Nonstandard Set Theories and

Information Management

VAROL AKMAN

akmantroycsbilkentedutr

Department of Computer Engineering and Information Science Bilkent University Bilkent

Ankara Turkey

MUJDATPAKKAN

pakkantrb ounbitnet

Department of Computer Engineering Bosphorus University Bebek Istanbul Turkey

Abstract The merits of set theory as a foundational to ol in mathematics stimulate its use in

various areas of articial intelligence in particular intelligent information systems In this pap er a

study of various nonstandard treatments of set theory from this p ersp ective is oered Applications

of these alternative set theories to information or knowledge management are surveyed

Keywords set theory knowledge representation information management commonsense rea

soning nonwellfounded sets hyp ersets

Intro duction

Set theory is a branch of mo dern mathematics with a unique place b ecause various other

branches can b e formally dened within it For example Book of the inuential works of

N Bourbaki is devoted to the theory of sets and provides the framework for the remaining

1

volumes Bourbaki said in Goldblatt

All mathematical theories may b e regarded as extensions of the general

theory of sets On these foundations I can state that I can build up the

whole of the mathematics of the present day

This brings up the p ossibility of using set theory in foundational studies in AI Mc

Carthy has emphasized the need for fundamental research in AI and claimed

that AI needs mathematical and logical theory involving conceptual innovations In an

op ening address McCarthy he stressed the feasibility of using set theory in AI b e

cause there is considerable b eauty economy and naturalness in using sets for information

mo deling and knowledge representation

In this pap er we rst give a brief review of classical set theory We avoid the technical

detailswhich the reader can nd in texts like Halmos Fraenkel et al and

Supp es and instead fo cus on the underlying concepts While we assume little or

no technical background in set theory p er se we hop e that the reader is interested in the

applications of this formal theory to the problems of intelligent information management

We then consider the alternative set theories which have b een prop osed to overcome the

limitations of the standard theory Finally we survey various nonstandard treatments

of set theory each innovating dierent asp ects such as urelements cumulative hierarchy

selfreference cardinality wellorderings and so on It is shown that such treatments

which are all very recent and sometimes esotericare quite useful to the I IS community

for there are assorted technical problems in information management eg commonsense

reasoning terminological logics etc that may prot from such nonstandard approaches

Early developments in set theory

G Cantors work on the theory of innite series should b e considered as the foundation

of the research in set theory In Cantors conception a set is a collection into a whole of

denite distinct ob jects of our p erception or our thought the elements of the set This

prop erty of deniteness implies that given a set and an ob ject it is p ossible to determine

if the ob ject is a memb er of that set In other words a set is completely determined by

its memb ers

In the initial stages of his research Cantor did not work from axioms However all of

his theorems can b e derived from three axioms Extensionality which states that two sets

are identical if they have the same memb ers Abstraction which states that for any given

prop erty there is a set whose memb ers are just those entities having that prop erty and

Choice which states that if b is a set all of whose elements are nonempty sets no two of

which have any elements in common then there is a set c which has precisely one element

in common with each element of b

The theory was so on threatened by the intro duction of some paradoxes which led to

its evolution In B Russell found a contradiction in G Freges foundational system

which was develop ed on Cantors naive conception van Heijenho ort This contra

diction could b e derived from the Axiom of AbstractionAxiom V in Freges systemby

considering the set of all things which have the prop erty of not b eing memb ers of them

selves This prop erty can b e denoted as x x or simply x x in the language of

rst order logic The Axiom of Abstraction itself can b e formulated as

xy x y x

where x is a formula in which y is free In the case of Russells Paradox x x x

and we have xy x y x x Substituting y for x we reach the contradiction

y y y y

Another antinomy o ccurred with the set of all sets V fx x xg The well

known Cantors Theorem states that the p ower set of V has a greater cardinality than V

itself This is paradoxical since V by denition is the most inclusive set This is the so

called Cantors Paradox and led to discussions on the size of comprehensible sets Strictly

sp eaking it was Freges foundational system that was overthrown by Russells Paradox

not Cantors naive set theory The latter came to grief precisely b ecause of the preceding

limitation of size constraint Later J von Neumann would clarify this problem as follows

Goldblatt Some predicates have extensions that are to o large to b e successfully

encompassed as a whole and treated as a mathematical ob ject

Paradoxes of the preceding sort were instrumental in new axiomatizations of the set

theory and in alternate approaches However it is b elieved that axiomatic set theory

would still have evolved in the absence of paradoxes b ecause of the continuous search for

foundational principles

Alternate axiomatizations

The new axiomatizations to ok a common step for overcoming the deciencies of the naive

approach by intro ducing classes NBG which was prop osed by von Neumann and

later revised and simplied by P Bernays and Godel was the most p opular

of these In NBG there are three primitive notions set class and memb ership Classes

are considered as totalities corresp onding to some but not necessarily all prop erties The

classical paradoxes are avoided by recognizing two typ es of classes sets and prop er classes

A class is a set if it is a memb er of some class Otherwise it is a prop er class Russells

Paradox is avoided by showing that the class Y fx x xg is a prop er class not a set

V is also considered as a prop er class The axioms of NBG are simply chosen with resp ect

to the limitation of size constraint

Other approaches against the deciencies of the naive approach alternatively played

with its language and are generally dubb ed typetheoretical Russell and Whiteheads

Theory of Types is the earliest and most p opular of these Whitehead Russell

In this theory a hierarchy of typ es is established to forbid circularity and hence avoid

paradoxes For this purp ose the universe is divided into typ es starting with a collection

M of individuals The elements of M are of typ e Sets whose memb ers are of typ e

are said to b e of typ e sets whose elements are of typ e are said to b e of typ e and so

n n+1

on The memb ership relation is dened b etween sets of dierent typ es eg x y

Therefore x x is not even a valid formula in this theory and Russells Paradox is trivially

avoided

Similar to the Theory of Typ es is Quines New Foundations NF which he invented to

overcome some unpleasant asp ects of the former Quine NF uses only one kind of

variable and a binary predicate letter for memb ership A notion called stratication is

