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Hierarchy of fuzzy cellular automata
Fuzzy Sets and Systems 62 (1994) 167-174 North-Holland
167
Hierarchy of fuzzy cellular automata A n d r e w I. A d a m a t z k y Biophysics Department, St. Petersburg University, St. Petersburg, Russia Received January 1992 Revised May 1993
Abstract: We elaborate notion of cellular automata in the directions of synchroneity/asynchroneity, stationarity/nonstationarity, and fuzzify the principle features of cellular automata as cell state transition, delay of transition and choice of a local transitions function. Through analyzing the properties in various combinations we divide a set of fuzzy cellular automata into 14 classes. To pelucide the whole structure of the classes and precise relations between them we prove the complexity hierarchy.
Keywords: Fuzzy cellular automata; complexity; hierarchy.
1. Introduction The notion of fuzziness was first applied to automata theory more than twenty years ago (it is held that there exist two pioneer w o r k s - Wee, in 1967 [17], and Wee and Fu, in 1969 [18]) and keenly interesting results were obtained by now [13, 9, 16]. It is assumed that an automaton is fuzzy on state transitions when an accurate a priory knowledge about the current state of the automaton and input signals cannot give us any opportunity to determine simply the next state. One of the recent streams is fuzzy neural networks [6, 11, 14] which can be considered as basis for implementing fuzzy automata [11] and as learning neuron-like automata networks [5, 14]. The work [11] describes parallel algorithms for realization of fuzzy automaton in fuzzy neurons; the producing procedure is written on OCCAM and it can be used in designing parallel fuzzy processors. Advanced investigations in VLSI technology allow to design fuzzy VLSI chips [15, 8, 7] which can approximate reasoning in control systems. It provides us with ample opportunity to make fuzzy massively parallel and neuronal computers. But good practice needs good theoretical foundations. The well-tried models for local parallel computing are cellular automata invented in their restricted form by von Neumann [10] (see, also [4, 19]). A multidimensional cellular automaton (CA) is a uniform multidimensional array of identical cells, each of which has a finite number of states and local and translationally invariant rule of state transitions. In this paper we discuss fuzzy multidimensional (a)synchronous (non)stationary CA and construct 14 classes of them. As far as we know there exist no published work concerning formal theory on fuzzy CA. Nonstationary (deterministic even) CA are not investigated also anywhere, except the author's works [1, 2]. In the mean time, the situation with deterministic asynchronous CA is most successful [3, 6, 12]. At present, regular CA are widely used in the simulation of such natural phenomena as fluid dynamic, diffusion, reaction-diffusion systems, populations, epidemics, etc. The common characteristics of the natural processes are their nonlinearity, complex interaction of the parameters, errors of measurements. Life is sufficiently complex and sometimes it cannot be understand enough to be simulated. Nevertheless, the fuzzy relations allow us to model the unknown (unexpected) functional dependences expressed in the form of qualitative connections. In most cases we do not know a priory any probabilistic distribution, therefore it is correct to use fuzzy distributions instead of probabilistic ones. Correspondence to: Dr. A. Adamatzky, Sankt-Petersburg University, Universitetskaya emb. 7/9, 199034 Sankt-Petersburg, Russia. 0165-0114/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0165-0114(93)E0224-G
A.L Adamatzky / Hierarchy of fuzzy cellular automata
168
To explain what kind of cellular automata will be considered, we briefly define asynchronous and nonstationary CA for the case of determinacy (we make them fuzzy in the next section). For a synchronous stationary CA the state taken by a cell at any time step is determined, accordingly to a local rule, by the states of cells in its neighborhood at the previous time step. Any cell of asynchronous stationary CA calculates its next state using neighborhood configuration but changes its present state to the next one in a some discrete time dealy determined on states of neighbors at a past step. The nonstationary CA can be characterized in the following way: every cell changes its state by the rule which is chosen from some set of local transitions rules; the result of this choice at given time step is determined by the configuration of the given cell neighborhood. In the mean time, asynchronous nonstationary cellular automata join properties of both above defined classes. The principal definitions will be given in Section 2. Section 3 presents results achieved.
2. Backgrounds Let us define multidimensional fuzzy cellular automaton (CA) ~ as a finite m-dimensional array A (m/> 2), each cell of which has q states of a finite nonempty set Q and such neighborhood u that Vx e A u(x) = (Yl . . . . . Yk), where Ix - Yjl ~
DSNr:/~,~:Qk × ~3(Qk, Q)_~ [0, 11.
The class is characterized by deterministic synchronous transitions with fuzzy choice of deterministic rule. ~ k is a set of all configurations of neighborhood of size k,
Qk=QxQx..-xQ, k times
[0, 1] is a unit real interval. Let t E ~, f t e ~ ( o k , Q), f t be a deterministic function of cell local transitions at step t. Through the text we mean x is a cell of array A, x t÷l is a state of cell x at time step t + 1 (x t e Q, x e A), and u(x) '-1, u(x) t are configurations of neighborhood u(x) at steps t - 1 and t, respectively (u(x) t-l, u(x)' E Qk). Then tz=(u(x) '-1, f ' ) is a grade of choosing function f ' at step t when configuration of neighborhood u(x) of cell x is equal to u(x) t-I at step t - 1, moreover cell x passes from state x t to state x '÷1 at time t according to function ff :x '÷1 = ft(u(x)t).
