A possibilistic argument for irreversibility

Fuzzy Sets and Systems34 (1990) 73-80 North-Holland

A POSSIBILISTIC A R G U M E N T

73

FOR IRREVERSIBILITY

V.N. B O B Y L E V Computing Center of the Academy of Sciences, Vavilova st. 40, Moscow GSP-1 117967, U.S.S.R.

Received February 1987 Revised April 1988 Abstract: Our aim is to explicate the idea of irreversibilityby a mathematicalrepresentation of

irreversible processes occurring in possibilisticdynamicalsystems. With this in mind, solutions of the Cauchyproblem in the space of fuzzypoints are examined. To meet the correspondence argument, solutions of the Cauchy problem in the space of concentrated points are examined too. Keywords: Differential equation; Cauchy problem; fuzziness; possibility; irreversibility; time

asymmetry.

1. Introduction The problem of irreversibility results from the fact that the empirical proposition "There exist irreversible processes" does not yet have generally accepted nomological foundations. The point is that standard classes of Cauchy problems in Euclidean space are temporally symmetrical (in that they are time reversal invariant) and therefore fail to specify irreversibility as a temporal asymmetry. The problem was a great worry to Boltzmann, was elicited by Reichenbach and now stimulates Prigogine. Presented here is another particular approach to irreversibility. We consider a temporally asymmetrical class of Cauchy problems and interpret it phenomenologically by making reference to possibilistic dynamical systems and irreversible processes occurring in them, so the existence of irreversible processes acquires new mathematical foundations. Together with 'possibility' a conscious subject comes into being, and the approach proves to be anthropologistic in substance and possibilistic in form, but not subjectivistic! - for the subject turns out to be an intersubject. Actually we consider two classes of Cauchy problems: in the space of fuzzy points and in its subspace of concentrated ones. The structure of the entire space is not completely Euclidean in that not all arithmetic operations are feasible. One can easily guess that the asymmetry originates in the absence of subtraction. The structure of the subspace is Euclidean and a temporal symmetry emerges to meet the correspondence argument. The proper fuzzy points approximate the concentrated points with any finite precision, yet the asymmetry turns into the symmetry only in the limit. 0165-0114/90/$3.50 ~ 1990, Elsevier Science Publishers B.V. (North-Holland)

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To illustrate what systems and processes are contemplated, visualize a rocket launched from a given point and flying away to an unknown target. If we envisage no instruments for observing the rocket, then the extent of our ignorance of its possible position must of necessity increase in the course of time. A similar example: if you know nothing about your old friend, then your uncertainty concerning his possible prosperity naturally grows as time goes on. It is clear that here the one-sided evolution originates in the absence of observation and occurs solely in our minds. According to M. Bunge (1973), any proper physical theory must not involve conscious subjects as points of reference. In contrast, the concept of possibility [16, 6] is human-centered first of all. Although this leads us to conclude that the systems under reference are outside the scope of physics, yet they have a thermodynamic analogue- Reichenbach's branch systems. On the other hand, those systems might have been spotted even in differential equations with set-valued solutions [10] or in interval differential equations [11]; unfortunately, the equations caused no reflections on the dynamics of mental observation. It is differential equations under fuzziness [1, 3-5, 8, 14, 15] that yield suggestive solutions. As another source of ideas the principle of accumulation of errors from the field of numerical methods for ordinary differential equations could be advanced. The one-sided sense of fuzzy trajectories of fuzzy dynamical systems [9, Lemma 5.3] might be used too.

2. Preliminaries As usual, R n denotes the n-dimensional Euclidean space. Let X, xo, y , z , f , f o e R " , x* • B* = {x: 0 < Ixl < 1}, 0 ~
a¢:0,

0
diam X =

y~Y

max Ix - Y l.

x,yEX

y¢Y

xeg

.I

A possibilistic argument for irreversibility

75

The resulting space (I2 with the above) will be denoted by the same letter g2. Let an r-family {Xr} contained in f2 be continuous from the left in the sense of h(., .), be decreasing and bounded in the sense of ___ and be such that NrXr = {X}, a singleton. This family will be treated as a generalized point of R n and be called the fuzzy point £, with levr £ = Xr and x being called its r-level set and its mean value. In the case of L_Jrx r = {x } we speak of the concentrated point £ as well. In addition, in order not to overload notation, we often substitute/u for £. The collection of all fuzzy points will be denoted by J/, including the collection of all concentrated points. We extend the structure of g2 to ~ in the levelwise manner: £ ~ )3 ¢:> levr £ ~_ levr )3 Vr, levr(£ + )3) = lev~ £ + levr )3, levr ao£ = ao levr £, h(£, )3) = sup h(lev~ £, levr)3), r

diam £ = sup diam levr £ = diam cl (_3 X~ r

r

(ci signifies the closure in Rn). If desired, diam can be complicated by introducing a weight factor into it as in [5]. The resulting space will be denoted by the same letter .,t/. Two other manners of treating fuzzy points should also be kept in mind so as to make use of the level sets' convexity and to incorporate their left-continuous monotonic dependence on r. Here we hint at a duality between fuzzy points and certain functionals on B* as well as a duality between fuzzy points and certain nonnegative functions on R ~, implying the following. Given/~, define the function x*~*(x*)=

max

(x,x*)

x ~ l e V l x . I gt

( ( . , .) signifies the inner product in Rn), which is called the support function of ~t. Then the transformation :t ~ / ~ * ( . ) is an isometrical isomorphism between d/ and a non-separable closed convex cone of the Banach space ~ of all bounded functionals on B* ('bounded functional' stands for 'real-valued function whose functional norm is finite'); the order ~_ and the diameter diam, too, can be spread to ~. The (inverse) transformation ~t*(.)---> ~ is determined by the formula levr g = {x: (x, x*) ~<~t*(x*) V