2

intro duced to maintain the hierarchy of typ es In NF Russells Paradox is avoided as in

the Theory of Typ es since the problematic w is not stratied

ZF set theory

ZermeloFraenkel ZF is the earliest axiomatic system in set theory The rst axiomati

zation was by E Zermelo A A Fraenkel observed a weakness of Zermelos

system and prop osed a way to overcome it His prop osal was reformulated by T Skolem

by intro ducing a new axiom ZF is carried out in a language which includes sets

as ob jects and for memb ership Equality is dened externally by the Axiom of Ex

tensionality which states that two sets are equal if and only if i they have the same

elements

ZFs essential feature is the cumulative hierarchy it prop oses Parsons The

intention is to build up mathematics by starting with and then construct further sets in a

stepwise manner by various dened op erators Hence there are no individuals urelements

in the universe of this theory The cumulative hierarchy works as follows Tiles

The Nul l Set Axiom guarantees that there is a set with no elements ie This is the

only set whose existence is explicitly stated The Pair Set Axiom states the existence of a

set which has a memb er when the only existing set is So the set fg can now b e formed

now and we have two ob jects and fg The application of the axiom rep etitively yields

any nite numb er of sets each with only one or two elements It is the Sum Set Axiom

which states the existence of sets containing any nite numb er of elements by dening the

S

union of already existing sets Thus ff fgg ff fgggg f fg f fggg However

it should b e noted that all these sets will b e nite b ecause only nitely many sets can b e

formed by applying Pair Set and Sum Set nitely many times It is the Axiom of Innity

which states the existence of at least one innite set from which other innite sets can b e

formed The set which the axiom asserts to exist is f fg f fgg f fg f fggg g

Figure The cumulative hierarchy

The cumulative hierarchy is depicted in Figure Thus the ZF universe simply starts

with the and extends to innity It can b e noticed that cumulative hierarchy pro duces

all nite sets and many innite ones but it do es not pro duce all innite sets eg V

While the rst ve axioms of ZF are quite obvious the Axiom of Foundation cannot

b e considered so The axiom states that every set has elements which are minimal with

resp ect to memb ership ie no innite set can contain an innite sequence of memb ers

x x x x Innite sets can only contain sets which are formed by a nite

3 2 1 0

numb er of iterations of set formation Hence this axiom forbids the formation of sets

which require an innity of iterations of an op eration to form sets It also forbids sets

which are memb ers of themselves ie circular sets Russells Paradox is avoided since

3

the problematic set x fxg cannot b e shown to exist The Axiom of Separation makes

it p ossible to collect together all the sets b elonging to a set whose existence has already

b een guaranteed by the previous axioms and which satisfy a prop erty

xux u x v x

The axiom do es not allow to simply collect all the things satisfying a given meaningful

description together into a set as assumed by Cantor It only gives p ermission to form

subsets of a set whose existence is already guaranteed It also forbids the universe of sets

to b e considered as a set hence avoiding the Cantors Paradox of the set of all sets The

Axiom of Replacement is a stronger version of the Axiom of Separation It allows the use

of functions for the formation of sets but still has the restriction of the original Axiom of

Separation It should b e noted that these two axioms are in fact not single axioms but

4

axiom schemes Therefore ZF is not nitely axiomatizable

The Power Axiom states the existence of the set of all subsets of a previously dened

set The formal denition of the p ower op eration P is P x fy y xg The Power

Axiom is an imp ortant axiom b ecause Cantors notion of an innite numb er was inspired

by showing that for any set the cardinality of its p ower set must b e greater than its

cardinality

The Axiom of Choice is not considered as a basic axiom and is explicitly stated when

used in a pro of ZF with the Axiom of Choice is known as ZFC

It should b e noted that the informal notion of cumulative hierarchy summarized ab ove

has a formal treatment The class WF of wel lfounded sets is dened recursively in ZF

starting with and iterating the p ower set op eration P where a rank function R is

5

dened for O r d the class of all ordinals

R α . . ω

(α) . R . 2 1

0

Figure The WF universe in terms of ordinals

R P R

S

R R when is a limit ordinal

S

WF fR O r dg

The WF universe is depicted in Figure which b ears a resemblance to Figure This is

justied by the common acceptance of the statement that the universe of ZF is equivalent

to the universe of WF

ZF and NBG pro duce essentially equivalent set theories since it can b e shown that

NBG is a conservative extension of ZF ie for any sentence if ZF j then NBG

j The main dierence b etween the two is that NBG is nitely axiomatizable whereas

ZF is not Still most of the current research in set theory is b eing carried out in ZF

Nevertheless ZF has its own drawbacks Barwise While the cumulative hierarchy

provides a precise formulation of many mathematical concepts it may b e asked whether

it is limiting in the sense that it might b e omitting some interesting sets one would like

to have around eg circular sets Clearly the theory is weak in applications involving

selfreference b ecause circular sets are prohibited by the Axiom of Foundation

Strangely enough ZF is to o strong in some ways Imp ortant dierences on the na

ture of the sets dened in it are o ccasionally lost For example b eing a prime numb er

2

b etween and is a dierent prop erty than b eing a solution to x x but

this distinction disapp ears in ZF Besides the Principle of Parsimony which states that

simple facts should have simple pro ofs is quite often violated in ZF Barwise For

example verication of a trivial fact like the existence in ZF of a b the set of all ordered

6

pairs hx y i such that x a and y b relies on the Power Set Axiom

Alternate approaches

Admissible sets

Admissible sets are formalized in a rst order set theory called KripkePlatek KP Bar

wise weakened KP by readmitting the urelements and called the resulting system KPU

Barwise Urelements are the ob jects or individuals with no elements ie they

can o ccur on the left of but not on the right They are not considered in ZF b ecause ZF (α) VM

M

Figure The universe of admissible sets

is strong enough to live without them But since KPU is a weak version of KP Barwise

decided to include them

KPU is formulated in a rst order language L with equality and It has six axioms

The axioms of Extensionality and Foundation are ab out the basic nature of sets The

7

axioms Pair Union and Separation treat the principles of set construction These

0

ve axioms can b e taken as corresp onding to ZF axioms of the same interpretation The

imp ortant axiom of Col lection assures that there are enough stages in the hierarchical

0

construction pro cess

The universe of admissible sets over an arbitrary collection M of urelements is dened

recursively

V

M

V P M V

M M

S

V V if is a limit ordinal

M M

S

V V

M M

This universe is depicted in Figure adapted from Barwise It should b e noticed

that the KPU universe is like the ZF universe excluding the existence of urelements since

it supp orts the same idea of cumulative hierarchy

8

If M is a structure for L then an admissible set over M is a mo del U of KPU of

M

the form U M A where A is a nonempty set of nonurelements and is dened

M

in M A A pure admissible set is an admissible set with no urelements ie it is a mo del

of KP

KPU is an elegant theory which supp orts the concept of cumulative hierarchy and re

sp ects the principle of parsimony The latter claim will b e proved in the sequel But it

still cannot deal with selfreference b ecause of its hierarchical nature

Hypersets

It was D Mirimano who rst stated the fundamental dierence b etween well

founded and nonwellfounded sets He called sets with no innite descending memb ership

sequence wellfounded and others nonwellfounded Nonwellfounded sets have b een exten

sively studied through decades but did not show up in notable applications until Aczel