(2)
9ArS:f:Qk--, Q,
x D--, [0, 1],
D is a set of delay, IDI = d. Every cell of CA of this class passes from state x' to state x t÷a, a ~ ~), and does not change its state x ~ in time interval It, t + a - 1]: x ~÷~= x', r c [0, a - 1], a ~ D. Symbol a depicts the delay of transition of cell x from state x t to state x '÷'~. And so, tz~(u(x) t, a) is a grade of delay a of state transitions when neighborhood configuration is equal u(x)'.
(3) (4)
DArter: IBA/-I~d:/x~,nr:Qk---~~(Q~, Q),
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
169
7r is a function of nonstationarity, which chooses local rule for cell x transition at step t on the neighborhood configuration U(x )t--I : T~(U(X ) t 1) = f t 6 ~ ( Q k , ~ ) .
(5)
DAdr%: ~,~, a :(~k__, ~} ~ N,
IDI = d,
6 is a deterministic delay function, Vu(x)' • Q', 6(u(x)') = a • D.
(6)
D:SS:re:Q* x Q--, [0, 1],
any CA of this class is synchronous stationary fuzzy, where for u(x)' • Qk, x,+l • Q tzf(u(x)', x '+~) is a grade of transiting cell x to state x '+1 when neighborhood u(x) has configuration u(x)' at time t. (7)
[FAd~:/L',6.
(8)
u:ArS: tz~, m.
(9)
Dz~Nf:/xn: Q* x ~(Q* x ~(Q*, Q), [0, 1])--> [0, 1],
the class is characterized by fuzzy local transitions of cells and fuzzy choice of fuzzy rule for the local transitions. (10)
UzS[Nd:/zF:Q*--, ~(Q* x Q, [0, 1]),
it is clear from tZF that CA of ~ 2 ~ d is a CA with fuzzy local transitions of cells and deterministic choosing fuzzy rules for local transitions. (11)
~:Ad~d: 6, tZF.
(12)
0:AfrNa:/z~,/XF.
(13)
OZAd~r:3,/~ri.
(14)
0zAft~y:/x~,/Zn.
The global state (global configuration, or configuration, shortly) of CA ~ is specified by the states of cells at a given time. It can be defined as a mapping c : A ~ Q. Configuration of ~ at time t will be depicted as c'. We consider finite CA of n cells only, so the set of all the configurations of ~ can be written as Q". CA ~ of n cells has exactly q" configurations, therefore graph G of the global transitions of t~ has q" vertices. Let us assign any vertex of G with configuration c • Q". Graph G is full-connected and oriented, and any edge ( q , C2) , Cl, C2 • Q n is assigned with weight M(Cl, C2) • [0, 1], which is a grade of global transition from configuration c~ to configuration c2. The grade of the global transitions of CA ~ can be calculated differently for classes of UzCA. For example,
D~Nf: ml(c',
c '+~) = max
min
[.£/r(U(X)t-l, ft),
x ~ A f ' ~'#(Q*,Q)
DAft: Mffd, c '+") = max min ~(u(x)', a), x~
aeD
DAlai: M3(c t, c '+") = max{Mffc', d+"), M2(c', c'+~)}, DAd[~f: ml(c', c'+~), a = 6(u(x)'), Y$$: M4(c', c '+1) = max min #f(u(x)', x'+l), Xe/5~
ff:Ad~: M4(c', c'+"),
xt+leQ
a = 6(u(x)').
In the remainder of this section we discuss a tiny example of a model of some natural phenomena in fuzzy cellular automata. Let us focus our attention on the quite simple chain of neuron-like elements. This is a one-dimensional lattice where every neuron has synaptic input from the two neighboring
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
170
Fig. 1. neurons and affects on them through its own axonal terminal in the same time (Figure 1, a neighborhood of a single neuron is encircled with a dotted line). In agreement with the expert physiologists we have decided that it is sufficient to assume at least two state of the neuron: rest state and excitatory state. Let us denote them 0 and 1, respectively. In the whole, we guess that a neuron excites when it has mediatory or electrical influence from its neighbors, i.e. when at least one of the closest neurons is excited. It is easy. However, in living neural chains (even in the so simple organisms as hydra or spongia) the interneuronal interactions are characterized by multimediatority and multifunctionality. If anybody desires to build a precise model of that neural chain he will meet with the numerous experiments which probably will give many different results because so many parameters as calcium concentration or liquid temperature must be accounted for. In other ways the investigator can ask a group of experts how they think the neuron state transitions can be implemented. H e orders all the neurons state transitions and all the configurations of neighborhood (tuples of states of the two closest neurons) in the vertical and draw a line segment in horizontal (see Table 1). The left top of the segment was assigned with a linguistic characteristic 'Never', the right one with 'Always' and in the middle of the segment he place a sentence 'God knows'. After that an expert checks the most appropriate grades of the state transitions as he has decided (Table 1). When the table has been completed we divide a scale 'Never'-'God knows'-'Always' into the ten intervals from 0 to 1 and construct the fuzzy matrix of the grades of the local transitions (Table 2). In the same way, one can construct the fuzzy models in the situations of asynchronous transitions or for nonstationary rules for the neuron state transitions. 3. R e s u l t s
In this section we will construct power hierarchy of ~:CA and after that space and time complexity hierarchy for simulating elements of 0:CA. Notice that we mean simulating ~ is a computation of weights of vertices of the global transitions graph G. Table 1 xr---~x '÷+
0~1
