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Representations of (M, R)-systems by categories of automata
Bulletin of Mathematical Biology, Vol. 44, No. 5, pp. 661-668, 1982. Printed in Great Britain.
0092-8240/82/050661-08503.00/0 Pergamon Press Ltd. © I982 Society for Mathematical Biology
R E P R E S E N T A T I O N S OF (M, R ) - S Y S T E M S BY C A T E G O R I E S OF A U T O M A T A • M. W. WARNER Department of Mathematics, The City University, Northampton Square, London EC1, U.K. Arbib in a paper entitled 'Categories of (M, R)-Systems' represents both simple (M, R)systems and those with varying genome as subcategories of the category of automata. An alternative characterisation of general (M, R)-systems as automata is proposed and two theorems on (M,R)-automata are proved. The two categories of automata, namely Arbib in a paper entitled 'Categories of (M,R)-Systems' represents both simple (M, R)systems with variable genetic structure, are compared.
1. Introduction. 1.1. Category theory has been used by Rosen (1964, 1966) Arbib (1966), Demetrius (1966), Baianu (1973, 1980) and Marinescu (1974) to represent biological systems. We consider the metabolismrepair systems of Rosen (1966) and representations of them as a subcategory of the category of automata (Rosen, 1964; Arbib, 1966; Baianu, 1973, 1980; Baianu and Marinescu, 1974). Arbib describes the subcategory M1 representing (M, R)-systems with fixed genome, then proceeds to a more general subcategory representing (M, R)-systems with variable genetic structure. We suggest an alternative description for this subcategory and present the corresponding representation theorem. 1.2. Arbib raised the question of finding a natural categorical characterisation for variable genome (M,R)-systems in which the morphisms allow some measure of incorporation of metabolic and genetic structure. This is achieved by our redefined subcategory M2. 1.3. Section 2 recapitulates the description of Arbib's representation of simple (M, R)-systems (with fixed genome) by the category M1 of simple (M,R)-automata. Sections 3 and 4 establish a category M2 of (M,R)automata representing (M, R)-systems of variable genetic structure. We end with a comparison of these categories, followed by a brief reference to related work on this topic and to Rosen's discussion on the general question of realizability of (M, R)-automata. 2. Representations of Simple (M, R)-Systems by Automata. 2.1. Definition 1. A category c~ is a class s¢ of objects A, B . . . . . together with a class of 2 / o f morphisms H~(A, B) . . . . , A, B E eg. Each 661
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He(A,B) is the set of morphisms from A to B. If /3 E He(B, C), a E He(A, B), then the composition of /3 by a is obtained from the function He(B, C ) x H e ( A , B ) to He(A, C), and is denoted by /3a. Whenever compositions make sense we have associativity, and He(A, A) contains an identity element, 1A. 2.2. Definition 2. A category ~g'= (M', At') is a subcategory of the category ~ if M' C M, and He,(A, B) C__He(A, B) for all (A, B) E A' x A'. The composition of morphisms in c~, is the same as in ~g, and the identity 1A is the same in ~' as in ~ for all A E M'. If, furthermore, He,(A, B) = He(A, B) for all A, B E M' x M', then rg, is a full subcategory of cC 2.3. Let rg be a category whose objects M are sets A, B , . . . and whose sets of morphisms H(A, B ) , . . . are subsets of the corresponding sets of all set-theoretic functions from A to B. Let M be closed under cartesian products. Definition 3. An automaton on the category cg is a sextuple A -- (X, Y, Q, ~, A), where X (the input set), Y (the output set), and Q (the state set) are objects of ~ , and 8: X x Q ~ Q , h: X x Q ~ Y , the next-state function and next-output function, respectively, are morphisms of At. We assume throughout that A is onto since there is no interest in outputs which are not achieved. Definition 4. A morphism ~b: A ~ A' from A to A' = (X', Y', Q',/Y, h') is a triple (a, /3, ~/), where a: X - ~ X ' , /3: Q ~ Q ' , y: Y-~ Y' and the diagram 8xX XxQ
• QxY
axB[
[Bxr • Q'x Y'
X'x O'
B'x X'
commutes. The identity morphism in H(A, A) has (a,/3, y) all as identity mappings, while the composition of ~b E H(A, A') with g,'E H(A', A") composes the corresponding triples (a,/3, y), (a',/3', 3") respectively to give ~ = (a'a, /3'/3, 3/30 E H(A, A") and the commutative diagram Xx 0
8xx
(a'a~#'#)[
"'Q x Y
I (#'#xr'y) V
X " x O"
- O"x Y" B" x X"
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
663
Here the double primes are taken to refer to the objects and morphisms of A". The class of automata thus forms a category (Glushkov, 1961; Rosen, 1964; C~z~nescu, 1967), denoted by M(C~). 2.4. Definition 5. A simple (M, R)-system on the category cg is a quadruple (A, B , / , @) where A, B are objects of ~/, H(A, B) is also an object of J , and f ~ H(A, B), dp E H(B, H(A, B)). A simple (M,R)-system may be represented by an automaton with Q = H(A, B), X = A and Y = B. Then if g E H(A, B), a E A, we have A: X x Q ~ Y given by A(a,g)=g(a), and 6: X x Q ~ Q by 6(a,g)= ¢}(g(a)) E H(A, B), since g(a) E B and @: B ~ H(A, B). We stipulate an initial state/o: A ~ B. Arbib (1966) proved that simple (M, R)-systems are represented by the objects of a subcategory MI(Cg) of M(~g) whose morphisms are triples (a, /3, y) with a,/3 surjective and 3' bijective. The proof followed from the representation of a simple (M, R)-system as an automaton with outputdependent state function, i.e. 6: X × Q ---> Q factorises through A: X × Q ---> Y by means of a function to: Y ---> Q which represents ~.