This is probably due to the fact that the classical wellfounded universe was a

rather satisfying domain for the practicing mathematicianthe mathematician in the

street Barwise Aczels work on nonwellfounded sets evolved from his interest in a

{c,d} b

c d

Figure a fb fc dgg in hyp erset notation

Ω

Figure The Aczel picture of

mo deling concurrent pro cesses He adopted the graph representation for sets to use in his

theory A set a fb fc dgg can b e unambiguously depicted as in Figure in this repre

sentation Aczel where an arrow from a no de x to a no de y denotes the memb ership

relation b etween x and y ie x y

A set pictured by a graph is called wellfounded if it has no innite paths or cycles

and nonwellfounded otherwise Aczels AntiFoundation Axiom AFA states that every

graph wellfounded or not pictures a unique set Removing the Axiom of Foundation FA

from the ZFC and adding the AFA results in the Hyperset Theory or ZFC AFA ZFC

without the FA is denoted as ZFC What is advantageous with the new theory is that

since graphs of arbitrary form are allowed including the ones containing prop er cycles

one can represent selfreferring sets Barwise For example the graph in Figure

is the picture of the unique set fg

The picture of a set can b e unfolded into a tree picture of the same set The tree whose

9

no des are the nite paths of the apg which start from the p oint of the apg whose edges

0

are pairs of paths hn n n n n i and whose ro ot is the

0 0

path n of length one is called the unfolding of that apg The unfolding of an apg always

0

pictures any set pictured by that apg Unfolding the apg in Figure results in an innite

tree analogous to fff ggg

According to Aczels conception for two sets to b e dierent there should b e a genuine

structural dierence b etween them Therefore all of the three graphs in Figure depict

the unique nonwellfounded set

Aczel develops his own Extensionality concept by intro ducing the notion of bisimula Ω Ω Ω

Ω

ΩΩ

Ω

Figure Other pictures of

A F A universe

well - founded

universe

Figure The AFA universe

tion A bisimulation b etween two apgs G with p oint p and G with p oint p is a

1 1 2 2

relation R G G satisfying the following conditions

1 2

p Rp

1 2

if nRm then

0 0

for every edge n n of G there exists an edge m m of G such that

1 2

0 0

n Rm

0 0

for every edge m m of G there exists an edge n n of G such that

2 1

0 0

n Rm

Two apgs G and G are said to b e bisimilar if a bisimulation exists b etween them

1 2

this means that they picture the same sets It can b e concluded that a set is completely

10

determined by any graph which pictures it

The AFA universe can b e depicted as in Figure extending around the wellfounded

universe b ecause it includes the nonwellfounded sets which are not covered by the latter

Equations in the AFA universe Aczels theory includes another imp ortant

useful feature equations in the universe of Hyp ersets

0

b e the Let V b e the universe of hyp ersets with atoms from a given set A and let V

A A

0 0

universe of hyp ersets with atoms from another given set A such that A A and X is

0

dened as A A The elements of X can b e considered as indeterminates ranging over

the universe V The sets which can contain atoms from X in their construction are called

A

Xsets A system of equations is a set of equations

fx a x X a is an Xset g

x x

0

for each x X For example cho osing X fx y z g and A fC Mg thus A

fx y z C Mg consider the system of equations

x fC y g

y fC z g

z fM xg

A solution to a system of equations is a family of pure sets b sets which can have only

x

sets but no atoms as elements one for each x X such that for each x X b a

x x

Here is a substitution operation dened b elow and a is the pure set obtained from a

by substituting b for each o ccurrence of an atom x in the construction of a

x

The Substitution Lemma states that for each family of pure sets b there exists a unique

x

op eration which assigns a pure set a to each X set a viz

a f b b is an Xset such that b ag f x x a X g

11

The Solution Lemma can now b e stated Barwise Moss If a is an X set

x

then the system of equations x a has a unique solution ie a unique family of pure

x

sets b such that for each x X b a

x x x

This lemma can b e stated somewhat dierently Pakkan Letting X again b e

the set of indeterminates g a function from X to P X and h a function from X to A

there exists a unique function f for all x X such that

f x ff y y g xg hx

Obviously g x is the set of indeterminates and hx is the set of atoms in each Xset a

x

of an equation x a In the ab ove example g x fy g g y fz g g z fxg and

x

hx fCg hy fCg hz fMg and one can compute the solution

f x fC fC fM xggg

f y fC fM fC y ggg

f z fM fC fC z ggg

As another example due to Barwise Etchemendy it may b e veried that the

system of equations

x fC M y g

y fM xg

z fx y g

has a unique solution in the universe of Hyp ersets depicted in Figure with x a y b

and z c

This technique of solving equations in the universe of hyp ersets can b e useful in mo del

ing information which can b e cast in the form of equations Akman Pakkan eg

situation theory Barwise Perry databases etc since it allows us to assert the

existence of some graphs the solutions of the equations without having to depict them

with graphs We now give an example from databases

AFA and relational databases Relational databases emb o dy data in tabular

forms and show how certain ob jects stand in certain relations to other ob jects As an c

a b

CM

Figure The solution to a system of equations

FatherOf MotherOf John Bill Sally Tim John Kitty Kathy Bill Tom Tim Kathy Kitty

BrotherOf

Bill Kitty

Figure A relational database SizeOf FatherOf 3 MotherOf 3 BrotherOf 1

SizeOf 4

Figure The SizeOf relation for the preceding database

example adapted from Barwise the database in Figure includes three binary

relations FatherOf MotherOf and BrotherOf A database model is a function M with

domain some set Rel of binary relation symb ols such that for each relation symb ol R Rel

M

R is a nite binary relation that holds in mo del M

0

If one wants to add a new relation symb ol SizeOf to this database then Rel Rel

M

0

fSizeOf g A database mo del M for Rel is correct if the relation SizeOf contains all

pairs hR ni where R Rel and n jRj the cardinality of R Such a relation can b e seen

in Figure It may app ear that every database for Rel can b e extended in a unique way

0

to a correct database for Rel Unfortunately this is not so

Assuming the FA it can b e shown that there are no correct database mo dels Be

cause if M is correct then the relation SizeOf stands in relation SizeOf to n denoted by