0---*0
u(xy
O0 O1 10 11 O0
Never 0
0.l
0.2
0.3
0.4
God knows 0.5 0.6
Always 1
x x x x x
11
1---->0
0.9
x
10
O0 O1 10 11 00 O1 10 11
0.8
x
O1
1--,1
0.7
x
× × × × × × × ×
A.1. Adamatzky / Hierarchy of fuzzy cellular automata
171
Table 2 u(x)'
000 001 010 011 100 101 110 111
x' 0
1
0.9 0.6 0.2 0.4 0.4 0.5 0.7 0.9
0.1 0.9 0.2 0.7 0.5 1 0.4 0.6
Let 151 be a number of elements in some class 5 of ~:CA, 151 is finite. For any two classes 5 ' and 5" we will write 5'---~ 5" when 15'1 >> 15"1 and there exists no class 5 of ~:CA such that 15'1 >> 151 >> 15"1, T h e o r e m 1. Let g:CA be a superclass of all the finite multidimensional fuzzy cellular automata, each cell
o f which has q states, k neighbors and d delays of state transitions. Then ~:CA can be characterized by the following power hierarchy:
Proof. One of the ways to formalize fuzziness is the forming a fuzzy set by expanding a two-element set of values of indicator function {0, 1} up to continuum [0, 1]. Interval [0, 1] is continual but the real state of affairs is that it can be reduced to some finite subset. Moreover, to solve the most applied problems it is sufficient to solve a finite number of linguistic variable values which are normalized in interval [0, 1]. We will write 0 meaning it is a finite subset of [0, 1] which has not more than o) elements, o) e N. Like this, for instance, a class U:SNi can be defined using degree/Xn:Q ~ x ~ ( Q * × ~(Q~, Q), D)~ 0 and 0:55 by Qk × Q___, D. Now that D is introduced, one can calculate the number of elements in the classes of 0:C~. Let a = qk, /3 = qqk and d = qDI. Then any class of ~CA has the following power: (1) IDSNil = w ~t~, (2) IDa51 = (qooa) ~, (3) [DAyNfl : w ~(t3+a), (4) ID/~fNa[ = (oJa/3) ", (5) [DAaNfl = (wt~d) ~, (6) 1~551 = o F , (7) [FAaSI = (wqd) ", (8) I~Aig I -- ,o =(a+q), (9) 11:SNil-- ,o °°~°~, (10) 10:gNal = ¢-o~r?q, (11) ID:~dN~I= (o)C'qd)'~, (12) l~2~fN,ll = ¢.0°'2q+ad, (13) IB:AaNII = (w'~"d) ~, (14) IU:NNI] = ,~(~o~o+a). The following assumption will be principal throughout the paper: o)>>/3 >> d ~ a >> q. The relation a = qk >> q, /3 = qqk >> Ot and o) >>/3 are clear but d ~ qk means that any cell switches its states with one of d delays, when number d of delays has the same order with the number of pair different configurations of neighborhood of size k. Results of the theorem can be obtained immediately from the aforesaid. E3. The local transitions functions and grades of transitions and delays for cells of CA being in 0:CA can be described by matrices. Thus, /z= of CA of DSNy is represented by matrix M = (mo), 1 <~j <~ qqk, 1 ~< i ~< qk, m0 ~ D, Dis a finite subset of [0, 1], where i is an index of neighborhood configuration in the lexicographic list of elements of Qk and j is an index of local transition rule of ~ ( Q k X Q). We mean that spatial complexity of ~ ~ ~CA is determined on the size of matrices (which describe
172
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
local rules, delays and grades) and complexity of matrix elements. In the case of DSN~ we have, for instance, that the space complexity is O(toqq~+k), where to is a complexity of M elements. Depict space complexity of class ~ of FCA by ~(~). Let two classes ~' and ~" of ~CA be in relation ~'--+ 5" iff ~(~") >> ~ ( ~ ' ) (i.e. space complexity of ~" is much more greater than one of ~') and there exists no such class ~ of FCA that ~(~")>> ~(~)>> ~ ( 5 ' ) . We will write ~'--~" when ~ ( ~ ' ) and ~(~") have the same order. Notice that relation--+ is not commutative, irreflexive and associative but -is commutative, reflexive and associative. 2. Let FCA be a superclass of all finite multidimensional fuzzy cellular automata, where each cell has q states, k neighbors, d delays of state transitions. Let each of the images of all membership functions have no more than ~o elements and let constraint to >> qq* >> d ~ qk >> q be held. Then ~:CA has the space complexity hierarchy shown in Figure 2. Theorem
qk
Proof. Let us show the upper bounds for space complexity of classes of IzCA: a = q~, /3 = q , (1) ~ ( D ~ f ) = O(to/3a), (2) ~ ( D A I 5 ) = O(a(dw + q)), (3) ~(DAyt~I) = O(toa(d +/3)), (4) ~(DAy~d) = O(a(dw +/3)), (5) ~(~)Ad~i) = O ( a ( d +/3w)), (6) ~([F~5) = O(waq), (7) ~(~:Ad~) = O(a(qw + d)), (8) ~(~:A~) = O(aw(d + q)), (9) ~ ( ~ z ~ ) = O((xO.)a/3+l), (10) ~(Bzg3Qa) = O(aw"q), (11) ~(~:AaNd) = O(a(d + w'~q)), (12) ~(~S/~f~]d) = O(olo.)(d ~r- Dr*q)), (13) ~(FAdFqf) = O ( a ( d + toufl+l)), (14) ~ ( F & ~ e ) = O(c~w(d + w't~)). They can be calculated by analogy with space complexity of [D~Q~. [] Let Z(~) be the time needed to compute grade of transition of cell x E A of CA ~ (lying in class 5) from state x' to state x '+1 when configurations of its neighborhood at time t - 1 and t are equal to u(x) '-~ and u(x)', respectively. We can point out that ~(~) is a principal measure to compare complexity bounds for simulating any two different classes of DzCA. Moreover, elements of all the classes of FCA contain the same number of cells and they have the same dimensions. Thus we have an opportunity to reject the time needed for scanning configurations of ~ or neighborhood for each cell of A. The time complexity for computing grades will nevertheless be taken into account.
