A Description Oriented Logic for Building ... - Semantic Scholar

A Descript ion Oriented Logic for Building Knowledge Bases GIUSEPPE ATTARDI and MARIA SIMI Several knowledge represent at ion syst em s have been based We discuss t he advantages of using a logic system for knowledge represent at ion which is based on descriptions, rat her than predicates, and which embodies t wo fundament al ideas for struct uring knowledge t hat are dist illed from semant ic net works and frame based languages: inherit ance and att ribut ions. Tax onomic reasoning on a lat t ice of descriptions combined wit h deduct ion st rat egies defined at t he met alevel provides t he knowlegde base with t he capabilit y t o deal with complex problem solving t asks.

on creat ing and m aintaining t ax onom ic st ruct ures of concept s. The use of such t ax onom ies has been usually eit her delegat ed t o specific procedures supplied by t he user [Brachm an and Schm olze 85, Mylopoulos and Wong 1980], or lim it ed t o very sim ple t asks as det erm ining where t o find at tribut es, values or propert ies of fram es (e.g. KEE [Fikes and Kehler 85]. These fram e- based represent at ion syst em s oft en resort t o hybrid solut ions where a fram e st ruct uring facilit y is combined

I.

INTRODUCTION

wit h eit her a procedural language (e.g. LOOPS [St efik et al. 83])

The abilit y t o represent knowledge is oft en considered essent ial t o build syst em s wit h reasoning capabilit ies. We discuss t he advantages of using a logic syst em for knowledge represent at ion which is based on descript ions, rat her t han predicat es, and which includes som e fundam ent al st ruct uring capabilit ies. Ex am ining t he m ost com m on knowledge represent at ion form alism s, we not ice t he following.

syst em s like Prolog, com pared t o syst em s based on full predicat e logic, t rade expressiveness of language for a m ore t ract able proof procedure. Fram e based languages concent rat e on epist em ological and

st ruct uring

m echanism s

but

deduct ive

capabilit ies are eit her ill defined or procedurally defined. The sam e is t rue for object - orient ed languages.

languages:

from

sem ant ic net works and

inherit ance

and

at tribut ions.

fram e based The

result ing

language is sim ple and underst andable, yet it ret ains t he full power of t he predicat e calculus or of a simple set t heory. The logic of Om ega has been provided wit h a form al sem ant ics and has been proved t o be sound and consist ent [At tardi and Sim i 81b]. Current work on Om ega is pursuing t hese fundam ent al lines

tax onomic reasoning , knowledge base functionalities and metalevel t ransfer .

of invest igat ion:

Manuscript received April 15, 1985; revised April 14, 1986. This research was support ed in part by ESPRIT, project P440 and

by

M.P.I.,

project

ASSI

and

Om ega ex plores t he idea of tax onom ic reasoning , t hat is t o base all reasoning on t raversal/ inst ant iat ion of t he latt ice of descript ions. All knowledge in Om ega is represent ed in a single lat t ice: from fact ual knowledge, t o general rules, t o dependencies and constraint s. When a descript ion needs t o be accessed t o answer a problem , all relevant and relat ed facts

is

a

is m ade of m arker propagat ion algorit hm s when t raversing t he net work, t o est ablish when a solut ion has been found or t o det erm ine when t he search has run int o a cyclic pat h. For a m ore det ailed

account

of

t he t echniques of

t ax onom ic

reasoning we refer t o [Att ardi et al. 1986]. We t ake t he view t hat complex problem solving t asks should be addressed dist inguishing a knowledge base com ponent and a m et hodological com ponent . The m et hodological com ponent is what drives t he problem solving t ask, by issuing queries t o

Om ega embodies t wo ideas for st ruct uring knowledge t hat are dist illed

perform ed by m eans of such addit ional com ponent s.

and assert ions can be found direct ly connect ed t o it . Heavy use

Am ong syst em s wit h sound logical foundat ions, Horn clause

adequacy

or a product ion syst em (e.g. KEE). Most of t he reasoning is

cont ribut ion

to

workpackage 4 of EEC Cost - 13 project n. 21 "Advanced Issues in Knowledge Represent at ion". G. Att ardi is wit h DELPHI SpA, I- 55049 Viareggio, It aly. M. Sim i is wit h t he Dipart im ent o di Inform at ica, Universit à di Pisa, Pisa, It aly.

t he knowledge base, act ing and int eract ing wit h t he ex t ernal word. This component can be considered by and large algorit hm ical in nat ure. The knowledge base on t he ot her hand is where deduct ion t akes place, applying deduct ion rules t o find answers. Tax onom ic reasoning is what provides t he knowledge base wit h t he fundam ent al reasoning capabilit ies, on t op of which m ore sophist icat ed st rat egies can be programm ed t o handle complex problem solving t asks. Rat her t han being able t o perform arbit rary complex t heorem proving t asks, an Om ega knowledge base is expect ed t o be able t o provide imm ediat e answers t o elem entary quest ions, of t he kind t hat hum ans solve wit h no effort . The deduct ion is perform ed t raversing t he lat t ice, but it is cont rolled by user specified st rat egies which det erm ine which part of t he lat t ice t o consider, and how t o m ove around it . To t ailor reasoning st rat egies t o specific applicat ions, strat egies can be program m ed in t he m etalanguage of Om ega. Met alanguage is also used as a foundat ion for t he viewpoint m echanism .

A

viewpoint

is

a

collect ion

of

st at em ents

0018- 9219/ 86/ 1000- 1335$01.00 © 1986 IEEE PROCEEDINGS OF THE IEEE, VOL. 74, NO. 10, OCTOBER 1986

1335

represent ing t he assumpt ions of a t heory. Mult iple viewpoint s

t o build m ore complex descript ions, like in t he following

provide t he abilit y t o handle different sit uat ions arising eit her

ex amples:

from hypot het ical reasoning or for evolut ion over t im e of

(t rue or f alse)

sit uat ions, as well as reasoning about beliefs. A form al discussion of viewpoint s appears in [6].

(an Even- Number) and (not zero)

The abilit y t o m anufact ure a m et adescript ion out of any

Special descript ions are Not hing and Som et hing, denot ing

descript ion provides a m ean t o Om ega for accessing it s int ernal st ruct ure, for inst ance t o int errogat e t he syst em about

t he empt y set and t he universe respect ively.

t he concept s t hat are present , or t o ex am ine and ex plore t he goal st ruct ure t hat has been generat ed in t he course of a deduct ion, for t he purpose of generat ing an ex planat ion of t he conclusion.

B.