SizeOf SizeOf n But this is not true in ZFC b ecause otherwise hSizeOf ni SizeOf

If Hyp erset Theory is used as the metatheory instead of ZFC in mo deling such

databases then the solution of the equation

M M

x fhR jR ji R Rel g fhx jRel j ig

which can b e found by applying the Solution Lemma is the desired SizeOf relation

Commonsense set theory

If we want to design articial systems which will work in the real world they must have a

go o d knowledge of that world and b e able to make inferences out of their knowledge The

common knowledge which is p ossessed by any child and the metho ds of making inferences

from this knowledge are known as common sense Any intelligent task requires it to some

degree and designing programs with common sense is one of the most imp ortant problems

in AI McCarthy claims that the rst task in the construction of a general intelligent

program is to dene a naive commonsense view of the world precisely enough but also

adds that this is a very dicult thing He states that a program has common sense if

it automatically deduces for itself a suciently wide class of immediate consequences of

anything it is told and what it already knows

It app ears that in commonsense reasoning a concept can b e considered as an indivis

ible unit or as comp osed of other parts as in mathematics Relationships again as in

mathematics can also b e represented with sets For example the notion of so ciety can

b e considered to b e a relationship b etween a set of p eople rules customs traditions etc

What is problematic here is that commonsense ideas do not have very precise denitions

since the real world is to o imprecise For example consider the denition of so ciety

adapted from Websters Ninth New Col legiate Dictionary

So ciety gives p eople having common traditions institutions and collective

activities and interests a choice to come together to give supp ort to and b e

supp orted by each other and continue their existence

It should b e noted that the notions tradition institution and existence also

app ear to b e as complex as the denition itself So this denition should probably b etter

b e left to the exp erience of the reader with all these complex entities

Nevertheless whether or not a set theoretical denition is given sets are useful for

conceptualizing commonsense terms For example we may want to consider the set of

traditions disjoint from the set of laws one can quickly imagine two separate circles

of a Venn diagram We may not have a wellformed formula w which denes either

of these sets Such a formation pro cess of collecting entities for further thought is still

imp ortant and simply corresp onds to the set formation pro cess of formal set theories ie

the comprehension principle

Having decided to investigate the use of sets in commonsense reasoning we have to

concentrate on the prop erties of such a theory Instead of directly checking if certain

settheoretic technicalities have a place in our theory we rst lo ok from the commonsense

reasoning p oint of view and examine the settheoretic principles which cannot b e excluded

from such reasoning

Desirable properties

We rst b egin with the general principles of set formation The rst choice is whether

to allow urelements This seems like the right thing to do b ecause in a naive sense a

set is a collection of individuals satisfying a prop erty This is what exactly corresp onds

to the unrestricted Comprehension Axiom of Cantor However we have seen that this

leads to Russells Paradox in ZF The problem arises when we use a set whose completion

is not over yet in the formation of another set or even in its own formation Then we

are led to the question when the collection of all individuals satisfying an expression can

b e considered an individual itself Since we are talking ab out the individuals as entities

formed out of previously formed entities the notion of cumulative hierarchy immediately

comes to mind

The cumulative hierarchy is one of the most common construction mechanisms of our

intuition and is supp orted by many existing theories viz ZF and KPU It can b e illus

trated by the hierarchical construct in Figure where we have bricks as individuals and

make towers out of bricks and then make walls out of towers and so on Perlis

In the cumulative hierarchy any set formed at some stage must b e consisting of the ure

lements if included in the theory and the sets which have b een formed at some previous

stage Sho eneld but not necessarily at the preceding stage as in Russells Typ e

Theory

At this p oint the problem of sets which can b e memb ers of themselves arises since

such sets are used in their own formation Circularity is obviously a common means of

commonsense knowledge representation For example nonprot organizations are sets of

individuals and the set of all nonprot organizations is also a set all these are expressible

in the cumulative hierarchy But what if the set of all nonprot organizations wants to

b e a memb er of itself since it also is a nonprot organization This is not an unexp ected

event Figure b ecause this umbrella organization may b enet from having the status

of a nonprot organization eg tax exemption etc Perlis bricks

towers made of bricks

walls made of towers

rooms made of walls

buildings made of rooms

. .

.

Figure A construction a la cumulative hierarchy

Red Cross

? UNICEF Amnesty NPO NPO

International

Figure Nonprot organizations

Thus we conclude that a p ossible commonsense set theory should also allow circular

sets This is an imp ortant issue in representation of metaknowledge and is addressed in

Feferman and Perlis In these references a metho d which reies creates

a syntactic term from a predicate expressiona w into a name for the w asserting that

the name has strong relationship with the formula is presented In this way any set of

w are matched with a set of names of w thereby allowing selfreference by the use of

names In Feferman the need for typefree admitting instances of selfapplication

frameworks for semantics is esp ecially emphasized However such formalizations which

also capture the cumulative hierarchy principle are not very common Among theories

revised so far Aczels theory is the only one which allows circularity By prop osing his

AntiFoundation Axiom Aczel overro de the FA of ZF which prohibits circular sets but

preserved the hierarchical nature of the original axiomatization

Applications

Situations for knowledge representation Situations are parts of the reality

that can enter into relations with other parts Barwise Perry Their internal

structures are sets of facts and hence they can b e mo deled by sets There has b een a

considerable deal of work on this esp ecially by Barwise himself Barwise a He

used his Admissible Set Theory Barwise as the principal mathematical to ol in

the b eginning However in the handling of circular situations he was confronted with

problems and then discovered that Aczels theory could b e a solution Barwise c

Circular situations are common in our daily life For example consider the situation in

which we utter the statement This is a b oring situation While we are referring to a

situation say s by saying this situation our utterance is also a part of that situation

As another example one sometimes hears radio announcements concluding with This

announcement will not b e rep eated If announcements are assumed to b e situations

then this one surely contains itself

Barwise dened the op eration M to mo del situations with sets taking values in hy

12

p ersets and satisfying

if b is not a situation or state of aairs then M b b

if hR a ii then M hR b ii which is called a state model where b is a

function on the domain of a satisfying bx M ax

if s is a situation then M s fM s j g

Using this op eration Barwise proves that there is no largest situation corresp onding

to the absence of a universal set in ZF

We also see a treatment of selfreference in Barwise Etchemendy where the

authors concentrate on the concept of truth In this study two conceptions of truth are