~ f ~f
-~- 0:AfNd
"--..
2->-.<:C .--"?;>< .......
.........
,'
a-T-_-. . . . . . . . .
;->-32\Fig. 2.
:
",,
A.I. Adamatzky
Hierarchyof fuzzy cellular automata
:Z
\-'
&f
173
"
x
~
--2:
>..,~/"
Fig. 3.
Theorem 3. Let ~:CA be a superclass of all finite multidimensional fuzzy cellular automata, where each
cell has q states, k neighbors, d delays of state transitions and maximum of delays is A. Let each of the images of all the membership functions have no more than to elements and let constraint to >> qq~ >> A >> d ~ qk >> q be held. Then ~:CA has the time complexity hierarchy shown in Figure 3. Proof. The relations - - and ---~ (which form the skeleton of the diagram) have been introduced before Theorem 2. We mean that the time needed to compute the grade of state transition of a single cell is a sum of steps for searching required elements of matrices which represent local functions of transitions, delays and grades. The upper bounds for the classes of ~:CA can be shown in the following manner:
a = qk, /3 = qqk, (2) ~ ( D ~ I $ ) = O(2a + d + d), (1) ~ ( D ~ N I ) = O(2a +/3), (4) ~(DA¢~d) = O(3a + d + A), (3) g(~)AiN~) = O(2a + d +/3 + k), (6) ~(Dz55) = O ( a + q), (5) g(D&,NI) = O(2a +/3 + a), (8) g(IFA~-5) = O(2a + q + d + a), (7) X(~&d~) = O(2a + q + A), (10) ~(Dz~Nd) = O(2a + q), (9) g ( ~ : 5 ~ ) = O(3a +/3), (11) ~(~-AdN,,)=O(3a + q + k), (12) ~(FAIMd) = O(3a + q + d + Zl), (14) g(DzAj-Ny): O(4a + d +/3 + a). (13) g(~/~aNl) = O(4a +/3 + A), The above stated bounds can be simply modified from the space complexity ones shown in Theorem 2. An upper limit A of the state transitions delays (A = max{a ]a e ~)}) should be added to the complexity bounds of such classes of DzCAthat have an asynchronous mode because of the behavior of an arbitrary cell x c A must be observed in the time interval [0, a], a ~
174
A.L Adamatzky / Hierarchy of fuzzy cellular automata
Acknowledgements It is a pleasure to thank anonymous referee for the helpful comments and suggestions.
