a no de is here denoted by r fy xg S

For x U and y U E if

r r r estx ss r esty Subsume r s

For x y E if all of the following hold

ss r esty r r r estx Subsume r s

r r r estx ss r esty Subsume r s

dd dependsy cc dependsx Subsume c d

Otherwise fail

Figure Subsumption Pro cedure

applies when a universal structured variable no de subsumes another no de This corresp onds to a

description like any rich man subsuming John if John is known to b e a man and rich Such a vari

able will subsume another no de if and only if every restriction on the variable can b e derived in

the current mo del for the no de b eing subsumed Subsumption consequently requires derivation

which is dened in For the examples shown here standard rstorder logical derivation will b e

assumed Case allows a more general universal variable no de to subsume a less general exis

tential variable no de For this to happ en for every restriction in the universal variable no de there

must b e a restriction in the existential variable no de and the former restriction must subsume the

latter restriction For example the variable no de corresp onding to every rich girl would subsume

some rich happy girl but not some girl Case allows one existential variable no de to subsume

another The requirement for this case is essentially that the variables b e notational variants of

each other This is b ecause it is not in general p ossible for any existential variable to subsume

another except when they are structurally identical The reason this case is needed rather than

just excluding it entirely is that for a rule no de corresp onding to every boy loves some girl to

subsume every rich boy loves some girl the existential variable no de corresp onding to the some

girl in the rst rule no de must subsume the existential variable no de corresp onding to the some

girl in the second rule no de see Section for numerous examples of this sort of subsumption in

natural language

The commonsense nature of the subsumption cases can most easily b e illustrated by examples

some of which for conciseness are given in natural language in Figure The examples illustrate

Case Subsumejohn john

Case Subsumefmember fjohn billg class fmangg

fmember fjohng class fmangg

SubsumeAll rich men are happy

All young rich men that own a car are happy

Case SubsumeAll dogs Fido Provided dogFido

SubsumeAll things have a mass Fido has a mass

Case SubsumeEvery girl Every pretty girl

SubsumeEvery pretty girl Every pretty girl that owns a dog

SubsumeEvery girl Some pretty girl

SubsumeEvery pretty girl Some pretty girl that owns a dog

Case SubsumeEvery b oy that loves a girl

Every b oy that loves a girl and that owns a dog

Figure Examples of Subsumption cases refer to cases of Figure

the idea that more general quantied descriptions should subsume less general quantied descrip

tions of the same sort In Figure a more detailed example for a particular mo del is given No de

M represents the prop osition that al l men are mortal M the prop osition that Socrates is a

man and M the prop osition that Socrates is mortal V is the structured variable representing

any man It then follows that M is a sp ecial case of M directly by subsumption since V subsumes

Socrates Note that the restrictions on subsumption involving variables is stricter than Reduce

which only requires that the wires of one no de b e a subset of the other

As with reduction Axiom a prop osition that is subsumed by a b elieved prop osition is also

a b elieved prop osition This can b e stated as a more general form of Axiom

Axiom S ubsumen n B el iev en B el iev en

1 2 1 2

Summary

We have formally sp ecied the subsumption mechanism in the ANALOG system The subsump

tion mechanism takes advantage of the conceptual completeness of the structured variable rep

resentation to allow the kinds of common sense description subsumption relationships that are

p ervasive in natural language

In the following mo del A B M R U E

A fmember class allg

B fman mortal Socratesg

M fM M Mg

R fMg

U fVg

where

M fmember fVg class fmangg

M fmember fVg class fmortalgg

M fmember fSocratesg class fmangg

M fmember fSocratesg class fmortalgg

V fall fMgg

The resulting subsumption

subsumeM M T

Figure Example of Subsumption for a Particular Mo del

ANALOG for Natural Language Pro cessing

So far we have motivated some asp ects of the logic underlying the ANALOG KRR system and

formalized some imp ortant concepts such as subsumption asso ciated with the logical system

At this junction we will attempt to illustrate the utility of the system in the context of sp ecic

examples of natural language pro cessing

ANALOG includes a generalized augmented transition network GATN natural language

parser and generation comp onent linked up to the knowledge base based on A GATN gram

mar sp ecies the translationgeneration of sentences involving complex noun phrases intofrom

ANALOG structured variable representations

We present three demonstrations of the NLP comp onent of ANALOG The rst illustrates

the representation and use of complex noun phrases the second illustrates the use of nonlinear

quantier scoping and structure sharing and the last is a detailed presentation with most of the

underlying ANALOG representations of a demonstration that illustrates the use of rules and valid

and useful answers to questions The last two demonstrations also have examples of subsumption