13

examined primarily on the basis of the notorious Liar Paradox The authors make use

of Aczels theory for this purp ose A statement like This sentence is not expressible in

English in ten words would b e represented in Aczels theory as in Figure adapted

from Barwise Etchemendy where hE p ii denotes that the prop osition p has

the prop erty E if i and it do es not have it if i which is the case in Figure if

we take E to b e the prop erty of b eing expressible in English in ten words p = = >

{E} {E,}

E

{p} {p,0}

0

p

Figure The Aczel picture of This sentence is not expressible in English in ten words

Common knowledge mutual information Two card players P and P

1 2

are given some cards such that each gets an ace Thus b oth P and P know that the

1 2

following is a fact

Either P or P has an ace

1 2

When asked whether they knew if the other one had an ace or not they b oth would

answer no If they are told that at least one of them has an ace and asked the ab ove

question again rst they b oth would answer no But up on hearing P answer no P

1 2

would know that P has an ace Because if P do es not know P has an ace having heard

1 1 2

that at least one of them do es it can only b e b ecause P has an ace Obviously P would

1 1

reason the same way to o So they would conclude that each has an ace Therefore

b eing told that at least one of them has an ace must have added some information to

the situation How can b eing told a fact that each of them already knew increase their

information This is known as Conways Paradox The solution relies on the fact that

initially was known by each of them but it was not common know ledge Only after it

b ecame common knowledge it gave more information Barwise d

Hence common knowledge can b e viewed as iterated knowledge of of the following

form P knows P knows P knows P knows P knows P knows and so

1 2 1 2 2 1

on This iteration can b e represented by an innite sequence of facts where K is the

relation knows and s is the situation in which the ab ove game takes place hence s

hK P si hK P si hK P hK P sii hK P hK P sii However considering the

1 2 1 2 2 1

system of equations

x fhK P y i hK P y ig

1 2

y s fhK P y i hK P y ig

1 2

0 0

the Solution Lemma asserts the existence of the unique sets s ands s satisfying these

Figure Finite cardinality without wellordering

equations resp ectively where

0 0 0

s fhK P s s i hK P s s ig

1 2

0

Then the fact that s is common knowledge can more eectively b e represented by s

which contains just two infons and is circular

Possible memb ership One further asp ect to b e considered is p ossible mem

b ership which might have many applications mainly in language oriented problems This

concept can b e handled by intro ducing partial functionsfunctions which might not have

corresp onding values for some of their arguments A commonsense set theory may b e

helpful in providing representations for dynamic asp ects of language by making use of

partiality For example partiality has applications in mo dality the part of linguistics

which deals with mo dal sentences ie sentences of necessity and p ossibility dynamic

pro cessing of syntactic information and situation semantics Mislove et al

We had mentioned ab ove that situations can b e mo deled by sets Consider a situation

s in which you have to guess the name of a b oy viz

s j The boys name is Jon or the boys name is John

This situation can b e mo deled by a set of two states of aairs The problem here is that

neither assertion ab out the name of the b oy can b e assured on the basis of s b ecause of the

disjunction A solution to this problem is to represent this situation as a partial set one

with two p ossible memb ers In this case s still supp orts the disjunction ab ove but do es

not have to supp ort either sp ecic assertion Another notion called clarication which is

a kind of general informationtheoretic ordering helps determine the real memb ers among

0 0

p ossible ones If there exists another situation s where s j The boys name is Jon then

0

s is called a clarication of s

Cardinality and wellordering Imagine a b ox of black and white balls

Figure We know that there are balls in the b ox or formally the cardinality of

the set of balls in the b ox is After shaking the b ox we would say that that the balls

in the b ox are not ordered any more or again formally the set of the balls do es not have

a wellordering This however is not true in classical set theory b ecause a set with nite

cardinality must have a wellordering Zadrozny

While the formal principles of counting are precise enough for mathematics we can

observe that p eople also use other quantiers like many or more than half for counting

purp oses in daily sp eech For example if asked ab out the numb er of balls in the b ox in

Figure one might have simply answered Many balls So at least in principle dierent

counting metho ds can b e develop ed for commonsense reasoning It is natural to exp ect

for example that a system which can represent a statement like A group of kids are

shouting should probably decline to answer questions such as Who is the rst one

Zadrozny

We also exp ect our theory to ob ey the parsimony principle This is a very natural

exp ectation from a commonsense set theory We have observed that the pro of of the ex

istence of a simple fact like the Cartesian pro duct of two sets a b required the use of

6

the Power Axiom in ZF The set obtained in this manner just consists of pairs formed

of one element of the set a and one element of the set b To prove this the strong Power

Axiom should not b e necessary We observe this in KPU set theory where the pro of is

14

obtained via denitions and simple axioms Barwise

Applications of nonstandard theories in I IS

Cowen showed in a landmark pap er that researchers in b oth set theory and com

puter science are studying similar ob jects graphs conjunctive normal forms set systems

etc and their interrelations The moral is he b elieves advances in one area can then

often b e of use in the other Gilmore provides evidence within the framework of his

NaDSet theory that the absolute character of ZF or NBG set theories is not easy to sup

p ort cf his remark The ad ho c character of the axioms of ZermeloFraenkel set theory

and the equivalent classset theory of GodelBernays naturally leads to skepticism ab out

their fundamental role in mathematics Set theory has also b een the sub ject of research

in automated theorem proving Brown gave a deductive system for elementary set

theory which is based on truthvalue preserving transformations Quaife presented

a new clausal version of NBG comparing it with the one given in Boyer et al

and claimed that automated development of set theory could b e improved We will now

review some essential research eorts by Perlis Zadrozny Mislove et al Dionne et al

Barwise and Yasukawa and Yokota

Perliss work

Perliss approach was to develop a series of theories towards a complete commonsense set

theory Perlis He rst prop osed an axiom scheme of set formation for a naive set

theory which he named CST

0

y xx y x I ndx

Here is any formula and I nd is a predicate symb ol with the intended extension indi

viduals However I nd can sometimes b e critically rich ie if is the same with I nd

itself then y may b e to o large to b e an individual This is the case of Cantors Axiom

of Abstraction Therefore a theory for a hierarchical extension for I nd is required To

supp ort the cumulative hierarchy Perlis extended this theory to a new one called CST

1

using the socalled Ackermanns Scheme

HC y HC y xx HC x z HC z xx z x

1 n

Here HC x can b e interpreted as x can b e built up as a collection from previously

obtained entities CST is consistent with resp ect to ZF Unfortunately it cannot deal

