Fuzzy sets in topoi

Fuzzy Sets and Systems 8 (1982) 93--99 North-Holland Publishing Company

FUZZY

SETS

93

IN TOPOI*

C.V. N E G O I T A Bucharest, Romania Received October 1980 Revised December 1980

1. Introduction

Recently, speaking about the current interest in fuzzy optimization [1], ~i noticed that fuzzy set theory is closely connected with time-variant systems, and that this connection has to be explored in depth. My conclusion was that by using, fuzzy sets a time-contraction is taking place, and, that this contraction is z. consequence of our method of analysis. This paper aims to explore further this topic.

2. Sets t h r o u g h time

Let us consider denumerable strings Xo ~ X 1 "-~ X 2 ...L> . . . of functions between sets, and strings of subsets So ~ $1 ~ $2---> • " • such th~,t Si c Xi. Then, the elements of Xo can be classified not in two but in many ways with respect to S~: elements Xo now in So, elements Xo not in So but with t"Xoe S,, and elements for which even this is false. This classification can be described by a characteristic function F:X---> H, H ={0, 1, 2 , . . . , o0}. Fxo= n is the time till xo lands in the subset S,. Clearly, the dynamics t nuances the set Xo, or, in other words, t induces a fuzzy set on Xo. Therefore, if S and X are the two diagrams So--->S;--* • • • Xo---->X1 --->" • ", then S c X means X ~ / 2 w h e r e / 2 is a Heyting algebra. S, X a n d / 2 are objects in a topos defined by the pullback S-->I

X - ~ /2 where 1 is the terminal,/2 the classifier and 1 --~/2 is the monic 'true'. In this way one can reconstruct a subobject of X from its characteristic function, starting from the special subobject 'true'. The characteristic functions required for the category of sets and for any other topos could be effectively described by pullback. * Paper presented at the Round 'Fable on Fuzzy Sets, Universit6 de Lyon (June 1980).

0165-0114/82/0000-0000/$02.75 © 1982 North-Holland

94

C. V. Negoim

In the naive theory of fuzzy sets, a fuzzy set was viewed as a function. No set or subset can be identified. In the framework of the theory of topoi, whose details I will not bother to describe here, objects and subobjects are concrete things, having the same nature. They are described by topoi having different subobject classifiers. A fuzzy set is therefore a subobject in a topos. In this framework, new tools for the study of time-variant systems and context-freedom seem to be at hand [2, 3].

3. Diachronism and synchronism It is worth noting that the predicates of X are arrows X---> ~ in the system and not metamathematical formulae about it. In topos theoretic jargon we say that usual logic is internalized [4]. The axiomatic description of topoi provides a formutation of axiomatic set theory wholly different from the usual set-theoretic axioms on the membership relation, and its further study can cast considerable light on fuzzy sets. The topos-theoretic outlook consists in the rejection of the idea that there is a fixed universe of constant sets within which mathematics can and should be developed, and the recognition that the notion of 'variable structure' may be more conveniently handled within a universe of continuously variable sets [5]. It is this generalization of ideas from constant to variable sets which lies at the heart of topos theory, and I think that through this path one can gain a great deal of understanding regarding time-variant systems. The underlying mathematical idea is to exploit the link between diagrams and fuzzy logic. In this way one can correlate historism and structural analysis. "Through time' means 'diachronic', of or being in development. Synchronism means the freezing in order to see the invariants. Proper to synchronism is the subordination of explanation to description or even the reduction of the former to the second. The diachronic perspective, on the contrary, considers the temporal dimension. Virtually all work on fuzzy sets has been confined exclusively to synchronic approaches. What about a diachronic approach? Before we examine such a question we might do well to review what we know about fuzzy sets in topoi. The expert in the theory of fuzzy sets is already familiar with the representation theorem [6]. According to this theorem a fuzzy subset is equivalent with a family of sets (levels). Embedding fuzzy sets in topoi is a step further. We are speaking now of chains, about sets through time. The diachronic aspect is emphasized.

4. State of the art As far as I know, this is the first survey on fuzzy sets in topoi, so I want to avoid getting entangled in detailed proofs, in order to concentrate on the main aspects of the subject. The aim is to give the reader some sort of idea as to what has happened so far, and where the subject is likely to go.

Fuzzy sets in topoi

95

The subject of fuzzy sets in topoi has three beginnings. The curtain rises in 1977 when Arbib [7] warned that we should be concerned with a deeper exploration of what fuzzy set theory is or ought to be. Boyd [8] considers a poset P and an order relation '~<'. Let A =(A,~)~p, a family of sets such that for every a <~/3 there is a map Ao '

>A.

with the property a ~
Given a second family B, a morphism f ' A ~ B is a family (f,~),~e such that the diagram L

----" B.

Aa

to

; Bo

commutes. The category with objects A, B and morphisms f is a category of presheaves on P, denoted P'. Now, if P = [0, 1] we have an elementary topos [0, 1]^ whose truth values are in the interval [0, 1]. Therefore, if we consider a family ~ = (X,)~ where X is t~fe same set for all a, then ~ is an object in [0, 1]^ and a characteristic morphism ~A __~ ~t0.11" is equivalent with a fuzzy subset

A:x

[0, 1].