St atements The most elem ent ary sent ence in Om ega is a predicat ion. A

predicat ion relat es a subject t o a predicat e by t he relat ion is.

The design of Om ega has it s origins in t he st udies perform ed at MIT Art ificial Int elligence Laborat ory on t he languages AMORD [11], Et her [15], and Om ega it self [4, 5, 17]. Om ega is also used as Knowledge Represent at ion syst em in a couple of Esprit project s, and in part icular in Project P440 on "Message Passing Archit ect ures and Descript ion Syst em s". In t he following sect ion we present an inform al int roduct ion t o t he language and t he ax iom at ic t heory of Om ega. In sect ion 3 we will discuss how an Om ega knowledge base can be organized and how deduct ion can be implem ent ed.

For inst ance t he predicat ion Bost on is (a Cit y) is underst ood t o assert t hat t he individual nam ed Bost on belongs t o t he class of cit ies. Predicat ion can be used t o relat e arbitrary descript ions. For inst ance t he sent ence: (a Hum an) is (a Mort al) st at es t he fact t hat any individual of class hum an is also an individual of class m ort al.

II.

THE DESCRIPTION LANGUAGE OMEGA

Not e t hat descript ions can consist ent ly be int erpret ed as

Om ega is a logic syst em , which consist s of a language, an ax iom syst em and a set of inference rules. In t his sect ion we inform ally int roduce t he language by m eans of ex am ples. Occasionally som e ax iom will be present ed to clarify cert ain

set s

of

individuals

Socrat es is (a Mortal)

A.

from

The

simplest

kind

of

descript ion

is

t he

individual

descript ion, like: Bost on or 3 Here "Bost on" and "3" are nam es describing individual ent it ies. An inst ance descript ion is t he basic indefinit e descript ion: it is m eant t o represent a collect ion of individuals. For inst ance (a Cit y) or (an Int eger) represent t he collect ion of individuals in t he class of cit ies and of int egers.1 The descript ion operat ors and, or and not , int erpret ed as set int ersect ion, union and complem ent respect ively, allow one

in

case

of

individual

One of t he fundam ent al propert ies of t he relat ion is is transitivity, t hat allows one t o conclude for inst ance t hat

propert ies of descript ions.

Descript ions

(singlet ons

descript ions), and is as t he subset relat ion am ong set s.

Socrat es is (a Hum an) and (a Hum an) is (a Mort al) Descript ions form a boolean latt ice, induced by t he part ial ordering relat ion is. The bot t om of t he latt ice is t he descript ion Not hing, a special const ant which plays t he role of t he null ent it y. The t op of t he latt ice is t he descript ion Som et hing, anot her special const ant which represent s t he m ost generic, universal descript ion. Composit e st at em ent s can be built by combining st at em ent s wit h t he logical connect ives ∧, ∨, ¬ and →, as in: Tom is ((a Cat ) or (a Dog)) ∧ ¬(Tom is (a Dog)) → Tom is (a Cat ) The difference between descript ion operat ors and st at em ent connect ives is illust rat ed by t he following ex am ples: (t rue ∧ false) is false (t rue and false) is Not hing Since t rue, f alse are t wo individuals, represent ing t he

1

The fact t hat t he singular form is used should not m islead

boolean const ant s, in t he latt er sent ence "(t rue and false)" is a

t he reader. An indefinit e descript ion like (a Cit y) does not

descript ion denot ing t he intersect ion of two disjoint set s,

represent an unspecified elem ent of t he class of cit ies, but

t herefore it is equivalent t o Not hing.

rat her t he whole collect ion of cit ies.

ATTARDI AND SIMI: LOGIC FOR KNOWLEDGE BASES

1336

C.

Attributions

(with owner (a Person)))

An inst ance descript ion can have attribut ions att ached t o it ; t his serves t he purpose of specializing a class by describing som e of it s propert ies.

Propert ies or att ribut es are t o be considered always relat ive t o a concept , and not as propert ies of an individual. In ot her words, an individual may have som e at tribut es when it is

For ex ample, in:

considered as m ember of a class and different at tribut es as m ember of anot her class.

(a Car (wit h colour red)) t he at tribut ion (wit h colour red) restrict s t he descript ion to represent just t hose cars which have color red. Here "color" is an at tribut e nam e for t he concept "car", and "red" is t he corresponding at tribut e. Att ribut ions provide also a m ean t o relat e descript ions, as for inst ance in:

It follows t hat att ribut es are not

allowed t o m igrat e from one concept t o anot her as it is t he case wit h m ost knowledge represent at ion syst em s [12, 20, 18, 10],

t hus avoiding

conflict s generat ed

am ong classes does not m ean t hat att ribut es of t he superclass are also at t ribut es of t he subclass. If t his is t he case it can be

(a St udent (wit h age = x )) is (a Person (wit h age = x ))

The at tribut e "owner" est ablishes a relat ionship bet ween m y- car and an It alian who is it s owner. Att ribut ions em bed a form of ex ist ent ial quant ificat ion; t he previous ex ample should indeed be read as saying t hat t here ex ist s an individual, who is an It alian, who is t he owner of m ycar. According t o t his sem ant ics, when t he value of an at t ribut e is Not hing, t he whole descript ion collapses t o Not hing (ax iom of strictness. For ex am ple:

where ident ifiers st art ing

This property provides a way to perform t ype checking or to discover inconsist encies in values of at t ribut es. If we want ed to describe t he set of cars wit h no owner, we would say inst ead:

tradit ional

inherit ance

m echanism

Most

knowledge

representat ion

m eans of predicat es. The relat ional dat a base m odel in comput er science uses a sim ilar concept of relat ions index ed by att ribut e nam es. The sem ant ics of bot h Predicat e Logic and relat ional dat abases depend on t he fact t hat relat ions or predicat es have a fix ed number of argum ent s. In Om ega att ribut es can be added or om it t ed from a descript ion, wit h no fix ed lim it . By adding one at t ribut ion we obt ain a m ore specific it s

ex t ension;

om itt ing

an

att ribut ion we obtain a m ore general descript ion. So for

syst em

provide

t wo

CAR: Number_of_weels: 4 engine: fault y or not - fault y colour: red or black or blue colour: red colour: red engine: fault y. A red car, described by t he concept RED- CAR, would inherit from CAR it s at tribut es. The sam e is true for MY- CAR. Therefore t he inheritance m echanism allows one t o deduce, for ex ample, t hat t he number of weels of MY- CAR is 4. In Om ega t he sam e ex ample is ex pressed as follows:

(wit h engine (fault y or not - fault y))

(a Car (wit h owner (an It alian)) (wit h make VW)) is

(wit h colour (red or black or blue)))

(a Car (wit h owner (an It alian))) Om ega provides t he capabilit y t o supply knowledge in an Therefore

increm ent al

will

(a Car) is (a Car (wit h number- of- weels 4)

inst ance, according t o t he ax iom of Omission :

descript ions and

We

between a class and it s inst ances. So for ex am ple one would

MY- CAR INSTANCE- OF CAR

fashion.

provided.

describe t he class of cars, or rat her t he concept of a car, as

Predicat e logic ex presses relat ionships of t his kind by

increm ent al

is

inherit ance relat ions: IS- A am ong classes and INSTANCE- OF

RED- Car IS- A CAR

rest rict

"= " denot e universally

illust rat e t his wit h an ex ample.