1

with selfreferring sets

Perlis nally prop osed CST which is a synthesis of the universal reection theory of

2

GilmoreKripke Gilmore which forms entities regardless of their origins and self

referential asp ects GK set theory has the following axiom scheme where each w x

has a reication x with variables free as in and distinguished variable x



y x y

15

where y do es not app ear in There is also a denitional equivalence denoted by

axiom

w z xx w x z

GK is consistent with resp ect to ZF Perlis Perlis then prop osed the following

axioms to augment GK

Ext x y ext x ext y

Ext x ext x

Ext x HC y x ext y

A y y HC xx x HC ext HC xx x

ext 1 n

These axioms provide extensional constructions ie collections determined only by

their memb ers Thus while GK provides the representation of circularity these axioms

supp ort the cumulative construction mechanism This theory can deal with problems like

nonprot organization memb ership describ ed earlier Perlis

Zadroznys work

Zadrozny do es not b elieve in a sup er theory of commonsense reasoning ab out sets but

rather in commonsense theories involving dierent asp ects of sets He thinks that these can

b e separately mo deled in an existing set theory In particular he prop osed a representation

scheme based on Barwises KPU for cardinality functions hence distinguishing reasoning

ab out wellorderings from reasoning ab out cardinalities and avoiding the b ox problem

mentioned earlier Zadrozny

Zadrozny interprets sets as directed graphs and do es not assume the FA A graph in

his conception is a triple hV S E E i where V is a set of vertices SE is a set of edges and

E is a function from a subset of SE into V V It is assumed that x y i there exists

an edge b etween x and y He denes the edges corresp onding to the memb ers of a set as

EM s fe ES v E e v sg

In classical set theory the cardinality of a nite set s is a onetoone function from

a natural numb er n onto a set ie a function from a numb er onto the no des of the

graph of the set However Zadrozny denes the cardinality function as a onetoone order

preserving mapping from the edges EM s of a set s into the numerals N ums an entity

of numerals which is linked with sets by existence of a counting routine denoted by and

which can take values like or aboutve or many The last element

of the range of the function is the cardinality The representation of the four element set

k fa b fx y g dg with three atoms and one twoatom set is shown in Figure adapted

from Zadrozny The cardinality of the set is aboutve ie the last element

of N ums which is the range of the mapping function from the edges of the set The

cardinality might well b e if N ums was dened as Zadrozny then proves two k

1 2 3 about-five . . .

c a b d

x y

Figure Onetoone order preservation

imp ortant theorems in which he shows that there exists a set x with n elements which do es

not have a wellordering and there exists a wellordering of typ e n ie with n elements

the elements of which do not form a set

More recent work of Zadrozny treating dierent asp ects of computational mereology

visa vis set theory can b e found in Zadrozny Kim In this work it is shown

that it is p ossible to write formal sp ecications for the pro cess of indexing or retrieving

multimedia ob jects using set theory and mereology Mereology is understo o d as a theory

of parts and wholes viz it tries to axiomatize the prop erties of the relation PartOf The

pap er argues that b oth set theory and mereology have sp ecic roles to play in the theory

of multimedia for they b oth provide us with a language to address multimedia ob jects

alb eit with complementary roles the former is used to index the latter to describ e knowl

edge ab out indexes This observation leads to a simple formalism regarding the storage

and retrieval of multimedia material

Mislove et als work

Mislove Moss and Oles develop ed a partial set theory ZFAP based on

protosets which is a generalization of HFthe set of wellfounded hereditarily nite sets

16

A protoset is like a wellfounded set except that it has some kind of packaging which can

hide some of its elements There exists a protoset which is empty except for packaging

From a nite collection x x one can construct the clear protoset fx x g which

1 n 1 n

has no packaging and the murky protoset x x which has some elements but also

1 n

packaging For example a murky set like contains and as elements but it might

contain other elements to o We say that x is claried by y x v y if one can obtain y

from x by taking some packaging inside x and replacing this by other protosets

Partial set theory has a rst order language L with three relation symb ols for ac

tual memb ership for p ossible memb ership and set for set existence The theory



set

consists of two axioms and ZFA the relativization of all axioms of ZFA ZF Aczels

AFA to the relation set The two axioms are i Pict which states that every partial set

has a picture a set G which is a partial set graph corresp onding to the accessible p ointed

graph of Aczel and such that there is a decoration d of G with the ro ot decorated as x

and ii PSA which states that every such G has a unique decoration Partial set theory

ZFAP is the set of all these axioms ZFAP is a conservative extension of ZFA

Dionne et als work

Dionne Mays and Oles prop ose a new approach to intensional semantics of term

subsumption languages Their work is inspired by the research of Wo o ds who

suggested that a more intensional view of concept descriptions should b e taken In general

most of the work in semantics of term subsumption languages adopts an extensional view

Thus concepts are interpreted as sets of ob jects from some universe Roles of concepts

are interpreted as binary relations over the universe Concept descriptions are complex

predicates and one inquires whether a given instance satises a complex predicate

Dionne et al consider a small subset of KREP a KLONE style of language Brach

man Schmolze They oer a general discussion of cycles viz how they arise and

how they are handled in KREP The restricted language they use allows simple concepts

concepts formed by conjunctions and roles of concepts whose value restrictions are other

concepts In their vision a knowledge base is a set of p ossibly mutually recursive equa

tions involving terms of this concept language Essential use is made of Aczels theory

of nonwellfounded sets esp ecially the bisimulation relation A socalled concept algebra

is develop ed as a new approach to semantics of term subsumption languages It is shown

that for the ab ove subset of KREP these algebras accurately mo del the pro cess of sub

sumption testingeven in the presence of cycles Concept algebras also allow for multiple

denitions ie concepts with dierent names that semantically stand for the same con

cept

Barwises work

Barwise b attempted to prop ose a set theory Situated Set Theory not just for use

in AI but for general use He mentioned the problems caused by the common view of set

theory with a universal set V but at the same time trying to treat this universe as an

extensional whole lo oking from outside which he names unsituated set theory His

prop osal is a hierarchy of universes V V V which allows for a universe of

0 1 2

a lower level to b e considered as an ob ject of a universe of a higher level He leaves the

axioms which these universes have to satisfy to ones conception of set b e it cumulative

or circular There are no paradoxes in this view since there is always a larger universe one

can step back and work in Therefore the notions of set prop er class and the set

theoretic notions ordinal cardinal are all context sensitive dep ending on the universe

one is currently working in This prop osal supp orts the Reection Principle which states

that for any given description of the sets of all sets V there will always b e a partial

universe satisfying that description

Barwise c also studied the mo deling of partial information and again exploited

Hyp erset Theory for this purp ose For this purp ose he used the ob jects of the universe