References [1] A.I. Adamatzky, Neural nets identification, M. Novak and E. Pelikan, Eds., Theoretical Aspects of Neurocomputing (World Scientific, Singapore, 1990) 79-93. [2] A.I. Adamatzky, Identification of nonstationary cellular automata, J. of Comput. Sci. and Technol. 7(4) (1992) 379-384. [3] V. Golze, (A-)synchronous (non-)deterministic cell spaces simulating each other, J. Comput. Syst. Sci. 17 (1978) 176-193. [4] J. Gruska, Synthesis, structure and power of systolic computations, Theoret. Comput. Sci. 71 (1990) 47-77. [5] S.C. Lee and E.T. Lee, Fuzzy sets and neural networks, J. Cybern. 4 (1974) 83-103. [6] R.J. Lipton, R.E. Miller and L. Snyder, Synchronization and computing capabilities of linear asynchronous structures, J. Comput. Syst. Sci. 14 (1977) 49-72. [7] M.A. Manzoul, Faults in fuzzy logic systolic arrays, Int. J. Cybern. Sys. 21 (1990) 513-524. [8] M.A. Manzoul and S. Tayal, Systolic VLSI array for multi-variable fuzzy control systems, Int. J. Cybern. Sys. 21 (1990) 27 -42. [9] M. Mizumoto, J. Toyoba and T. Tanaka, Some consideration on fuzzy automata, J. Comput. Syst. Sci. 3 (1969) 409-422. [10] J. von Neumann, Theory of Self-Reproducing Automata (University of Illinois Press, Urbana and London, 1966). [11] V. Olej, J. Chmurny and M. Lehotsky, Fuzzy automaton based on fuzzy neurons, NEURONET'90, Proc. Int. Symp. Neural Networks and Neural Computing, Prague (1990) 264-266. [12] L.A. Prieze, A note on asynchronus cellular automata, J. Comput. Syst. Sci. 47 (1978) 237-252. [13] E.S. Santos, Maximin automata, Inform. Control 13 (1968) 363-377. [14] L.C. Shine and R.O. Grondin, On designing fuzzy learning neural-automata, Proc. lEE First Int. Conf. Neural Networks 2 (1987) 299-307. [15] M. Togai and H. Watanabe, Expert system on a chip: an engine real time approximating reasoning, IEEE Expert 1 (1986) 55-62. [16] W. Wechler, The Concept of Fuzziness in Automata and Language Theory (Akademie-Verlag, Berlin, 1978). [17[ W.G. Wee, On generalization of adaptive algorithm and application of fuzzy sets concept to pattern recognition (Ph.D. Thesis, Purdue University, June, 1967). [18] W.G. Wee and K.S. Fu, A formulation of fuzzy automata and its application as model of learning systems, IEEE Trans. Syst. Man Cyber. 5 (1969) 215-223. [19] S. Wolfram, Computation theory of cellular automata, Comm. Math. Phys. 96 (1984) 15-57.
167
Hierarchy of fuzzy cellular automata A n d r e w I. A d a m a t z k y Biophysics Department, St. Petersburg University, St. Petersburg, Russia Received January 1992 Revised May 1993
Abstract: We elaborate notion of cellular automata in the directions of synchroneity/asynchroneity, stationarity/nonstationarity, and fuzzify the principle features of cellular automata as cell state transition, delay of transition and choice of a local transitions function. Through analyzing the properties in various combinations we divide a set of fuzzy cellular automata into 14 classes. To pelucide the whole structure of the classes and precise relations between them we prove the complexity hierarchy.
Keywords: Fuzzy cellular automata; complexity; hierarchy.
1. Introduction The notion of fuzziness was first applied to automata theory more than twenty years ago (it is held that there exist two pioneer w o r k s - Wee, in 1967 [17], and Wee and Fu, in 1969 [18]) and keenly interesting results were obtained by now [13, 9, 16]. It is assumed that an automaton is fuzzy on state transitions when an accurate a priory knowledge about the current state of the automaton and input signals cannot give us any opportunity to determine simply the next state. One of the recent streams is fuzzy neural networks [6, 11, 14] which can be considered as basis for implementing fuzzy automata [11] and as learning neuron-like automata networks [5, 14]. The work [11] describes parallel algorithms for realization of fuzzy automaton in fuzzy neurons; the producing procedure is written on OCCAM and it can be used in designing parallel fuzzy processors. Advanced investigations in VLSI technology allow to design fuzzy VLSI chips [15, 8, 7] which can approximate reasoning in control systems. It provides us with ample opportunity to make fuzzy massively parallel and neuronal computers. But good practice needs good theoretical foundations. The well-tried models for local parallel computing are cellular automata invented in their restricted form by von Neumann [10] (see, also [4, 19]). A multidimensional cellular automaton (CA) is a uniform multidimensional array of identical cells, each of which has a finite number of states and local and translationally invariant rule of state transitions. In this paper we discuss fuzzy multidimensional (a)synchronous (non)stationary CA and construct 14 classes of them. As far as we know there exist no published work concerning formal theory on fuzzy CA. Nonstationary (deterministic even) CA are not investigated also anywhere, except the author's works [1, 2]. In the mean time, the situation with deterministic asynchronous CA is most successful [3, 6, 12]. At present, regular CA are widely used in the simulation of such natural phenomena as fluid dynamic, diffusion, reaction-diffusion systems, populations, epidemics, etc. The common characteristics of the natural processes are their nonlinearity, complex interaction of the parameters, errors of measurements. Life is sufficiently complex and sometimes it cannot be understand enough to be simulated. Nevertheless, the fuzzy relations allow us to model the unknown (unexpected) functional dependences expressed in the form of qualitative connections. In most cases we do not know a priory any probabilistic distribution, therefore it is correct to use fuzzy distributions instead of probabilistic ones. Correspondence to: Dr. A. Adamatzky, Sankt-Petersburg University, Universitetskaya emb. 7/9, 199034 Sankt-Petersburg, Russia. 0165-0114/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0165-0114(93)E0224-G
A.L Adamatzky / Hierarchy of fuzzy cellular automata
168
To explain what kind of cellular automata will be considered, we briefly define asynchronous and nonstationary CA for the case of determinacy (we make them fuzzy in the next section). For a synchronous stationary CA the state taken by a cell at any time step is determined, accordingly to a local rule, by the states of cells in its neighborhood at the previous time step. Any cell of asynchronous stationary CA calculates its next state using neighborhood configuration but changes its present state to the next one in a some discrete time dealy determined on states of neighbors at a past step. The nonstationary CA can be characterized in the following way: every cell changes its state by the rule which is chosen from some set of local transitions rules; the result of this choice at given time step is determined by the configuration of the given cell neighborhood. In the mean time, asynchronous nonstationary cellular automata join properties of both above defined classes. The principal definitions will be given in Section 2. Section 3 presents results achieved.