(a Car) and not (a Car (wit h owner Som et hing))

we

wit h

quant ified variables. Nevert heless all t he funct ionalit y of t he

follows:

(a Car (wit h owner Not hing)) is Not hing

i.e.

hom onym ous

ex plicit ly st at ed as in:

m y- car is (a Car (wit h owner (an Italian)))

descript ion,

by

at tribut es and m ult iple inheritance. Therefore t he relat ion is

Om ega

refinem ent

handles of

part ial

(a Red- Car) is (a Car (wit h colour red)) m y- car is (a Car (with colour red) (with engine fault y))

concept s is

possible. For example we can say: (a Car) is (a Car (wit h number- of- weels 4)) and lat er assert :

Reasoning according t o t he ax iom s and inference rules of Om ega, we could deduce: m y- car is (a Car (with number- of- weels 4) (wit h engine fault y)

(a Car) is (a Car (wit h owner (a Person)) By t he ax iom of merging t he following can be concluded: (a Car) is (a Car (wit h number- of- weels 4)

PROCEEDINGS OF THE IEEE, VOL. 74, NO. 10, OCTOBER 1986

(wit h colour red)) More precisely t he ax iom s used are Omission , Transitivity and Merging .

1337

where t he first number in each t uple is t he product of t he

Not e however t hat redefinit ion is not allowed in subclasses but only refinem ent of at tribut e values. For som e of the

second and t hird numbers. In t his case, t he denot at ion of (a Product ) would be all

problem s relat ed t o overriding or cancellat ions of propert ies in

int egers, in fact any int eger num ber can be t he result of a

fram e based languages see [7]. Several individual descript ions are allowed as att ribut es for t he sam e att ribut e nam e, as in:

product . The denot at ion of (a Product (of fact or 1 2)) will be inst ead t he set of even num bers, in fact only even num bers can be obt ained as t he result of a product in which one of t he

(a Car (wit h driver John) (wit h driver Jane)) As a consequence it seem s convenient t o int roduce a not at ion for att ribut ions allowing a unique individual value for t he at tribut e, like for ex ample in t he at tribut e mot her of a

fact ors is 2. The wit h kind of at tribut ion, corresponding t o binary relat ions, is defined as a special case: (a c (wit h a δ)) sam e (a c (of a δ))

person. We call t his kind of att ribut ions projective. We will ex press t his fact by using an att ribut ion of t ype wit h- unique.

where "sam e" is defined as is in bot h direct ions.

So we will say:

In addit ion, wit h- unique and with- every are wit h's wit h

(a Person (with- unique m ot her Sarah))

addit ional constraint s.

In addit ion a not at ion will be int roduced for constraining

D.

values for t he att ribut e, like for ex ample when we want to describe a person who owns just am erican cars. For t his purpose we introduce t he wit h- every t ype of at tribut ions.

The use of descript ion variables enables general fact s t o be assert ed. For ex am ple, from : (a Teacher (wit h subject = x )) is (an Ex pert (with field = x ))

(a Driver (wit h- every car (an Am erican- car)))

one can deduce t hat :

For wit h- unique and wit h- every a stronger ax iom t han

merging ex ist s, ex ample:

to

recompose

part ial

descript ions.

Variables

(a Teacher (wit h subject m usic)) is (an Ex pert (wit h field

For

m usic)) An int erest ing ex am ple is t he following:

(a Person (with- unique m ot her Sarah) (wit h m ot her (a Doct or))) is

((= a is (a Nat ural)) ∧

(a Person (with m ot her Sarah and (a Doct or)))

(0 is = a) ∧ (= n is = a) → ((a Successor (with- unique pred = n)) is = a)) →

(a Product (with- every fact or 1 (an Int eger))

((a Nat ural) is = a)

(wit h fact or 1 2)) is (a Product (with- every fact or 1 (an Int eger)) (wit h fact or 1 2 and (an Int eger))) Despit e t he proliferat ion of t he not at ion, it t urns out however t hat t here is only one prim it ive kind of at tribut ion. The ot her t ypes of at tribut ions have been defined in t erm s of t he prim it ive one and t heir propert ies derived as t heorem s.

This st at em ent is a form ulat ion of t he f ull second order induct ion principle for nat ural num bers. It can be read as follows: if = a is a set of numbers cont aining 0, and for each subset of = a, t he set of it s successors is cont ained in = a, t hen = a cont ains all t he nat ural num bers.

E.

Description abstraction

funct ion associat es t o a concept a n- ary relat ion; for ex am ple

Descript ion abst raction is a powerful const ruct provided in Om ega. In t he following ex am ple it is used t o revert a t eacherst udent relat ion:

t he set of t uples of t he relat ion associat ed t o t he concept

(any = x such- t hat (Carl is (a Teacher (wit h st udent = x ))))

The prim it ive t ype of att ribut ion is called of and it s form al sem ant ics is given in t erm s of n- ary relat ions. A sem ant ic

"Product " could be:

denot es t he set of individuals for which t he predicat ion

< 0 < fact or 1 0> < fact or 2 0> >

(Carl is (a Teacher (wit h st udent = x )))

< 0 < fact or 1 0> < fact or 2 1> >

is t rue, i.e. all t he st udent s of Carl.

< 0 < fact or 1 0> < fact or 2 2> > … < 1 < fact or 1 1> < fact or 2 1> > < 2 < fact or 1 2> < fact or 2 1> > < 2 < fact or 1 1> < fact or 2 2> > < 3 < fact or 1 3> < fact or 2 1> > < 3 < fact or 1 1> < fact or 2 3> > …


An

ex t ensive

discussion

on

t he

inference

rules

for

descript ion abstract ion can be found in [At t ardi and Sim i 81b].

F.