Ix*l = r}.

For details and the methodology, see [2]. To minimize ~: as a Banach space, we can take the cone's closed linear span (in ~ ) or, what is the same up to an isometrical isomorphism and a completion, a certain normed space of equivalence classes [12]. As regards the other duality, define the function x--->/~(x) = sup{r: x ~ levr gt}

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V.N. Bobylev

(supposing sup 0 = 0), which is called the membership function of p. Then the transformation ~---~(.) is a bijection between J~ and the collection of all unimodal normalized support-bounded upper-semicontinuous quasi-concave nonnegative functions on R n ('unimodal function' stands for 'function whose point of global maximum exists and is unique'). The (inverse) transformation ~(.)---> ~ is determined by the formula leVr ~ = {X: p(X)/> r}. It is easy to establish the required duality [8]. For the methodology, see [13]. Time-dependent vectors, sets and fuzzy sets are called vector(-valued), multivalued and fuzzy functions of time, respectively. Let xt e R n be a vector function of time, of t, and :~, • ~ a fuzzy one. In the case of :~, • D '¢t we speak of the function :?t as well. For (continuous) multivalued functions of time one defines the conical tl derivative d/dt and the Riemann integral St0 dt as induced by the structure o f / 2 so as to extend them to fuzzy functions of time, say, in levelwise manner. However, when reformulated in terms of support functions the definitions look more operational: given/zt, we define d

*

a

,

,

( ~ ~,) (x*)=-fit lit (x ) iff the functional of x* on the right-hand side exists and is a support function (otherwise, the conical derivative of/~, does not exist); similarly, ~, at

(x*) = (R)

~?(x*) dt.

Incidentally, the given concept of derivative and integral [1, 3] is equivalent in the class of Cauchy problems to be considered to the concept of Hukuhara derivative and Aumann integral [8]. Finally, let x--*f(x)• R ~ and ~---~](2)• ~ be Lipschitzian mappings. In the case of ] ( i ) = ] ( i ) • ~ V2 we speak of the mapping 2---~.f(x) as well. In addition, we substitute ~ for ], as we substitute ~ for i. Time-dependent mappings are omitted here without loss of generality.

3. Cauchy problem in fuzzy medium "The irreversibility that we observe is a feature of theories that take proper account of the nature and limitation of observation." (I. Prigogine, 1980) The notation

dl~ = q~(iz)" l~lt=o= #o, dt will be called a fuzzy Cauchy problem, (q0(/~), flo)- Its solution should be thought

A possibilistic argument for irreversibility

77

of as a differentiable fuzzy function of time, #,, such that d dt #' = qg(#,) Vt,

#t=o = #o,

and be sometimes specified as a fuzzy solution. According to a standard scheme (based on Banach's contraction principle and Gronwall's inequality), the solution exists, is unique, can be found by the method of successive approximations and continuously depends on the problem. Notice that the mean values' structure plays no part in the proof [1, 8]. A fuzzy solution will be called irreversible if it strictly increases in diameter (of the solution's values) with time.

Proposition. In the class of fuzzy solutions under examination every one of them increases in the sense of diam with increasing time, and some are irreversible.

Proof. According to the scheme, the solution #t of (q~(#), #o) obeys the equality ~Ut, = #to "~

(P(]'gt) dt

Vtl t> to

(which, in terms of support functions, follows from the Newton-Leibniz formula for continuously differentiable scalar functions of time). On the other hand, the structure of d~ (in fact, of I2) does not admit a genuine subtraction; this appears as follows: 2=)~+~

~

diam2~>diamy

(V2,)~,~).

Therefore the composition diam #, has to increase in t. In the examples to be cited it increases linearly, exponentially and logistically, in a word, increases strictly (and smoothly).

Example 1. Given q0o, put q0(#) = q0o, #t = Xo + tq0o. Then #, is the solution of (q0(#), Xo) because a d --fft(a~ + Y) = (-~tat)2. Taking into account that diam(2 + 3~) = diam 2,

diam ao~ = ao diam 2,

we obtain diam #t = t diam q0o. For the solution to be irreversible it remains to observe that diam2=0

¢~ 2 e ~

and to suppose q0o ¢ ~.

V.N. Bobylev

78

Example 2. Given a, put tp(/z) = a/z,

/Z, = ea'/zo.