2. Backgrounds Let us define multidimensional fuzzy cellular automaton (CA) ~ as a finite m-dimensional array A (m/> 2), each cell of which has q states of a finite nonempty set Q and such neighborhood u that Vx e A u(x) = (Yl . . . . . Yk), where Ix - Yjl ~
DSNr:/~,~:Qk × ~3(Qk, Q)_~ [0, 11.
The class is characterized by deterministic synchronous transitions with fuzzy choice of deterministic rule. ~ k is a set of all configurations of neighborhood of size k,
Qk=QxQx..-xQ, k times
[0, 1] is a unit real interval. Let t E ~, f t e ~ ( o k , Q), f t be a deterministic function of cell local transitions at step t. Through the text we mean x is a cell of array A, x t÷l is a state of cell x at time step t + 1 (x t e Q, x e A), and u(x) '-1, u(x) t are configurations of neighborhood u(x) at steps t - 1 and t, respectively (u(x) t-l, u(x)' E Qk). Then tz=(u(x) '-1, f ' ) is a grade of choosing function f ' at step t when configuration of neighborhood u(x) of cell x is equal to u(x) t-I at step t - 1, moreover cell x passes from state x t to state x '÷1 at time t according to function ff :x '÷1 = ft(u(x)t).
(2)
9ArS:f:Qk--, Q,
x D--, [0, 1],
D is a set of delay, IDI = d. Every cell of CA of this class passes from state x' to state x t÷a, a ~ ~), and does not change its state x ~ in time interval It, t + a - 1]: x ~÷~= x', r c [0, a - 1], a ~ D. Symbol a depicts the delay of transition of cell x from state x t to state x '÷'~. And so, tz~(u(x) t, a) is a grade of delay a of state transitions when neighborhood configuration is equal u(x)'.
(3) (4)
DArter: IBA/-I~d:/x~,nr:Qk---~~(Q~, Q),
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
169
7r is a function of nonstationarity, which chooses local rule for cell x transition at step t on the neighborhood configuration U(x )t--I : T~(U(X ) t 1) = f t 6 ~ ( Q k , ~ ) .
(5)
DAdr%: ~,~, a :(~k__, ~} ~ N,
IDI = d,
6 is a deterministic delay function, Vu(x)' • Q', 6(u(x)') = a • D.
(6)
D:SS:re:Q* x Q--, [0, 1],
any CA of this class is synchronous stationary fuzzy, where for u(x)' • Qk, x,+l • Q tzf(u(x)', x '+~) is a grade of transiting cell x to state x '+1 when neighborhood u(x) has configuration u(x)' at time t. (7)
[FAd~:/L',6.
(8)
u:ArS: tz~, m.
(9)
Dz~Nf:/xn: Q* x ~(Q* x ~(Q*, Q), [0, 1])--> [0, 1],
the class is characterized by fuzzy local transitions of cells and fuzzy choice of fuzzy rule for the local transitions. (10)
UzS[Nd:/zF:Q*--, ~(Q* x Q, [0, 1]),
it is clear from tZF that CA of ~ 2 ~ d is a CA with fuzzy local transitions of cells and deterministic choosing fuzzy rules for local transitions. (11)
~:Ad~d: 6, tZF.
(12)
0:AfrNa:/z~,/XF.
(13)
OZAd~r:3,/~ri.
(14)
0zAft~y:/x~,/Zn.
The global state (global configuration, or configuration, shortly) of CA ~ is specified by the states of cells at a given time. It can be defined as a mapping c : A ~ Q. Configuration of ~ at time t will be depicted as c'. We consider finite CA of n cells only, so the set of all the configurations of ~ can be written as Q". CA ~ of n cells has exactly q" configurations, therefore graph G of the global transitions of t~ has q" vertices. Let us assign any vertex of G with configuration c • Q". Graph G is full-connected and oriented, and any edge ( q , C2) , Cl, C2 • Q n is assigned with weight M(Cl, C2) • [0, 1], which is a grade of global transition from configuration c~ to configuration c2. The grade of the global transitions of CA ~ can be calculated differently for classes of UzCA. For example,
D~Nf: ml(c',
c '+~) = max
min
[.£/r(U(X)t-l, ft),
x ~ A f ' ~'#(Q*,Q)
DAft: Mffd, c '+") = max min ~(u(x)', a), x~
aeD
DAlai: M3(c t, c '+") = max{Mffc', d+"), M2(c', c'+~)}, DAd[~f: ml(c', c'+~), a = 6(u(x)'), Y$$: M4(c', c '+1) = max min #f(u(x)', x'+l), Xe/5~
ff:Ad~: M4(c', c'+"),
xt+leQ
a = 6(u(x)').