Viewpoints Reasoning

on

a

knowledge

base

m ay

oft en

require

considering different sit uat ions, eit her pert aining t o separat e worlds (real or hypot het ical), or arising because of changes over t im e.

1338

A viewpoint capabilit y is provided in Om ega t o enable t he

(inherit ance and at tribut ions) can be ex ploit ed t o build a

deal wit h such circum stances. For inst ance t o perform t he

knowledge base organized as a net work where each node

diagnosis of a piece of equipm ent , one m ight proceed by a

corresponds t o a descript ion and links t o t he is relat ionship. In

m et hod of hypot hesize and t est . Each hypot hesis is assert ed in

t his sect ion we discuss som e of t he t echniques t hat are used in

a new viewpoint , and it s consequences are ex am ined. If the

t he implem ent at ion of an Om ega knowledge base.

consequences are in disagreem ent wit h t he observat ions on t he equipm ent , t hen t he hypot hesis m ust be discarded,

A.

ret urning t o t he previous sit uat ion and select ing a different hypot hesis. The viewpoint m echanism of Om ega relies fully on a logical basis. In fact a viewpoint is defined as a collect ion of Om ega assert ions. The fact s t hat hold in a cert ain viewpoint are t hose t hat derive logically from t he assert ions in t he viewpoint by applicat ion of deduct ion rules. In t he em pt y viewpoint , cont aining no assert ions, only fact s t hat

are taut ologies hold. For

t his reason

it

is called

t aut ological viewpoint . A new viewpoint can be creat ed by ex t ending a previous viewpoint wit h an addit ional assert ion, or by combining t wo viewpoint s. As a consequence viewpoint s form t hem selves a (tangled) t ree st ruct ure, whose root is t he t aut ological viewpoint . Every st at em ent is assert ed in Om ega wit hin a cert ain viewpoint , eit her im plicit ely, or ex plicit ely, wit h t he form :

The ax iom at ic t heory consist s in a set of ax iom s defining a calculus of descript ions and, in close correspondence, a set of ax iom s defining a calculus of st at em ent s. For ex ample t he t ransit ivit y of is has a dual correspondent in t he transit ivit y of t he logical im plicat ion.

of

ax iom s

δ1 is δ2 , δ2 is δ3 δ1 is δ3 defines

t he

behaviour

of

att ribut ions, ex amples of which (om ission, st rict ness, m erging, fusing) have been present ed. The consist ency of t he Omega logic syst em has been est ablished in [5], t oget her with a result about complet eness: a form ula is valid is and only if it is a form al t heorem . Since t his logic syst em is a second order syst em , and in it a finit e cat egorical set of ax iom s can be form ulat ed for the nat ural num bers, t his result m ay appear t o contradict Gödel incom plet eness t heorem . This difficult y is overcom e by using a wider class of m odels t hen t he st andard m odels t o int erpret t he formalism [16]. IV.

perform ed exploit ing t he sam e st ruct ure. Anot her charact erist ic of t he im plem ent at ion is t o provide m eans t o att ach to t he lat t ice object s which are ex t ernal t o t he Om ega syst em , t hereby providing ways t o int erface Om ega t o ex t ernal data bases, or t o t he underlying Lisp syst em . The latt ice of classificat ion is designed t o cont ain all kinds of Om ega descript ions, from at om ic individuals, t o inst ance

not general ones, i.e. universally quant ified assert ions.

Mont ague 64].

set

new descript ions, access t o descript ions, and select ion of st at em ent s t hat are applicable during deduct ion can all be

in t he lat t ice enables t o overcom e t he lim it at ions of t radit ional

and Sim i 81b]. in t he st yle of nat ural deduct ion [Kalish and

furt her

A m ajor goal of t he implem entat ion has been t o design t he struct ure of t he latt ice so t hat operat ions like classificat ion of

sem ant ic net works, which can only represent specific fact s and

An ax iom at ic t heory of Om ega was present ed in [At tardi

A

several ways: as an efficient access pat h t o descript ions, and as a struct ure t o be ex plored during reasoning and search.

descript ions wit h individual at tribut e values t o stat em ent

THE AXIOMATIC THEORY

σ1 → σ2 , σ2 → σ3 σ1 → σ3

The Om ega knowledge base is arranged as a lat t ice of descript ions. The st ruct ure of t he lat t ice can be exploit ed in

cont aining variables. The inclusion of st at em ent s wit h variables

R is (a Resist or (wit h working YES)) in t est - 3

III.

Classificat ion

THE OMEGA KNOWLEDGE BASE

Classificat ion algorit hm algorit hm is invoked when a new descript ion has t o be insert ed in t he latt ice. The form al sem ant ics of Om ega [5] defines t he is relat ionship between descript ions. The problem of det erm ining subsum pt ion, and t herefore t he is relat ionship, is undecidable in Om ega and com plex in any language wit h a significant ex pressive power [9]. Therefore t he classificat ion algorit hm implem ent ed rest ricts t he att ent ion t o t hat part of t he is relat ion which is ex plicit ely represent ed by links or pat hs in t he latt ice and which is induced by a few st ruct ural propert ies of descript ions. Direct access point s in t he network are available t o reach individual const ant s and t he m ost generic represent at ive of inst ance descript ions wit h t he sam e concept (t he one wit h no at tribut ions). For inst ance, when t rying t o locat e (a Man (with dog Fido)), one st art s from t he node (a m an) which is direct ly accessible t hrough t he concept "m an". The m ost int erest ing t ask of the classificat ion algorit hm is t he classificat ion of inst ance descript ions. To classify an inst ance descript ion δ, t he classificat ion algorit hm st art s from

1)

The classificat ion

t he descript ion, which is direct ly accessible from t he concept nam e of δ and searches t he lat t ice downward. For each child s of t his descript ion, a t est is done t o det erm ine whet her s is m ore general t hen δ (i.e. δ is s). In part icular, given t wo inst ance descript ions: (a C [att r1 δ1] [att r2 δ2] [att r3 δ3])2

Om ega is primarily a calculus of descript ions. Descript ions and t he struct uring m echanism s embedded in t he logic


2

The not at ion [at tr δ] is an abbreviat ion for (wit h- unique at tr δ)

1339

(a C [at tr1 δ4] [at tr3 δ5])

As an ex ample, consider descript ion δ1[δ] which would appear in t he latt ice as a descendant of δ1[= x ]. When we ask

we can prove t hat

for a parent of δ1[δ], t he descript ion

and t his causes = x t o be inst ant iat ed wit h δ and t he

δ5])

descript ion δ2[δ] t o be ret urned. Wit h t his schem a unificat ion is not used for select ing am ong

if t he following holds:

applicable rules but just t o instant iat e an applicable rule.