T h e n / z t is the solution of (tp(/z),/zo), with diam/zt = eat diam/Zo. If/zo q ~ the solution is irreversible.

Example 3. Given a, put f ( £ ) = max{0, a - Ixl}~ (max enables one to avoid a negative h o m o t h e t i c here). If [Xol < a the solution of (f($), ~0) assumes the form

£t = at£o,

at = a(Ixol + (a - Ixol)e-at)-l;

accordingly, diam £'t = at diam -loIn m o r e detail: a, really is nonnegative and strictly increasing, and ld atdtat = a - at Ixol = a - Ix, I, so that

d

(Xd)

dt x, = ~ a t

,f, = (a - I x , I)~,

= max{0, a - Ix, I}~, -- f(~,). If additionally Xo ~ ~ the solution is irreversible.

4. Correspondenceargument "Theoretical reversibility arises from . . . idealizations . . . that go beyond the possibilities of measurement performed with any finite precision." (I. Prigogine, 1980) T h e notation dx

~ =f(x), xl,=o=Xo, is called a vector C a u c h y problem, ( f ( x ) , Xo). Its solution is thought of as a differentiable vector function, x,, such that d ~xt=f(x,)

Vt,

X,=o=Xo,

and is sometimes specified as a vector solution. It is well known that the solution

A possibilistic argument for irreversibility

79

exists, is unique, can be found by the method of successive approximations, continuously depends on the problem and is reversible in that it remains a vector solution under time reversal, t ~ 1 - t. A fuzzy solution will be called reversible if it remains a fuzzy solution under time reversal. (Thereby reversibility and irreversibility do not overlap in the class of fuzzy solutions covered.)

Proposition. The solution of (f (x), Xo) is reversible. In point of fact, when x, is the solution of (f(x), x0), then Jet is the solution of (f(x), YCo). 5. Interpretation On reflection, we propose the following conception of solving the problem of irreversibility. There exists a subject through the medium of which, in his mind's eye, solutions of fuzzy Cauchy problems represent processes occurring in possibilistic dynamical systems, the underlying systems being visualized as deterministic dynamical systems. Irreversible solutions represent irreversible processes. The present man is a subject of that nature, a storage medium of possibility.

Comment. Based on the solution #, of (qg(#), #o), the phrase "to represent a process occurring in a possibilistic dynamical system" means that the system's deterministic position and velocity at time t are represented by arbitrary points located independently of one another in levr #t and levs q0(#t), the level sets of #, and cp(#,), respectively, with some grades of possibility (r and s at least) of the representation, provided the system emanates from #o. Once again, but in terms of membership functions: the system's deterministic position and velocity at time t are represented by x and f, arbitrary points in R n, furnished with the possibilistic grades #,(x) and tp(#~)(f) respectively. Thus deterministic dynamical systems give way and rise to possibilistic dynamical systems. Recall the rocket flying to an unknown target in the absence of observation.

Acknowledgment The author wishes to thank Academician N.N. Moiseev, Professor V.R. Khachaturov and especially Dr. V.A. Sokol for their encouragement as well as Professor O. Kaleva, Dr. M.A. Vedjushkin and an anonymous referee for useful comments. References [1] V.N. Bobylev, Fuzzy Cauchy problem, Report 3848-83DEP, Computing Center of the USSR Academy of Sciences, Moscow (1983) (in Russian).

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[2] V.N. Bobylev, Support function of a fuzzy set and its characteristic properties, Math. Notes 37 (1985) 281-285. [3] V.N. Bobylev, Cauchy problem under fuzzy control, BUSEFAL 21 (1985) 117-126. [4] V.N. Bobylev, On the evolution operator for fuzzy-initiated linear systems, BUSEFAL 32 (1987) 92-96. [5] V.N. Bobylev, On the states of a dynamical system under fuzzy control, BUSEFAL 33 (1987) 126-133. [6] D. Dubois and H. Prade, Fuzzy Sets and Systems - Theory and Applications (Academic Press, New York, 1980). [7] O. Kaleva, A differential and integral calculus for fuzzy mappings and fuzzy differential equations, Report 48, Tampere University of Technology (1984) (see [8]). [8] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and System 24 (1987) 301-317 (a slightly modified version of [7]). [9] P.E. Kloeden, Fuzzy dynamical systems, Fuzzy Sets and Systems 7 (1982) 275-296. [10] A.J. Br. Lopes Pinto, F.S. De Blasi and F. Iervolino, Uniqueness and existence theorems for differential equations with compact convex valued solutions, Boll. Un. Mat. ltal. 3 (1970) 47-54. [11] S.M. Markov, Existence and uniqueness of solutions of the interval differential equation X ' = F(t, X), C.R. Acad. Bulgare Sci. 31 (1978) 1519-1522. [12] M.L. Puri and D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal Appl. 91 (1983) 552-558. [13] D. Ralescu, A survey of the representation of fuzzy concepts and its applications, in: M.M. Gupta, R.K. Ragade and R.R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 77-91. [14] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319-330. [15] S. Seikkala, On differential equations with fuzzy initial value and coefficients, Preprint, University of Oulu, Linnanmaa (1986). [16] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.