V of hyp ersets over a set A of atoms to mo del nonparametric ob jects ie ob jects with

A

complete information and the set X of indeterminates to represent parametric ob jects

ie ob jects with partial information The universe of hyp ersets on A X is denoted as

V X analogous to the adjunction of indeterminates in algebra

A

objectrefAkman et al

date

kindpaper

authorsV Akman M Pakkan M Surav

titleLectures in Commonsense Set Theory

journalProc Natl Acad Sci Utopia

volume

pages

year

Figure An ob ject in QU I X O T E

For any ob ject a V X Barwise calls the set

A

par a fx X x TC ag

where TC a denotes the transitive closure of a the set of parameters of a If a V

A

then par a since a do es not have any parameters Barwise then denes an anchor

as a function f with domainf X and rangef V A which assigns sets to

A

indeterminates For any a V X and anchor f af is the ob ject obtained by replacing

A

each indeterminate x par a domainf by the set f x in a This is accomplished by

solving the resulting equations by the Solution Lemma

Parametric anchors can also b e dened as functions from a subset of X into V X to

A

assign parametric ob jects not just sets to indeterminates For example if ax is a para

metric ob ject representing partial information ab out some nonparametric ob ject a V

A

and if one do es not know the value to which x is to b e anchored but knows that it is of

the form by another parametric ob ject then anchoring x to by results in the ob ject

aby which do es not give the ultimate ob ject p erhaps but is at least more informative

ab out its structure

Yasukawa and Yokotas work

Yokota and Yasukawa prop osed a knowledge representation language called

QU I X O T E for deductive and ob jectoriented databases In QU I X O T E an ob

ject consists of an identier oid and prop erties each attribute of which is a triple

hl abel oper ator v al uei

In Yasukawa Yokota a partial semantics for the semantics of ob jects in

QU I X O T E is given This work is along the lines of Dionne et al and will now b e

summarized Consider Figure where the left hand side of is an oid and the right

hand side is the related prop erties An ob ject consists of an oid and its prop erties and

can b e written as a set of attribute terms with the same oid as follows

ol a l b ol a ol b

1 2 1 2

An oid can b e dened intensionally by a set of rules as follows note the Prologlike style

pathf r om X to Y ar cf r om X to Y

pathf r om X to Y ar cf r om X to Z pathf r om Z to Y

Here pathf r om X to Y is transitively dened from a set of facts such as

ar cf r om a to b etc and the oid is generated by instantiating X and Y as a

result of the execution of the program This guarantees that an ob ject has a unique oid

even if it is generated in dierent environments Furthermore circular paths can b e de

ned X ol X X jfX ol X g This denotes that a variable X is an oid with

the constraint X ol X The semantics of oids is dened on a set of lab eled graphs

as a sub class of hyp ersets Thus an oid is mapp ed to a lab eled graph and an attribute

is related to a function on a set of lab eled graphs Since the metatheory is ZFC AFA

all of the familiar settheoretic techniques to deal with circular phenomena are brought to

b ear

Conclusion

Set theory can b e useful in intelligent information management The metho dology may

change of course A universal set theory answering many technical questions in infor

mation management can b e develop ed by means of prop osing new axioms or mo difying

existing ones Alternatively and more conservatively dierent settheoretic concepts may

b e examined and mo died based on existing set theories No matter what prop osal is

followed we b elieve that further research in this eld should b e to the advantage of the

I IS community

Acknowledgments

We would like to thank an anonymous referee of the Journal of Intel ligent Information

Systems for critical comments which were crucial in revising the original manuscript Larry

Kerschb erg and Maria Zemankova two of the EditorsinChief of JIIS were extremely kind

and accommo dating when we rst submitted the pap er We are also indebted to Wlo dek

Zadrozny IBM T J Watson Research Center for moral supp ort and technical advice

As usual we are solely resp onsible for this nal version

The rst authors research is supp orted in part by the Scientic and Technical Research

Council of Turkey under grant no TBAG

Notes

On the other hand another great mathematician of this century R Thom said in The

old hop e of Bourbaki to see mathematical structures arise naturally from a hierarchy of sets

from their subsets and from their combination is doubtless only an illusion Goldblatt

A w w is said to b e stratied if integers are assigned to the variables of w such that all

o ccurrences of the same free variable are assigned the same integer all b ound o ccurrences

of a variable that are b ound by the same quantier are assigned the same integer and for

every subformula x y the integer assigned to y is equal to the integer assigned to x

1 2 0 1

For example x y z x is stratied as x y z x

Let us recall Russells Paradox We let r b e the set whose memb ers are all sets x such that

x is not a memb er of x Then for every set x x r i x x Substituting r for x we

obtained the contradiction

The explanation is not dicult When we are forming a set z by cho osing its memb ers we

do not yet have the ob ject z and hence cannot use it as a memb er of z The same reasoning

shows that certain other sets cannot b e memb ers of z For example supp ose that z y

Then we cannot form y until we have formed z Hence y is not available and therefore cannot

b e a memb er of z Carrying this analysis a bit further we arrive at the following Sets are

formed in stages For each stage S there are certain stages which are b efore S

Stages are imp ortant b ecause they enable us to form sets Supp ose that x is a collection of

sets and is a collection of stages such that each memb er of x is formed at a stage which is

a memb er of If there is a stage after all of the memb ers of then we can form x at this

stage Now the question b ecomes Given a collection of stages is there a stage after all

of the memb ers of We would like to have an armative answer to this question Still

the answer cannot always b e yes if is the collection of all stages then there is no stage

after every stage in

A theory T any set of formulas closed under implication ie for any if T j then

0 0

T is nitely axiomatizable i there is a nite T T such that for every in T T j

A linear ordering of a set a is a wel lordering if every nonempty subset of a has a least

element Informally an ordered set is said to b e wel lordered if the set itself and all its

nonempty subsets have a rst element under the order prescrib ed for its elements by that

set An ordinal numb er stands for an order typ e which is represented by wellordered sets