Representation of Complex Noun Phrases

One of the most apparent advantages of the use of structured variables lies in the representation

and generation of complex noun phrases that involve restrictive relative clause complements The

restriction set of a structured variable typically consists of a type constraint along with prop erty

constraints adjectives and other more complex constraints restrictive relative clause comple

ments

parse

ATN parser initialization

Input sentences in normal English orthographic convention

Sentences may go beyond a line by having a space followed by a CR

To exit the parser write end

Every man owns a car

I understand that every man owns some car

Every young man owns a car

I understand that every young man owns some car

Every young man that loves a girl owns a car that is sporty

I understand that every young man that loves any girl owns some sporty car

Every young man that loves a girl that owns a dog owns a red car that is sporty

I understand that every young man that loves any girl that owns any dog owns

some red sporty car

Every young man that loves a girl and that is happy owns a red sporty car that wastes gas

I understand that every young happy man that loves any girl owns some

sporty red car that wastes gas

end

ATN Parser exits

Figure Examples of complex noun phrase use that correspond to structured vari

ables

In Figure user input is italicized the text at the b eginning is a standard message and will

b e omitted from the remaining gures Figure shows example sentences with progressively more

complex noun phrases b eing used These noun phrases are uniformly represented using structured

variables Parsing and generation of these noun phrases is simplied b ecause structured vari

ables collect all relevant restrictions on a variable into one unit a structured variable The parser

parses the users sentence and builds an ANALOG representation for the user input The resulting

representation is then passed to the generation comp onent which generates the output resp onse

sometimes prexed by the canned phrase I understand that If constraints on variables cor

resp onding to the complex noun phrases were represented using FOPL then it would b e dicult

to generate natural language noun phrases corresp onding to these variables This is b ecause the

constraints on variables would likely b e wellseparated from the variables in the antecedents of

rules involving these variables This is not the case in a structured variable representation

NonLinear Quantier Scopings and Structure Sharing

Since this representational formalism is grounded in an inherently nonlinear notation semantic

networks the representation of treelike quantier scopings is straightforward Thus sentences

involving branching quantiers can b e represented In addition the structure of ANALOGs se

mantic network representation allows structuresharing and indeed may require it to a high

degree One of the initial goals was the representation of coreference and structure sharing in

multisentential dialog Because the parsergenerator maintains a very simple discourse mo del we

cannot illustrate complex examples such as the branching quantier sentence of this structure

sharing in natural language although such representation may readily b e built A simpler ex

ample is the donkey sentence where a scop ed constituent of a noun phrase the donkey in Every

farmer who owns a donkey beats it is used in the main clause of the sentence Figure illustrates

a dialog involving questions ab out the donkey sentence

Every man that owns some donkey beats it

I understand that every man that owns some donkey beats some donkey

Fred is a boy

I understand that Fred is a boy

Every boy is a man

I understand that every boy is a man

Doc is a donkey

I understand that Doc is a donkey

Dumbo is a donkey

I understand that Dumbo is a donkey

Fred owns Doc

I understand that Fred owns Doc

Fred owns Dumbo

I understand that Fred owns Dumbo

Does Fred beat Doc

I dont know

Does any man beat some donkey

Yes every man that owns some donkey beats some donkey

Every man that owns any donkey beats it

I understand that every man that owns any donkey beats every donkey

Does Fred beat Doc

Yes Fred beats Doc

Does any man beat some donkey

Yes every man that owns some donkey beats some donkey

Does any man beat any donkey

Yes Fred beats Doc and every man that owns any donkey beats every

donkey and every man that owns some donkey beats some donkey

Who beats a donkey

Fred beats Doc and every man that owns any donkey beats every donkey

and every man that owns some donkey beats some donkey

Figure Example of structure sharing in donkey sentence

In Figure note that the system is initially unable to determine whether Fred b eats Do c

or Dumbo This is b ecause the initial rule every man that owns some donkey beats it is

satised in a mo del where only one of the donkeys is b eing b eaten After the system is told that

al l such donkeys are b eaten it do es determine that Fred b eats Do c Note that this determination

also requires that the description every man subsume Fred who is a b oy and consequently a

man This is an example of derived subsumption Also note that the answers to many questions

are often rules themselves e g Who beats a donkey has as one answer Every man that owns

some donkey This is discussed in the next section

Every man is mortal

I understand that every man is mortal

Who is mortal

Every man is mortal

Is any rich man mortal

Yes every rich man is mortal

John is a man

I understand that John is a man

Is John mortal

Yes John is mortal

Who is mortal

John is mortal and every rich man is mortal and every man is mortal

Are al l rich young men that own some car mortal

Yes every young rich man that owns some car is mortal

Any rich young man that owns any car is happy

I understand that every young rich man that owns any car is happy

Is John happy

I dont know

Young rich John owns a car

I understand that mortal rich young John owns some car

Who owns a car

Mortal rich young John owns some car

Is John happy

Yes mortal rich young John is happy

Figure Examples of questions that have rules as answers

Rules as Answers to Questions

Because the structure of the representation of rules is at that is there is not the articial

antecedentconsequent structure asso ciated with rstorder logicbased representations it is p ossi

ble to frame questions whose answers are rules and not just ground formulas Since the structure

of the question will mirror the structure of the rule any rule that is subsumed by a question is

an answer to that question Figure gives a sample dialog involving questions whose answers

are ground prop ositions e g Is John mortal as well as questions whose answers are rules e g

Who is mortal This dialog also illustrates the uses of subsumption Since we told the system

Every man is mortal it follows that any more sp ecically constrained man e g Every rich young

man that owns some car must also b e mortal Note that this answer a rule follows directly by

subsumption from a rule previously told to the system This is another way in which rules may

b e answers to questions M2!