Eytan [9] follows a different path. He defines a category Fuz H whose objects are applications in a Heyting algebra ¢:X~H,

~: Y---~H and whose morphisms ~ - ~ ~b are applications

f:XxY---~H such that

f(x, y) ~ ¢(X)A ~(y), f(x, y)Af(x, y ' ) ~ d ( y , y') d:XxX---~H.

for all x, y,

The catei~:c: y Fuz H of H-valued fuzzy sets is a topos if H is Boolean. ~ Now, if a t Observation due to A.M. Pitts from t h e University of Cambridge, who suggested H-Set, a category studied by Fourman and Scott [19], as a suitable topos for fuzzy mathematics. See also [21 ].

C. V. Negoita

96

H-set is a pair (X, d) such that

d(x, y)=d(y,x),

d(x, y)Ad(y,z)<~d(x,z),

one can make these the objects of a category Set H as follows: objects are H-sets and arrows (X, d ) ~ (Y, d) are maps f such that

d(x, x') ^ f(x, f(x', y), f(x, y ) ^ d(y, y')~< f(x, y'), [(x, y) ^ f(x, y') d(y, y') V¥ [(x, y) = d(x, x), composition (X, d) ~ (Y, d) -~ (Z, d) is defined by putting gf(x, z ) = V f(x, y ) ^ g(y, z). Y

Set H is a topos and Fuz H ~ Set H is a logical embedding between the topos of fuzzy sets and the topos of H-sets. In other words, we have a ~ubcategory of H A as defined by Boyd. Stefanescu [10] considers the category Setf whose objects are applications A'X~[O,

1],

/~ • Y--~ [0, 1]

and whose morphisms F:,~ --* B are applications F : X x Y-.[O, 1] such that

f(x, Vy f(x, y) ^ f(x, y') <~d(y, Y')B

={/3(y) 0

fory=y', otherwise,

A(x)= V f(x, y). Clearly, [0, 1] is a Heyting algebra and Set [0, 1] as defined by Eytan is equivalent with Setf. Let me pause to summarize. A topos is a category satisfying certain conditions. These conditions were concocted to described categories of presheaves. The category Set o where D is a diagram of the form •---~----~.---~. • • ts a particular case of the category 15 as defined by Boyd, if P is a chain. Set H used by Eytan, representing an object as a family, the family being a chain, is also a particular case of Set ° . There is another trend which starting from topos theory looks for a natural reason why the theory of fuzzy sets can be considered a generalized set theory. Let us consider the monic 1 ~ l't where a is an arbitrary truth value. By pullback F-l(a) ---, 1 X--~ 0 we can see that F-t(a) is a subobject of X. Therefore, for every a there is a subobject and for H o m O , O) a family of subobjects. This is the tooos-theoretic version of the representation theorem [11].

Fuzzy sets in topoi

97

Now, let us consider the set Sub X of the subobjects of X. This set is partially ordered by the evident inclusion of subobjects and there is a greatest lower bound, which is just the intersection of subobjects as found by pullback. The movement in this category can be regarded as a model for irreversible processes~ [12, 13, 14].

5. Applications All my instincts tell me that the pretty subject of fuzzy sets in topoi will be useful. For illustration let us consider a feedback loop. It is known [15] that to every multiple input and output sequential machine can be assigned a suitable biaction of the output monoids on the left and the input monoids on the right. A multiple input and output sequential machine with inputs X, Z and outputs Y, Z consists of a set Q of states and functions O x(XxZ)-,

Q,

Qx(XxZ)--,(Y×Z).

The free monoid generated by a set A is denoted A* and has as its elements finite sequences of elements of A, with concatenation as multiplication and the empty sequence as identity. A configuration of the above sequential machine is an element of Y*x Z * x Q:,< X * x Z*. We define an equivalence relation --- on configurations whereby each configuration is identified with its successor (if any) under the operation of the machine. Let C be the set of configurations and t~ the set of equivalence classes. Then C admits a left Y*x Z*, right X * x Z* action. This action is called the characteristic biaction of the machine, interpreted as a functor (Y* x Z*) °p x (X* x Z*) ---, Set which is a generalized predicate. Any functor is a diagram representing a process, a transformation. Note that ~:his is a functor defined on the category attached to a monoid. It defines a family o[ endomorphisms, applications C ~ C. We are facing an indexing problem. By using a fuzzy logic, a compact representation is possible. The framework is Set r', D = Y* x Z* x X* x Z*, that is a topos. An object in this topos is

where a, b, c are elements (words) of the monoid D, putting into evidence a characteristic function C"~ {0.1, 2 , . . . , x}. Thus, through this technique we have the opport, mity to capture and summarize changes simply by employing fuzzy logic. In other words, a dynamics analysis can be made codifyin.~g these changes. The technology of capturing and codifying the dynamics of the |~ehavioral process is that of topos theory. We have described a different approach to the feedback problem elsewhere in some detail [16]. According to the internal model principle [17] a structurally stable synthesis of a feedback controller claims that the feedback loop incorporates a model of the dynamic system which generates the exogenous signals to be

C. V. Negoim

98

processed. Bearing in mind that any dynamic induces an action, the exosystem will be described as the action X ~ X and the controller as W ~ W. The internal model principle means that the diagram

X----~ X W----~ W commutes. This principle is valid also for fuzzy systems [18] and in this case the commutative diagram is

F(X)

, F(X)

F( W)

>F( W)

where F(X) means the set of all fuzzy sets with X as support. If [ ~ F(X), then there are two possibilities to define the map T~:

f inf Ta(D(y)=~;E,_,(,)f(x) if a-l(Y) ~ O, otherwise,

• f sup f(x) ifa-~(y)~), T~([)(Y) = ~Oa-'(Y)

x,y~X.

otherwise.