First call a set a transitive if xx a x a Then a set is an ordinal numb er or

ordinal if it is transitive and wellordered by It should b e noted that in case of nite sets

the notions of cardinal numb er and ordinal numb er are the same The class of all ordinal

numb ers is denoted by O r d The relationship b etween two ordinals and is

dened i If then is called a successor ordinal else it is a limit ordinal

The Axiom of Innity guarantees the existence of limit ordinals other than In fact the

set of natural numb ers is the next limit ordinal Figure

We have to prove the theorem

cxx c y z y a z b x hy z i

to show that the Cartesian pro duct set exists The main p oint of the pro of is that if x hy z i

and if y a and z b then x P P a b Then by the Axiom of Separation

cxx c x P P a b y z y a z b x hy z i

The theorem to b e proved is equivalent to without the statement x P P a b Then

we must show that the equivalence in still holds when that statement is eliminated

Given it follows that

x c

implies

y z y a z b x hy z i

To prove the converse implication we must show that implies x P P a b since

it is obvious by that implies By and the denition of ordered pairs x

ffy g fy z gg and by the hyp otheses y a and z b we have

fy g fa bg and fy z g fa bg

Then by the following theorem which can b e proved by the Power Axiom

b P a b a

we conclude

fy g P a b and fy z g P a b

Thus ffy g fy z gg P a b ie x P a b and again by we have x P P a b

The collection of formulas of a language L is the smallest collection containing the

0

atomic formulas of L and inductively dened as

a If in in then is also in

b If are in then and are also in

c If is in then u v and u v are also in for all variables u and v

A structure for a rst order language L is a pair hM I i where M is a nonempty set called

the domain of the structure and I is an interpretation function assigning functions and

predicates over M to the symb ols in L

Aczel uses tagged graphs to represent sets ie each childless no de in the graph is tagged

by an atom or A pointed graph is a tagged graph with a sp ecic no de n called its

0

point A p ointed graph is accessible denoted as apg if for every no de n there is a path

n n n A decoration of a graph is a function D n for each no de n dened

0 1

as

tag n if n has no children

D n

fD m m is a child of ng otherwise

A picture of a set is an apg which has a decoration in which the set is assigned to the p oint

The uniqueness prop erty of AFA leads to an intriguing concept of extensionality for hy

p ersets The classical extensionality paradigm that sets are equal i they have the same

memb ers works ne with wellfounded sets However this is not of use in deciding the equal

ity of say a f ag and b f bg b ecause it just asserts that a b i a b a triviality

Barwise Etchemendy However in the universe of hyp ersets a is indeed equal to b

since they are depicted by the same graph To see this consider a graph G and a decoration

0

D assigning a to a no de x of G ie D x a Now consider the decoration D exactly

0 0

the same as D except that D x b D must also b e a decoration for G But by the

0 0

uniqueness prop erty of AFA D D so D x D x and therefore a b

The Solution Lemma is an elegant result but not every system of equations has a solution

First of all the equations have to b e in the form suitable for the Solution Lemma For

example the pair

x fy z g

y f xg

cannot b e solved since it requires the solution to b e stated in terms of the indeterminate z

As another example the equation x P x cannot b e solved b ecause Cantor has proved in

ZFC that there is no set which contains its own p ower setno matter what axioms are

added to ZFC

A state of aair aka infon hR a ii is a triple where R is an nary relation a is an

appropriate assignment of ob jects and i is the p olarity if there is at least one instance of

R holding of a and otherwise By a state of aair a state that aairs may or may not

b e in is meant When i that state of aair is called a fact and the p olarity is usually

omitted For example the state of aair hsleeping Tom gardeni is a fact if Tom is sleeping

in the garden

According to the Liar Paradox also known as the Epimenides Paradox Epimenides the

Cretan said All Cretans are liars Now this statement cannot b e true since this would

make Epimenides a liar leading to the falsity of his statement The statement cannot b e

false either since this would imply that Cretans are not liars hence what Epimenides says

should b e true leading to a contradiction

The predicate of a b u which is dened as u is an ordered pair hy z i with y a and z b

in KPU is Hence Separation can b e used once it is known that there exists a set

0 0

c with hy z i c for all y a and z b This follows from Collection as follows Given

0

y b there exists a set d hy z i So by Collection there exists a set w such that

0 y

hy z i w for all z b Applying Collection again we have

y 0

y a w z b d w d hy z i

S

so there is a c such that for all y a z b hy z i w for some w c Thus if c c

1 1 1

then hy z i c for all y a and z b

The is the notation for rst writing all symb ols in terms of and then passing nega

tions in through to predicate letters and nally replacing each o ccurrence of a subformula

x y in the result by x y

The transitive closure of a set denoted by TC a is dened by recursion as follows

S

0

a a

S S S

n+1 n

a a

S S

n

TC a f a n g

S S

2

a For any innite cardinal m the set H m is dened as Hence TC a a a

H m fx jTC xj mg The elements of H m are said to b e hereditarily of cardinality

less than m fH n n g is the set of hereditarily nite sets Hence every element

of a hereditary set is a hereditary set

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Authors biographies

Varol Akman is asso ciate professor of Computer Engineering and Information Science

at Bilkent University Ankara Turkey His research is concentrated in mathematical logic

and commonsense reasoning and additionally philosophy and natural language semantics

Prior to Bilkent he held a senior researcher p osition with the Centrum vo or Wiskunde

en Informatica Amsterdam and a p ostdo ctoral p osition with the Vakgro ep

Informatica Rijksuniversiteit Utrecht the Netherlands Akman received a

BS degree in Electrical Engineering high honors and an MS degree in Com

puter Engineering from Middle East Technical University Ankara In

he was a Fulbright scholar and a research assistant at Rensselaer Polytechnic Institute

New York and received a PhD degree in Computer and Systems Engineering Akman

was conferred the research prize of the Scientic and Technical Research Council

of Turkey He also received the Prof Mustafa N Parlar Foundations scientic research

award in Akmans do ctoral dissertation was published in by SpringerVerlag

He is also the senior editor of a SpringerVerlag volume on design He is an editorial

b oard memb er of the journal Computers Graphics and the Bilkent University Lecture

Series in Engineering research monographs marketed by SpringerVerlag He is in the

program committees of various conferences and is a memb er of American Asso ciation for

Articial Intelligence and Europ ean Foundation for Logic Language and Information

Mujdat Pakkan is a do ctoral student in the Department of Computer Engineering

Bosphorus University Istanbul Turkey His areas of interest include knowledge repre

sentation and natural language semantics Pakkan received a BS degree in Computer

Engineering honors from Aegean University Izmir Turkey He then received an

MS degree in Computer Engineering and Information Science from Bilkent Univer

sity Ankara Pakkans masters thesis studies nonwellfounded sets from a computational

p ersp ective