δ1 is δ4

The abilit y t o handle st at em ent s wit h variables m eans for

δ3 is δ5

inst ance t hat Prolog clauses can be direct ly m apped t o Om ega

t hat is, if t wo inst ance descript ions are com pared during t he classificat ion process, t he corresponding att ribut e values are also com pared, according t o t he m onot onicit y of att ribut ions. If (δ is s) is not true, t hen none of t he children of s could be m ore specific t han δ eit her, so it rem ains just t o consider whet her t he opposit e is relat ion holds bet ween s and δ, so t hat δ could be classified as a parent of s. On t he ot her hand, if (δ is s), we recursively classify δ under s. Since in general a descript ion m ight have several children, a simple- m inded classificat ion algorit hm would have t o search t hrough all t he children in som e order and since a descript ion could be a child (or a parent ) of m ore t han one descript ion, all children will have t o be considered. This is a pot ent ial source of com binat orial ex plosion. Therefore we have devised a schem a by which children of an inst ance descript ion are ordered in such a way t hat t he pat h to an already ex ist ing descript ion (or t o t he place in which to insert

a

new

descript ion)

can

be

det erm ined

wit hout

backtracking. 2)

δ1[= x ] is encount ered,

(a C [att r1 δ1] [att r2 δ2] [att r3 δ3]) is (a C [at tr1 δ4] [att r3

Stat em ent s with variables We describe t he t echnique t hat is used t o let descript ions

cont aining variables appear sim ilar t o ot her descript ions, so

predicat ions wit h variables. A backward chaining st rat egy suit able for em ulat ion of Prolog in Om ega is ex pressed quite easily. Composit e stat em ent s In Om ega composit e st at em ent s can be built using t he tradit ional logical operat ors (∧, ∨, … ) st art ing from t he elem ent ary

is

predicat ion.

A

calculus

of

st at em ent s

is

ax iom at ized side by side wit h the calculus of descript ions. We will define a sort of canonical form for st at em ents (sim ilar t o clausal form of first order predicat e calculus) and discuss how a st at em ent ex pressed in t his form can be assim ilat ed in t he net work. The

t heoret ical

result s

present ed

here

const it ut e

a

necessary st ep t owards t he const ruct ion of t he knowledge base. Definit ion: Not Empt y[δ] ↔ ∃ i. Individual[i] ∧ i is δ Lemm a: ¬ (δ1 is δ2 ) ↔ Not Empt y[δ1 and not δ2 ] Given t hat not corresponds t o set complem ent and and t o set int ersect ion t his is a rat her obvious set t heoret ic st at em ent. What is relevant here is t hat logical negat ion can be ex pressed in t erm s of t he descript ion operat ors (not and and) plus t he ex ist ent ial quant ifier.

t hat t hey can be classified and dealt sim ilarly t o ordinary

Theorem :

descript ions.

Each Om ega st at em ent can be reduced t o t he following

A predicat ion stat em ent is represent ed by classifying it s subject and predicat e "at t he right place" in t he latt ice, and connect ing t he t wo part s by an is link. Suppose we have δ1[= x ] is δ2[= x ] t he right place t o put δ1[= x ] in t he lat t ice is where δ1[Som et hing] would appear, so t hat it can be reached and inst ant iat ed m oving upwards in t he latt ice from descript ions below it , i.e. from less general descript ions. Wit h a sim ilar argum ent t he right place for δ2[= x ] is where δ2[Not hing] would go.3 The advant age of t his solut ion is t hat t he unificat ion process can be split in two part s: half of t he job is done by t he classificat ion algorit hm which places a descript ion cont aining variables connect ed only t o t hose descript ions which can possibly mat ch it , t he second part is t he inst ant iat ion of variables wit h suit able values, which is done when t raversing t he lat t ice in eit her direct ion.

canonical form : σ1 ∧ σ2 ∧ … ∧ σk- 1 → σk where each σi is an elem entary predicat ion of t he form (δ1 is δ2 ) or a Not Empt y predicat e. A st at em ent of t he form σ1 ∧ σ2 ∧ ... ∧ σk- 1 → δ1 is δ2 Will be assim ilat ed in t he net work of descript ions as follows. The t wo nodes corresponding t o δ1 and δ2 will be connect ed by a special condit ional link (is- cond) whose associat ed condit ions will be t he st at em ent s σ1 , σ2 , ... σk- 1 . The m eaning is t hat t he link can be followed only if we are able t o verify t he associat ed

condit ions,

which

in

t urn

are elem ent ary is

predicat ions or Not Empt y predicat es. In addit ion Not Empt y[δ] can be assert ed by int roducing a fake individual connect ed by an is link t o δ and checked by searching for som e individual st art ing from δ and following is links downwards.

3

Act ually t he const ruct ion of t he descript ion t o classify is

slight ly m ore complex , in t he sense t hat one has t o t ake int o account whet her t he variable appears inside som e negat ion.


1340

B.

Tax onomic Reasoning

This is just one st rat egy t hat can be applied t o solve

Taxonomic reasoning is a deduct ion process based on t raversal of t he lat t ice of descript ions. Algorit hm s for t ax onom ic reasoning rely upon t he st ruct ure built by t he classificat ion. As a trivial ex ample of tax onom ic reasoning, we can consider t he classical syllogism :

inherit ance problem s, and in fact , ex ploit ing t he fact t hat with at tribut ions can have m ult iple values, we can also add t he following st rat egy: ↑(= d1 is = d2) is (a Predicat ion (with backward- strat egy (foreach d' being t he son of = d2

(a Man) is (a Mort al) Socrat es is (a Man)

(goal ↑ (= d1 is d') (new- at t empt )))))

The t hree descript ions will be linked in t he lat t ice wit h one anot her, and t o det erm ine whet her Socrat es is a m ort al, t wo links m ust be traversed t o det erm ine t he ex ist ence of a path bet ween t he t wo descript ions. It has been argued in [Aït - Kaci and Nasr 85] t hat int egrat ion of such kind of connect ion can be very beneficial also in a t radit ional int o a logic program m ing syst em in t erm s of improved efficiency due t o a signif icant reduct ion in t he num ber of resolut ion/ unificat ion st eps. In our approach, since not only const ant t erm s, but also assert ions wit h variables are arranged in t he st ruct ure of t he latt ice, all deduct ions can be perform ed in t his way, wit hout need

for

full

unificat ion,

just

by a process of

lat t ice

t raversal/ m arking/ inst ant iat ion.