Member Class

Member Class men M1 V1 mortal

All

Figure Representation of sentence Every man is mortal

M3

Member Class

V2 mortal

Figure Representation of sentence Who is mortal

A Detailed Demonstration Examination

In this section we present the representations and pro cessing asso ciated with the last demonstra

tion in detail All references to sentences will b e to the numbered sentences in Figure The

representation for sentence is that of Figure Sentence then asks who is mortal In a

standard FOPLbased system no answer could b e given b ecause there are as yet no instances of

men in the knowledge base This is contrary to the commonsense answer of sentence which

reiterates the rule of sentence This is p ossible in ANALOG b ecause the structure of the

representation of the question Who is mortal is similar to that of any of its answers Thus any

asserted prop osition that is subsumed by the question is a valid answer including rules

Sentence is an example of a question ab out a rule Since every man is mortal is b elieved

the system was told this in sentence it follows that any more restricted sort of man is also

mortal The subsumption pro cedure sp ecies this explicitly The representation of sentence

in Figure is a less general form of sentence in Figure since V any man subsumes V

any rich man Since the rule in Figure is subsumed by a b elieved no de that of sentence

it follows by Axiom that sentence is b elieved thus the representation of the question itself

is a b elieved prop osition and the system answers yes to the question Sentence asserts that

John is a man and the result is the representation of Figure At this p oint the system knows

al l men are mortal and John is a man When the question of sentence whose representation M6!

Member Class

Class Member men M4 V3 mortal All

Object All

M5

Property

rich

Figure Representation of sentence Is any rich man mortal

M7!

Member Class

John man

Figure Representation of sentence John is a man

is in Figure is asked the system nds the rule of sentence and determines that it subsumes

sentence b ecause John is a man and again by axiom the result follows However note that

in this derivation the result is a ground formula rather than a rule Sentence illustrates the

retrieval of the systems information ab out who is mortal note the additional b elieved prop ositions

Sentence is an example of a more complex noun phrase in a rule The representation of is

in Figure and is subsumed by that of sentence or leading to the yes answer In Figure

V represents the arbitrary rich young man that owns some car and V represents some owned M8!

Member Class

John mortal

Figure Representation of sentence Is John mortal

M15! Class Member Member Class V5 M13 car Depends Some mortal Object2

Member Object1 Class Relation men M10 V4 M14 own All All

Object Object All All

M11 M12

Property Property

rich young

Figure Representation of sentence Are al l rich young men that own some car mortal

car of V In sentence Figure a new rule ab out rich young carowning men V b eing

happy M is introduced The question of sentence is John happy cannot b e answered

b ecause the system cannot determine that no de V subsumes John This is b ecause while John is

a man he is not known to b e young rich and owning a car requirements for this subsumption

Sentence informs the system of these requirements the systems understanding is veried by

question whose representation is shown in Figure Note that the structure of the question

involving two variables Who and a car is identical to that of the structure of its answer which

would not b e the case if constraints were separated from variables in the antecedents of rules as

is done in typical logics The question is asked again and b ecause the subsumption relationship

can b e determined is answered in the armative M21! Property Object Member Class V7 M19 car All happy Object2

Member Object1 Class Relation men M16 V6 M20 own All All

Object Object All All

M17 M18

Property Property

rich young

Figure Representation of sentence Any rich young man that owns any car is happy

M22

Property Object

happy John

Figure Representation of sentence Is John happy

Summary

We initially presented some broad goals for a knowledge representation and reasoning system for

natural language pro cessing We have describ ed in some detail such a system ANALOG is a

prop ositional semanticnetworkbased knowledge representation and reasoning system that sup

p orts many asp ects of NLP in particular the representation and generation of complex noun

phrases the representation of various types of quantied variable scoping a high degree of struc

ture sharing and subsumption of the sort typically asso ciated with ordinary natural language use

We have presented examples of natural language dialog and their asso ciated representations in the

ANALOG system that illustrate the utility of this formalism for NLP

The full ANALOG system has not b een describ ed here due to space limitations but includes M37

Object1 Object2 Relation

Member Class V2 own V8 M19 car

All

Figure Representation of sentence Who owns a car

facilities for derivation inference pathbased inference and b elief revision

Acknowledgments

Our thanks to Professor William J Rapap ort for his comments on various drafts of this pap er

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