BULLETI~ OF MATHEMATICAL BIOPHYSICS VOLUME 30, 1968

CELLULAR SYSTEMS AS SEQUENTIAL MACHINES: STABILITY P R O P E R T I E S

LLOYD DEMETRIUS

Committee on Mathematical Biology, The University of Chicago, Chicago, Illinois

In previous studies of (~J~, ~R)-systems (Rosen, 1961; Demetrius, 1966), it was assumed that changes in the structure of (gJ~, ~R)-systems which were induced by environmental alterna- tions occurred without error. Here, the effect of both "genet ic" and "metabol ic" mal- functions on the behavior of ( ~ , ~R)-systems is examined and a subclass of these systems whose behavior is invulnerable to such errors is specified.

The effect of environmental alterations on the structure of (~r~, ~R)-systems has been discussed by Rosen (1961). Results on the periodicity of structural alterations of (~ , 9~)-systems were derived on the assumption that the systems functioned without error. Here we investigate aberrations which may arise in the metabolic apparatus of the systems, and we indicate to what extent the stabili ty of the systems is altered by the malfunctions.

Considering the simplest (~ , 9~)-system {f, r defined over the set (A, B), where

f : A - - > B q~: B ----> H ( A , B),

the malfunctions we investigate may be formalized as follows: The system on receipt of a stimulus x I, does not make the response, say, f ( x l ) = Yl, as pre-

*Present address: College of Engineering, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley 94720.

117

118 L. D E M E T R I U S

scribed by its structure, bu t behaves as if it were stimulated by the input x2, and makes the transformation f ( x2) = Y2.

In this situation, we assume tha t the component {f} is only temporarily im- paired and that the next time it receives a stimulus it makes the prescribed transition without error. In the case of certain (~ , ~)-systems these transient malfunctions, although they are not due to any permanent failure of the system, will be perpetuated indefinitely and result in the complete disorder of the system. With (~ , ~)-systems having a particular structure, transient errors will only affect the behavior of the system over a finite number of environmental alterations, and the system eventually becomes well-behaved. In such cases the effects of the metabolic malfunctions can be erased by subject- ing the system to a finite sequence of environmental inputs. We describe systems whose original behavior can be recovered by such a process as error- correcting systems. The purpose of this report is to determine what class of (~ , ~)-systems are error-correcting.

The approach we adopt is closely related to studies of error-limiting devices in sequential machines. In these studies a sequential machine is regarded as the encoder for a communication channel, and the malfunctions that arise are due to errors in the input sequence to the encoder. Although in our discussion of (~ , ~)-systems we consider metabolic malfunctions, it is evident that , formally, errors due to the components {f} can be interpreted as errors in the sequence of environmental stimuli.

The notion of synchronizing input sequences is crucial to our analysis of transient errors in (~ , ~)-systems. I t was first introduced by Gilbert and Moore (1959) in studies of binary encoding for communication channels. The application to sequential machines was made by Neumann (1964). We recall tha t an input sequence �9 is a synchronizer for a state s~ iff, irrespective of the initial state, the application of the input sequence always results in the final state s~.

Thus the application of a synchronized sequence results in the resynchroniza- tion of the machine, and the subsequent output sequence is correct, even though transient errors occurred in the past history of the system. Machines with finite memory have the property that their behavior is dependent only on a finite number, say n, of previous inputs. The effects of transient errors may thus be erased by applying to the machine any sequence of inputs of length greater than n.

We formalize these notions in the following definitions and derive results which will be useful in our analysis.

Definition 1. A sequential machine is a 5-tuple ~ = (S, I , J , 8, 2) where S

CELLULAR SYSTEMS AS SEQUE:NTIAL MACHINES 119

is the set of states; I , the set of inputs; J , the set of outputs; 8, the transition function which is a mapping from S • I into S; and ~, the output function which is a mapping from S • I into J .

Defini t ion 2. An input sequence 5 is a synchronizer for the state st if 8(s, 5) = st, for all s �9 S.

Defini t ion 3. A sequential machine is strongly connected if, given any st, sj �9 S , there exists a sequence �9 such tha t 8(s,, 5) = %

Proposi t ion 1. I f ~ is strongly connected and there exists a synchronizer for the state st, then there exist synchronizers for all states s �9 S.

Proof. Since there exists a synchronizer for s~, then given any sj there exists a sequence i such tha t $(sj, 7) = s~. Since ~ is strongly connected, given any state s �9 ~q, there exists a sequence 9 such tha t 8(s~, Y) = s, 8(sj, Y~) = 8(s~, Y) = s and the sequence i~ is a synchronizer for the state s.

Defini t ion 3. A sequential machine ~ has finite memory of order n. I f given any states s~, sj E S, then 8(s~, 5) = 8(sj, 5) for all sequences ~ of length greater than n, and ~(s~, 7) # 8(sj, 7) for at least one sequence ~ such tha t l(~) = n [/(i) denotes the length of the sequence ~].

Proposi t ion 2. I f ~ has finite memory of order n, and ~ is strongly con- nected, then each state s has a synchronizer.

Proof. I f ~ has finite memory n, then there exists a state s~ E S with a synchronizer. But then all states have a synchronizer by Proposition 1.