C.

which would generat e one at t empt for each child d' of d2 t o prove t he subgoal t hat (d1 is d'). The t wo st rat egies above are all t hat is necessary t o sim ulat e t he backward chaining deduct ive st rat egy of Prolog. A Prolog

can

be

translat ed

int o

Om ega

in

a

strat egy. This part icular st rat egy is t he only one which is provided by Prolog. This strat egy works well in all sit uat ions where for inst ance an ex haust ive search of t he space of solut ion is desired. In m any ot her cases though t his m ay not be t he best strat egy.

D.

Met alevel and deduct ion strategies

program

straight forward way and t hen ex ecut ed according t o t he above

Metalevel and Viewpoints The following is a st rat egy corresponding t o t he ax iom of

Tax onom ic reasoning provides one level of deduct ive capabilit ies for Om ega. Since problem - solving m et hods oft en are t o be t ailored t o t he specific applicat ion, in Om ega t hey m ay be programm ed by writ ing st rat egies. St rat egies are procedural m echanism s t hat are at t ached to st at em ent s or classes of st at em ent s.

im plicat ion int roduct ion which is int erest ing since it involves hypot et hical reasoning. Inform ally it can be form ulat ed as follows:

if you want t o prove "σ1 → σ2", assume σ1 and prove σ2. The corresponding strat egy can be ex pressed as: ↑ (= s1 → = s2) is

To be able t o handle knowledge about strat egies, Om ega is provided wit h a m et a- level capabilit y [Att ardi- Sim i 1984]. The

(a St at em ent (wit h backward- st rat egy

synt act ical ent it ies of t he Om ega language, like const ants,

'(let ((nvp (creat e- viewpoint ))

inst ance descript ions or predicat ions can be referred t o in

(at t empt (new- at t empt )))

Om ega it self. For instance t he descript ion:

(vp- got o nvp) (assert = s1)

(d1 and d2)

(goal = s2 at t empt )))))

can be referred as:

The hypot het ical assumpt ion = s1 is form ulat ed here in a

(an And (wit h- unique arg1 d1) (wit h- unique arg2 d2))

new viewpoint nvp, which is a son of t he viewpoint where t he

or, wit h a short - hand not at ion, as:

goal was t ackled. If = s2 is proved in nvp, t hen t he original goal

↑(d1 and d2)

is no longer visible.

St rat egies are ex pressed

as att ribut ions of

succeds, and one ret urns t o t he original viewpoint , where = s1 st at em ent

descript ions, as in t he following ex am ple: ↑(= d is (= d1 and = d2)) is (a Predicat ion (wit h backward- st rat egy '(let ((at t empt (new- at t empt ))) (goal ↑(= d is = d1) at t empt )) (goal ↑(= d is = d2) at t empt ))) The above m et a- level st at ement says t hat one st rat egy

In t he implem ent at ion of Om ega, each st at em ent is t agged wit h t he viewpoint t o which it belongs, and is visible only from descendant

viewpoint s. To obt ain adequat e efficiency in

dealing wit h viewpoint s, viewpoint s are im plem ent ed using a bit array represent at ion for set s. Each bit however, rat her t hen represent

a single assert ion, represent s a collect ion

parent and son viewpoint .

which can be applied when t he goal is t o prove a st at em ent of t he form "d is (d1 and d2)", is t o generat e a single at t empt to prove t he t wo dist inct subgoals t hat (d is d1) and (d is d2).


of

assert ions describing t he increm ent al difference between a

1341

V.

AN EXAMPLE OF DIAGNOSYS

[swit ch (a Swit ch [working YES] [st at us (1 or 2)])]

To illust rat e t he use and t he reasoning capabilit ies of t he

[fan (a Fan [working YES])])

syst em , we present in t his sect ion a sm all protot ype of ex pert We will consider a very simple device, nam ely a hair- dryer. adm issible values for

all

[swit ch (a Swit ch [working YES] [st at us (1 or 2)])]

it s

[fan (a Fan [working YES])])

component s and at tribut es.

A.

(a Dryer [swit ch (a Swit ch [working YES] [st at us off])])

Struct ure

power swit ch, a fan and a resist or. Each of t hese com ponent s will be described in t urn. We use t he not at ion "[att r δ]" as an abbreviat ion for "(wit h- unique att r δ)".

(a Dryer [fan (a Fan [st at us (not spinning)])]) (a Dryer [swit ch (a Swit ch [working YES] [st at us (off or 1)])]) is (Ass. 9)

(a Dryer) is (a Dryer [power- cable (a Power- cable)]

(a Dryer [resist or (a Resist or [stat us cool])])

(Ass. 1)

(a Dryer [power- cable (a Power- cable [working YES] [st at us

[swit ch (a Swit ch)]

plugged])]

[fan (a Fan)]

[swit ch (a Swit ch [working YES] [st at us 2])]

[resist or (a Resist or [working YES])])

[resist or (a Resist or)]) (a Power- cable) is (a Power- cable [working (YES or (Ass. 2) (a Swit ch) is (a Swit ch [working (YES or NO)] (Ass. 3) [st at us (off or 1 or 2)]) (a Fan) is (a Fan [working (YES or NO)]

(a Dryer [resist or (a Resist or [stat us hot ])])

C.

Investigat ion We wish t o discover t he fault in a dryer which blows air but

does not produce heat . We will ask whet her a dryer like t hat can ex ist :

(Ass. 4) [st at us (spinning or (not spinning))])

(is- t here?

(a Resist or) is (a Resist or [working (YES or NO)]

(a Dryer

(Ass. 5)

[power- cable

[st at us (hot or cool)])

(a Power- cable [st at us plugged] [working = pw])]

Not e t hat a st at em ent like:

[swit ch (a Swit ch [st at us 2] [working = sw])] [fan (a Fan [st at us spinning] [working = fw])]

(a Dryer) is (a Dryer ...

[resist or (a Resist or [st at us cool] [working = rw])]))

provides a m ean t o assert propert ies valid for all dryers.