F(X) is a lattice, and therefore a category. The functor

F(X), a - l ( f ) = fa preserved order on F(X). 3;~tis functor has two adjoints, a fight one, Ta and a left a - ' : F ( X ) --,

one, T'~. Now, instead of Ta : F ( X ) ~ F(X) which is an application in the category Set, let us consider 3 a : P ( X ) ~ , P(X) which is a morphism in a topos E. Then, the subobjects of X are in bijection with F(X) and 3 a is the equivalent of T~. The logical functors have been internalized. The immediate conclusion is that topos theory is a good framework to describe the internal model principle both for machines and humans. We have done more than merely drawing analogies between animals and machines--like the naive cybernetician--but have developed a conceptual system at a level of generality which subsumes both animals and machines. This is precisely the role of a topos theory as viewed by Bainbridge'[15] which use it to study both flowcharts and networks. He says that the a priori distinction between software and hardware is the distinction between procedure interconnection (the semantics of flowcharts which is known to be described by regular expression calculus)and system interconneetion (the semantics of networks, described by a certain logical calculus, dual to a calculus of regular expression).

Fuzzy sets in topoi

99

Pulling back to a higher level of abstaction, hardware and software can be approached in the same framework. The price paid is the need for a fuzzy logic [20].

References [1] C.V. Negoita, The current interest in fuzzy optimization, Busefal 2 (1980) 39-54. [2] C.V. Negoita and R. Roman, On the logic of dynamic systems, Kybernetes 9 (198{)) 189-192. [3] A.F. Rocha, E. Francoza, M.I. Hadler and M.A. Balduiono, Neural languages, Fuzzy Sets and Systems 3 (1980 11-35.

[4] S. MacLane, Sets, topoi and internal logic in categories, in: H.E. Rose apd J.C. Shepherdson, Eds., Logic Colloquium '73 (North-Holland; Amsterdam, 1975) 119-134.

[5] P.T. Johnstone, Topos Theory (Academic Pre~s, New York, 1977). [6] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to System Analysis (Birkh~iuser Verlag, Basel, 1975).

[7] M. Arbib, Book review essay, Bull. Am. Math. Soc. 81 (1977) 947-951. [8] J.P. Boyd, Topoi as models of fuzzy sets, Paper presented at the 22nd Annual Meeting of the Society for General Systems Research (Washington, Feb. 1978).

[9] M. Eytan, Fuzzy sets: a topos-logical point of view, Fuzzy Sets and Systems 5 (1981} 47-67. [10] C.AI. Stefanescu, On the category Setf as a topos, Communication at the Seminar on Fuzzy Systems, Institute of Management and Informatics (Bucharest, 1977).

[11] C.V. Negoita and C.AI. Stefanescu, Fuzzy objects in topoi: a generalization of fuzzy sets, Bul. Inst. Politehn. Ia~i 24 (1978) 25-28.

[12] C.V. Negoita, On the stability of fuzzy systems, Proc. Int. Conf. on Cybernetics and Society (Tokyo, 1978).

[13] C.V. Negoita, On fuzzy systems, in: M. Gupta, R. Ragade and R. Yager, Eds., Advances in Fuzzy Sets Theory and Applications (North-Holland, Amsterdam, 1979).

[14] C.V. Negoita, Pullback versus feedback, Human Systems Management 1 (1980) 71-76. [15] E.S. Bainbridge, Feedback and generalized logic, Information and Control 11 (1976) 75-96, [16] C.V. Negoita, M. Kelemen and Al. Stefanescu, A generalization of the internal model principle for man-machine, Proc. Int. Symp. on Control Systems and Computer Science, Polytechnical Institute, (Bucharest, June 1979). [17] W.M. Wonham, Linear Multivariable Control: A Geometric Approach (Springer, Berlin, 1979) 2nd ed. [18] C.V. Negoita and M. Kelemen, On the internal model principle, Proc. of the 1977 IEEE Conf. on Decision and Control (New Orleans, 1977) 1343-1344. [19] M.P. Fourman and D.S. Scott, Sheaves and logic, in: Applications of Sheaves, Lecture Notes in Mathematics No. 753 (Springer, Berlin-New York, 1979) 302-401. [20] C.V. Negoita, Fuzzy Systems (Abacus Press, Tunbridge Wells, 1981). [21] A.M. Pitts, Fuzzy sets do not form a topos, Fuzzy Sets and Systems 8 (1982) 101-104 (this issueL