In the remainder of this section we discuss a tiny example of a model of some natural phenomena in fuzzy cellular automata. Let us focus our attention on the quite simple chain of neuron-like elements. This is a one-dimensional lattice where every neuron has synaptic input from the two neighboring
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
170
Fig. 1. neurons and affects on them through its own axonal terminal in the same time (Figure 1, a neighborhood of a single neuron is encircled with a dotted line). In agreement with the expert physiologists we have decided that it is sufficient to assume at least two state of the neuron: rest state and excitatory state. Let us denote them 0 and 1, respectively. In the whole, we guess that a neuron excites when it has mediatory or electrical influence from its neighbors, i.e. when at least one of the closest neurons is excited. It is easy. However, in living neural chains (even in the so simple organisms as hydra or spongia) the interneuronal interactions are characterized by multimediatority and multifunctionality. If anybody desires to build a precise model of that neural chain he will meet with the numerous experiments which probably will give many different results because so many parameters as calcium concentration or liquid temperature must be accounted for. In other ways the investigator can ask a group of experts how they think the neuron state transitions can be implemented. H e orders all the neurons state transitions and all the configurations of neighborhood (tuples of states of the two closest neurons) in the vertical and draw a line segment in horizontal (see Table 1). The left top of the segment was assigned with a linguistic characteristic 'Never', the right one with 'Always' and in the middle of the segment he place a sentence 'God knows'. After that an expert checks the most appropriate grades of the state transitions as he has decided (Table 1). When the table has been completed we divide a scale 'Never'-'God knows'-'Always' into the ten intervals from 0 to 1 and construct the fuzzy matrix of the grades of the local transitions (Table 2). In the same way, one can construct the fuzzy models in the situations of asynchronous transitions or for nonstationary rules for the neuron state transitions. 3. R e s u l t s
In this section we will construct power hierarchy of ~:CA and after that space and time complexity hierarchy for simulating elements of 0:CA. Notice that we mean simulating ~ is a computation of weights of vertices of the global transitions graph G. Table 1 xr---~x '÷+
0~1
0---*0
u(xy
O0 O1 10 11 O0
Never 0
0.l
0.2
0.3
0.4
God knows 0.5 0.6
Always 1
x x x x x
11
1---->0
0.9
x
10
O0 O1 10 11 00 O1 10 11
0.8
x
O1
1--,1
0.7
x
× × × × × × × ×
A.1. Adamatzky / Hierarchy of fuzzy cellular automata
171
Table 2 u(x)'
000 001 010 011 100 101 110 111
x' 0
1
0.9 0.6 0.2 0.4 0.4 0.5 0.7 0.9
0.1 0.9 0.2 0.7 0.5 1 0.4 0.6
Let 151 be a number of elements in some class 5 of ~:CA, 151 is finite. For any two classes 5 ' and 5" we will write 5'---~ 5" when 15'1 >> 15"1 and there exists no class 5 of ~:CA such that 15'1 >> 151 >> 15"1, T h e o r e m 1. Let g:CA be a superclass of all the finite multidimensional fuzzy cellular automata, each cell
o f which has q states, k neighbors and d delays of state transitions. Then ~:CA can be characterized by the following power hierarchy:
Proof. One of the ways to formalize fuzziness is the forming a fuzzy set by expanding a two-element set of values of indicator function {0, 1} up to continuum [0, 1]. Interval [0, 1] is continual but the real state of affairs is that it can be reduced to some finite subset. Moreover, to solve the most applied problems it is sufficient to solve a finite number of linguistic variable values which are normalized in interval [0, 1]. We will write 0 meaning it is a finite subset of [0, 1] which has not more than o) elements, o) e N. Like this, for instance, a class U:SNi can be defined using degree/Xn:Q ~ x ~ ( Q * × ~(Q~, Q), D)~ 0 and 0:55 by Qk × Q___, D. Now that D is introduced, one can calculate the number of elements in the classes of 0:C~. Let a = qk, /3 = qqk and d = qDI. Then any class of ~CA has the following power: (1) IDSNil = w ~t~, (2) IDa51 = (qooa) ~, (3) [DAyNfl : w ~(t3+a), (4) ID/~fNa[ = (oJa/3) ", (5) [DAaNfl = (wt~d) ~, (6) 1~551 = o F , (7) [FAaSI = (wqd) ", (8) I~Aig I -- ,o =(a+q), (9) 11:SNil-- ,o °°~°~, (10) 10:gNal = ¢-o~r?q, (11) ID:~dN~I= (o)C'qd)'~, (12) l~2~fN,ll = ¢.0°'2q+ad, (13) IB:AaNII = (w'~"d) ~, (14) IU:NNI] = ,~(~o~o+a). The following assumption will be principal throughout the paper: o)>>/3 >> d ~ a >> q. The relation a = qk >> q, /3 = qqk >> Ot and o) >>/3 are clear but d ~ qk means that any cell switches its states with one of d delays, when number d of delays has the same order with the number of pair different configurations of neighborhood of size k. Results of the theorem can be obtained immediately from the aforesaid. E3. The local transitions functions and grades of transitions and delays for cells of CA being in 0:CA can be described by matrices. Thus, /z= of CA of DSNy is represented by matrix M = (mo), 1 <~j <~ qqk, 1 ~< i ~< qk, m0 ~ D, Dis a finite subset of [0, 1], where i is an index of neighborhood configuration in the lexicographic list of elements of Qk and j is an index of local transition rule of ~ ( Q k X Q). We mean that spatial complexity of ~ ~ ~CA is determined on the size of matrices (which describe