Int errelations The proper working of each of t he part s of t he dryer

depends on t he working of ot her com ponent s. What we describe here is just t he proper behavior of t he syst em , not all it s possible m isbehaviors. We consider m ore convenient t o describe what is known of t he int ernal working of t he syst em , rat her t han trying t o form ulat e all possible condit ions of fault in t he syst em . Many ex pert syst em s for diagnosis do inst ead represent ex plicit ely t he inform at ion about fault s, and how sympt hom s are relat ed t o a fault or t o ot her sympt hom s. Trying t o const ruct such int ercausal relat ionships t urns out t o be t he m ost difficult t ask in building an ex pert syst em for diagnosis. In our approach, t his informat ion is deduced by t he syst em, which reasons about it s m odel of t he working of t he syst em , and draws conclusions about possible sources of fault . This approach follows m ore closely t he approach t hat an engineer would apply in t rying t o find t he cause of a fault . (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]


is

(Ass. 10)

[st at us

(plugged or unplugged)])

B.

is

(Ass. 8)

A hair- dryer consist s of a power cable, a t hree- posit ion

NO)]

is

(Ass. 7)

Let us describe how a hair- dryer works. First we describe struct ure, and

(Ass. 6)

(a Dryer [fan (a Fan [st at us spinning])])

syst em for t he diagnosis of hardware.

it s int ernal

is

(a Dryer [fan (a Fan [st at us spinning])])

If such a dryer can ex ist , t he query will find values for t he variables = pw, = sw, = fw, and = rw, which describe t he working condit ion of all component s of such dryer.

D.

Strat egy Our st rat egy for finding what is wrong wit h our dryer will be

t he following. First , we collect all we know about t his dryer, gat hering all inform at ion we have about dryers. This is done by t ax onom ic reasoning, by ex ploring t he inherit ance net work and m erging all inform at ion t hat is so collect ed. This effect is provided by t he basic query st rat egy is- t here?. As a result of t his st rat egy, som e variables present in t he query can receive a value. The second st ep will be t o sim plify as m uch as possible t he descript ions

obt ained,

applying

algebraic

propert ies

of

descript ions.

E.

Finding the solut ion We will follow here t he above st rat egy. The deduct ive st eps

t hat we obt ain are t he following:

1342

(a Dryer [power- cable (a Power- cable [working = pw] [st at us plugged])]

[swit ch

(a

Swit ch

[working

YES

[st at us

2])]

[fan (a Fan [working YES] [st at us spinning])] [resist or (a Resist or [working YES] [st at us cool])]) or

[swit ch

(a

Swit ch

[working

= sw]

[st at us

2])]

[fan (a Fan [working = fw] [st at us spinning])]

(a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])] [swit ch

[resist or (a Resist or [working = rw] [st at us cool])])

(a

Swit ch

[working

YES

[st at us

2])]

[fan (a Fan [working YES] [st at us spinning])] [resist or (a Resist or [working NO] [st at us cool])])

is (by Ax iom of Dropping) (a Dryer [fan (a Fan [working = fw] [st at us spinning])])

Analyzing t he t wo part s separat ely we obtain: 1. (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]

is (by assert ion 7)

[swit ch

(a

Swit ch

[working

YES [st at us

2])]

[fan (a Fan [working YES] (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]

[swit ch (a Swit ch [working YES] [st at us (1

[st at us spinning])]

or 2)]) [resist or (a Resist or [working YES] [st at us cool])]) is

Applying

(from assert ion 5) (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])] [swit ch

sim ilar

st eps

involving

assert ion

5,

t his

descript ion reduces t o: (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]

(a

Swit ch

[working

YES

[st at us

2])]

[fan (a Fan [working YES] [st at us spinning])]

[swit ch

(a

Swit ch

[working

YES [st at us

2])]

[fan (a Fan [working YES]

[resist or (a Resist or [working (YES or NO)] [st at us

[st at us spinning])]

cool])]) [resist or (a Resist or [working YES] [st at us (hot and

F.

cool)])])

Simplificat ion We can now sim plify t he descript ion we have obt ained so far

of our dryer. The sim plificat ion rules are t hose deriving from t he ax iom s of Om ega. For instance, we know t hat :

which f urt her reduces t o: (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]

(individual- 1 and individual- 2) is Not hing

[swit ch (a Swit ch [working YES [st at us 2])]

since t he int ersect ion of two singlet ons consist ing of


different individuals is t he empt y set . Sim ilarly:

[resist or (a Resist or [working YES] [st at us Not hing])])

(Nothing or δ) is δ According t o t he ax iom of st rict ness of att ribut ions, if t he value of an at tribut ion is Nothing, t hen t he whole descript ion is Not hing.

i.e. Nothing 2. (a Dryer [power- cable (a Power- cable [working YES]

(a Dryer [power- cable Not hing]) is Nothing

[st at us plugged])] [swit ch

Also we can dist ribut e an or over at tribut es like in: (a Resist or [working (YES or NO)]) sam e

((a

Resist or

[working YES]) or (a Resist or [working NO]))

such st rat egy consist s in dist ribut ing t he descript ion (YES or NO), obt aining: (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]


Swit ch

[working

YES [st at us

2])]

[st at us spinning])] [resist or (a Resist or [working NO] [st at us cool])])

St art ing wit h t he descript ion obt ained so far, simplificat ion proceeds by applying t he st rat egy of split by cases. Form ally

(a

[fan (a Fan [working YES]

When put t ing back t he two part s, t he first one, which is Not hing will vanish, leaving just t he following result : (a Dryer [power- cable (a Power- cable [working YES] [st at us plugged])]

1343

[swit ch

(a

Swit ch

[working

YES

[st at us

2])]

VI.


CONCLUSIONS We have argued in favor of a logic wit h st ruct uring

[resist or (a Resist or [working NO] [st at us cool])]) The variables present in t he query have been suit ably inst ant iat ed during t his process, t herefore t he answer is:

capabilit ies and about problem orient ed deduct ion st rat egies in order t o m anage t he complex ity, as opposed t o ot her approaches where t he ex pressiveness of language is reduced or where formal propert ies of t he syst em are not provided.

= pw: YES = sw: YES = fw: YES = rw: NO

Om ega is a formal calculus of descript ions, which offers

It is int erest ing t o not e t hat if we t ranslat ed t he above problem in Prolog, we would get som et hing like t his:

such st ruct uring capabilit ies.

We discussed how, ex ploit ing

such feat ures of t he logic, knowledge can suitably be arranged in a net work so t hat algorit hm s can be devised t o perform

Dryer (_c, _s, _f, _r) :- IsCable(_c), IsSwit ch(_s), IsFan(_f), IsResist or(_r), CableSwit chFan (_c, _s, _f),

t ax onom ic reasoning. Deduct ion strat egies can be defined at t he m et alevel in order t o be t ailored t o t he problem .