We note tha t if ~ is strongly connected and there exists a synchronizer for each state, then ~ is not necessarily of finite memory. For example, consider the machine whose state table is shown in Table 1. From the graph shown in Figure 1, we note tha t (0, 0) is a synchronizer for the state sl: (0, 1) is a syn-

TABLE I (sl, s2)

sl s~ s2 sn/x~,1 I ( s . s2) 82 81 sl

s I 82 Figure 1.

120 L. DEMETRIUS

chronizer for a~. 9X has infinite memory since sequences of the form (111. . . 1) do not distinguish between the states s~ and a2.

This concludes our remarks on finite memory in sequential machines. Let us consider the application of these notions to ( ~ , 9t)-systems. We recall tha t in Demetrius (1966) the term (!IX, ilt)-machine was used to denote those sequential machines which were derived from (Y)2, 9t)-systems. I t was shown that (!~, !R)-machines are characterized by the following property.

~(s~, x~ ) = ;~(sj, x~) ~ ~(a~, x ~ ) = ~(s s, x . ) .

~re now impose restrictions on the (YX, ~)-system (f, 4} and obtain the follow- ing.

Proposition 3. I f (f, r denotes an (9X, 9l)-system and 4: B --> H(A, B) is a one-to-one mapping, then

~(at, zm) = ~(as, x~) ~ ;t(a~, xm) = ;t(ss, x~).

Proof. From 8(si, xm) = 8(a s, x~), it follows that 4[f(xm)] -- r ]. But 4 is a one-to-one mapping, so that ;~(s i, xm)= 7t(Ss, xn). This result enables us to prove the following.

Proposition 4. Let (f, r denote an (OX, 9t)-system, and let the mapping 4 be one-to-one, then the equivalent (9X, !R)-machine A has infinite memory.

Proof. Suppose A has finite memory; then there exist states af, a s (s t ~ as) such, that 8(s~, ~) = ~(s:, ~) for all sequences �9 sufficiently long. Hence in the (~0l, 9t)-system (f, r we have

:,(~') = IA~); r = 4[fj(~)].

Then fi = Is, which implies tha t at = sj, contrary to our assumption. Hence, A has a finite memory.

We illustrate this result by the following example. Consider the (~ , !it)- system defined on the sets (A, B), where

A = (xl, x2), B = (0, 1) fl(xl) = O, f2 (x l )= O, f3 (x l )= l, A ( x l ) = 1, fl(x2) = O, f2(x2)= 1, fa(x2)= O, f4(x2)= 1,

4(0) = f l , 4 (1) = :2.

The equivalent (YJ~, iit)-maehine A is given in Table II. From the graph in Figure 2 we note tha t the sequences of the form

(xg. x g . . . x2) identify the states (s 1, s2). Hence, A has infinite memory. We

CELLULAR SYSTEMS AS SEQUENTIAL MACHINES 121

note, however, tha t (xl, xl) is a synchronizer for the state sl. Hence, the application of the input sequence (xl, Xx) will erase the effects of previous errors b y resynchronizing the machine to state s 1. Let us define an (!l~, ~)-system to be stable if the equivalent (~ , ~)-machine has finite memory and as unstable

TABLE II

( 8 1 • 82~ 83, 34) X 1 Z 2 X ~ 2

sl sl, 0 s . 0 (s. s 2 ~

s2 s~, 0 s2, 1 / \ (s. s2)

S3 82'1 81'0 X J ~

s~ s2, 1 s2, 1 s~ (s. s2)

Figure 2.

otherwise; then a sufficient condition that an (~ , !R)-system is stable is given by the following.

Proposition 5. Let {f, r denote an (~ , ~)-system over the set (A, B). I f r = f for all Yt e B, then {f, r is stable.

Proof. In the equivalent (~ , !R)-maehine A, we have that

a(s~, 2 ) = a(s j , 2 ) = s

for all sequences 2 and for s~ # % Hence, A has finite memory and {f, r is stable.

From our definition of stability, it follows that if transient errors arise in the metabolic apparatus of a stable (~ , ~)-system, then the system will function correctly if subjected to a sufficiently large number of environmental alterations. :From the example given in Table I, we observe that some unstable systems can eventually respond correctly if a particular sequence of environmental inputs is applied.

The author is indebted to Dr. R. Rosen and Dr. B. Foster for their critical comments on the manuscript. This work was supported by the Air Force Office of Scientific Research Grant. # AF-AFOSR-9-65.

122 L. DEMETRIUS

L I T E R A T U R E

Demetrius, L. 1966. "Abstract Biological Systems as Sequential Machines. Behavioral Reversibility." Bull. Math. Biophysics, 28, 153-160.

Gilbert, E. N. and Moore, E .F . 1959. "Variable-length Binary Encoding." BellSystems Tech. J . , 38, 933--968.

Neumann, P. 1964. "Error-limiting Coding. Using Information-lossless Sequential Machines." I .R .E . Trans. on Inf . Theory, 1O, 108-115.

Rosen, R. 1961. "A Relational Theory of Structural Changes Induced in Biological Systems by Alterations in Environment." Bull. Math. Biophysics, 23, 165-171.

RECEIVED 10-23-67