172
A.I. Adamatzky / Hierarchy of fuzzy cellular automata
local rules, delays and grades) and complexity of matrix elements. In the case of DSN~ we have, for instance, that the space complexity is O(toqq~+k), where to is a complexity of M elements. Depict space complexity of class ~ of FCA by ~(~). Let two classes ~' and ~" of ~CA be in relation ~'--+ 5" iff ~(~") >> ~ ( ~ ' ) (i.e. space complexity of ~" is much more greater than one of ~') and there exists no such class ~ of FCA that ~(~")>> ~(~)>> ~ ( 5 ' ) . We will write ~'--~" when ~ ( ~ ' ) and ~(~") have the same order. Notice that relation--+ is not commutative, irreflexive and associative but -is commutative, reflexive and associative. 2. Let FCA be a superclass of all finite multidimensional fuzzy cellular automata, where each cell has q states, k neighbors, d delays of state transitions. Let each of the images of all membership functions have no more than ~o elements and let constraint to >> qq* >> d ~ qk >> q be held. Then ~:CA has the space complexity hierarchy shown in Figure 2. Theorem
qk
Proof. Let us show the upper bounds for space complexity of classes of IzCA: a = q~, /3 = q , (1) ~ ( D ~ f ) = O(to/3a), (2) ~ ( D A I 5 ) = O(a(dw + q)), (3) ~(DAyt~I) = O(toa(d +/3)), (4) ~(DAy~d) = O(a(dw +/3)), (5) ~(~)Ad~i) = O ( a ( d +/3w)), (6) ~([F~5) = O(waq), (7) ~(~:Ad~) = O(a(qw + d)), (8) ~(~:A~) = O(aw(d + q)), (9) ~ ( ~ z ~ ) = O((xO.)a/3+l), (10) ~(Bzg3Qa) = O(aw"q), (11) ~(~:AaNd) = O(a(d + w'~q)), (12) ~(~S/~f~]d) = O(olo.)(d ~r- Dr*q)), (13) ~(FAdFqf) = O ( a ( d + toufl+l)), (14) ~ ( F & ~ e ) = O(c~w(d + w't~)). They can be calculated by analogy with space complexity of [D~Q~. [] Let Z(~) be the time needed to compute grade of transition of cell x E A of CA ~ (lying in class 5) from state x' to state x '+1 when configurations of its neighborhood at time t - 1 and t are equal to u(x) '-~ and u(x)', respectively. We can point out that ~(~) is a principal measure to compare complexity bounds for simulating any two different classes of DzCA. Moreover, elements of all the classes of FCA contain the same number of cells and they have the same dimensions. Thus we have an opportunity to reject the time needed for scanning configurations of ~ or neighborhood for each cell of A. The time complexity for computing grades will nevertheless be taken into account.
~ f ~f
-~- 0:AfNd
"--..
2->-.<:C .--"?;>< .......
.........
,'
a-T-_-. . . . . . . . .
;->-32\Fig. 2.
:
",,
A.I. Adamatzky
Hierarchyof fuzzy cellular automata
:Z
\-'
&f
173
"
x
~
--2:
>..,~/"
Fig. 3.
Theorem 3. Let ~:CA be a superclass of all finite multidimensional fuzzy cellular automata, where each
cell has q states, k neighbors, d delays of state transitions and maximum of delays is A. Let each of the images of all the membership functions have no more than to elements and let constraint to >> qq~ >> A >> d ~ qk >> q be held. Then ~:CA has the time complexity hierarchy shown in Figure 3. Proof. The relations - - and ---~ (which form the skeleton of the diagram) have been introduced before Theorem 2. We mean that the time needed to compute the grade of state transition of a single cell is a sum of steps for searching required elements of matrices which represent local functions of transitions, delays and grades. The upper bounds for the classes of ~:CA can be shown in the following manner:
a = qk, /3 = qqk, (2) ~ ( D ~ I $ ) = O(2a + d + d), (1) ~ ( D ~ N I ) = O(2a +/3), (4) ~(DA¢~d) = O(3a + d + A), (3) g(~)AiN~) = O(2a + d +/3 + k), (6) ~(Dz55) = O ( a + q), (5) g(D&,NI) = O(2a +/3 + a), (8) g(IFA~-5) = O(2a + q + d + a), (7) X(~&d~) = O(2a + q + A), (10) ~(Dz~Nd) = O(2a + q), (9) g ( ~ : 5 ~ ) = O(3a +/3), (11) ~(~-AdN,,)=O(3a + q + k), (12) ~(FAIMd) = O(3a + q + d + Zl), (14) g(DzAj-Ny): O(4a + d +/3 + a). (13) g(~/~aNl) = O(4a +/3 + A), The above stated bounds can be simply modified from the space complexity ones shown in Theorem 2. An upper limit A of the state transitions delays (A = max{a ]a e ~)}) should be added to the complexity bounds of such classes of DzCAthat have an asynchronous mode because of the behavior of an arbitrary cell x c A must be observed in the time interval [0, a], a ~
174
A.L Adamatzky / Hierarchy of fuzzy cellular automata
Acknowledgements It is a pleasure to thank anonymous referee for the helpful comments and suggestions.
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