CableSwit chResist or(_c, _f, _r) VII. AKNOWLEDGEMENTS IsCable(cable(_w, _s)) :- YesNo(_w), Plug (_s)

Carl Hewit t has been t he leading force in t he early st ages of

IsSwit ch(swit ch(_w, _s)) :- YesNo(_w), Pos (_s)

t he design of Om ega. Andrea Corradini, St efano Diom edi and

IsFan(fan(_w, _s)) :- YesNo(_w), Spin (_s)

Maurizio De Cecco of t he ESPRIT t eam at DELPHI have

IsResist or(resist or(_w, _s)) :- YesNo(_w), Temp (_s)

cont ribut ed t o t he im plem ent at ion of t he ideas present ed in

YesNo(YES) :-

t his paper. Cat iuscia Palam idessi e Sim one Mart ini helped t o

YesNo(NO) :-

clarify som e t heoret ical issues.

Plug(plugged) :Plug (not - plugged) :-

VIII. REFERENCES

Pos(0) :-

[1]

Pos (1) :-

Aït - Kaci,

H.

and

R.

Nasr

(1985)

LOGIN:

A

Logic

Programm ing Language wit h Built - in Inherit ance, MCC

Pos (2) :-

Technical Report number AI- 068- 85.

Spin(spinning) :-

[2]

Spin (not - spinning) :-

At t ardi, G., Corradini, A., De Cecco, M., Diom edi, S., Sim i, M. (1985a) The Om ega Knowledge Base Developm ent

Temp(hot ) :-

Environm ent . in "Esprit ' 85: a St at us Report on Cont inuing

Temp (cool) :-

Work", Nort h- Holland.

CableSwit chFan(cable(YES, plugged), swit ch(YES, 1),

[3]

fan (YES, spinning)) :-

At t ardi, G., Corradini, A., De Cecco, M., Sim i, M., (1985b) Building Ex pert Syst em s wit h Om ega. Technical Report

CableSwit chFan(cable(YES, plugged), swit ch (YES, 2),

ESP/ 85/ 2.

fan (YES, spinning)) :-

[4]

CableSwit chFan(_c, swit ch(YES, 0), fan(_w, not - spinning)) :CableSwit chResist or(_c, swit ch(YES, 0), resist or(_w, cool)) :CableSwit chResist or(_c, swit ch(YES, 1), resist or(_w, cool)) :-

At t ardi,

G.

and

Complet eness

of

M.

Sim i

Om ega,

(1981a) a

Consist ency

Logic

for

and

Knowledge

Represent at ion. IJCAI. Vancouver. [5]

At t ardi, G. and M. Sim i (1981b) Sem ant ics of Inherit ance

CableSwit chResist or(cable (YES, plugged), swit ch(YES, 2),

and Att ribut ions in t he Descript ion Syst em Om ega. AI

resist or(YES, hot )) :-

Mem o 642, M.I.T. [6]

If we ask t he following quest ion: :-

Dryer(cable(_cw,

plugged),

At t ardi,

G.

and

M.

Sim i

(1984) Metalanguage

and

Reasoning across Viewpoint s. Proc. of Sixt h European swit ch(_sw,

2),

fan(_fw,

spinning), resist or(_rw, cool)) t he Prolog int erpret er will not be able t o reach a conclusion because it is unable t o perform a case analysis on t he possible values for t he att ribut e "working" of t he resist or. This is due t o t he fact t hat negat ive inform at ion cannot be represent ed and used in t he Horn clause logic of Prolog. Furt her ex amples t he use of Om ega in t he const ruct ion of ex pert syst em s are report ed in [At t ardi et al. 1985a, Att ardi et al. 1985b]. An im plem ent at ion of t he Om ega syst em as described in t his paper has been done on a Sym bolics Lisp Machine, in t he fram ework of ESPRIT project P440.


Conference on Art ificial Intelligence, Pisa. [7]

Brachm an, R. J., (1985) "I lied about t he trees" or, Default s

Definit ions in Knowledge Represent at ion, AI Magazine, Vol.6, N.3. [8] Brachm an, R.J., Gilbert , V.P., Levesque, H.J., (1985) An Essent ial Hybrid Reasoning Syst em : Knowledge and Symbol Levels Account s of Krypt on. Proc. of 9th IJCAI, Los Angeles, 1985. [9] Brachm an, R.J., Levesque, H.J., (1984) The Tract abilit y of Subsumpt ion in Fram e Based Descript ion Languages, in Proc. of AAAI- 84 , Aust in, TX, August . [10] Brachm an, R.J., Schm olze (1985) An Overview of t he KLONE Knowledge Represent at ion Syst em . Cognit ive Science 9(2), April- June. and

1344

[11] deKleer. J., Doyle, J., Rich, C., St eele, G.L., Sussm an, G.J., (1978) AMORD: a Deduct ive Procedure Syst em . MIT AIMem o 435. [12] Fahlm an, S., (1979) NETL: A Syst em for Represent ing and Using Real World Knowledge. MIT Press. [13] Fikes, R. E. and T. P. Kehler, (1985) The Role of Fram eBased Represent at ion in Reasoning. CACM Vol. 28, No. 9. [14] Kalish and Montague (1964) Logic: Techniques of Form al Reasoning. Harcourt , Brace and World. [15] Kornfeld, W. (1982). Using parallel processing for problem solving. M.I.T. Ph.D. Thesis. [16] Henkin, L., Complet eness in t he Theory of t ypes, The

Journal of Symbolics Logic, Vol. 15, No. 2, 1950. C., Att ardi, G., Sim i M. (1980) Knowledge Embedding in t he Descript ion Syst em Om ega. Proc. of First AAAI Conference, St anford. Mart in, W., (1979) Descript ion and Specializat ion of Concept s. In Pat rick Winst on and Richard Brown, Eds. Art ificial Int elligence, MIT Press, Cam bridge. Mylopoulos, J., and Wong, H., (1980) Som e Feat ures of t he Tax is Dat a Model, Proc. of t he Conference on Very Large Data Bases, IEEE Comput er Societ y, pp. 399- 410. St eels L., (1979) Reasoning Modeled as a Societ y of Com m unicat ing Ex pert s, AI Lab Technical Report 542, MIT. St efik, M., Bobrow, D.G., Mit t al, S., Conway, L. (1983) Knowledge Program m ing in LOOPS: Report on an Ex perim ent al Course, The AI Magazine, Vol. 4, No. 3, Fall, pp 3- 14. Weyhrauch, R. (1980) "Prolegom ena t o a Theory of Form al Reasoning", Art ificial Intelligence 13, 1- 2, 133- 170.

[17] Hewit t ,

[18]

[19]

[20]

[21]

[22]


1345