8 XxQ
"Q
Y
In the diagram, 6 = toA. Such an automaton is called a simple (M, R)automaton. Here d~: B ~ H(A, B) is the genome of the system with .fo: A ~ B as initial metabolic component (initial state), so that in the environment a E A the metabolic component g: A ~ B changes to ~ ( g ( a ) ) E H(A, B). 2.5. The dual of an output-dependent state function would be a statedependent output function (s.d.o.), i.e. A: X × Q ~ Y factorises through 6: X × Q ~ Q, and the next output depends on the next state at each stage through a function ~': Q ~ Y. In the diagram, A = if6, the direction of ~" being opposite to that of to in the previous diagram. The category of s.d.o, automata occurs more frequently in automata theory where the output is often considered to be functionally dependent on the next state. The morphisms (a,/3, 3') of this
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X×Q
. Q
category have a, 3' surjective a n d / 3 bijective. This follows f r o m the next proposition. THEOREM 1. If A is an s.d.o, automaton and x]¢ = (c~, /3, 3/) is a morphism from A to an automaton A', then A' is an s.d.o, automaton if a, /3 (and hence 3') are onto and if/3 is also 1 - 1. Proof. L e t ~"--- 3"ff/3 1. Then, if x E X, q ~ Q,
('(tS'(a(x),/3(q))) = 3"~'/3 '/3/~(x, q) = 3,(6(x, q) = 3"A(x, q) = 3"(a(x), fl(q)).
3. Variable Genetic Structure. 3.1. Variable genetic structures have been c o n s i d e r e d by several authors (e.g. Rosen, 1973; Baianu, 1973, 1980). In Arbib's paper a m e t h o d is introduced of r e p r e s e n t i n g an (M, R)-system with varying g e n o m e by an a u t o m a t o n with a fixed next-state function. He considers (f,D) E H ( A , B ) x H ( B , H ( A , B)) to be a state of this a u t o m a t o n so that both genetic and metabolic structure can be altered by an e n v i r o n m e n t A (input) acting on a given state (f, D). Definition 6. A n (M, R)-system on ~ is a quintuple J = (A, B, f, D, ~), where A , B , H ( A , B ) , H ( B , H ( A , B ) ) are objects of ~/, while f @ H ( A , B), D E H ( B , H ( A , B)), and g E H ( B x H ( B , H ( A , B)), H(B, H ( A , B))). T h e n if /Co and Do are the initial metabolic and genetic c o m p o n e n t s of the system in the e n v i r o n m e n t a E A, t h e y are c h a n g e d to (Do(fo(a)), g(fo(a), Do)). 3.2. This (M, R)-system m a y be replaced by an a u t o m a t o n A = (X, Y,
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
665
Q, 8, A) by putting Q = H ( A , B) × H ( B , H ( A , B)) X=A Y = B x H ( B , H ( A , B))
A(a, (f, O)) = (f(a), O) 8(a, (f, O)) = O(f(a), g(f(a), 0)). Here we have differed from Arbib (1966) who takes Y = B as the output set and defines A(a, (/, O)) = f(a). 2. A n automation A on the category c~ represents an (M, R)-system if and only if the state-space Q can be written as a product Q~ × 02 and the output set Y as a product I11 × Q2 in such a way that 3.3. THEOREM
(i) A(x, ql, q2) = (2(x, ql), q2) for all x @ X, ql ~ Q1, q2 @ Q2, where 2: X x Q I ~ Yl, (ii) there exists X: Y1 x Q2--> QI x Q2 such that 6 = XA. Such an automaton is called an (M, R)-automaton. 3.4. Arbib's original theorem (Arbib, 1966, Theorem 3, p. 514) not only leaves the output set as YI but also introduces a function/3: I11 -> Q1 instead of ~rlX: I11 x Q2 ~ Q~, with 1r,: Q~ x Q2-~ Q1. However this does not provide an embedding of Q2 in H(Y~, QO, so it would seem that Arbib's automaton A does not provide an adequate representation for an (M, R)-system J defined as the quintuple (A, B, f, q~, 8). 4. The Subcategory of (M, R)-Automata. 4.1. By exhibiting the morphisms we show that (M, R)-automata form a subcategory M2(C~) of the category M(C~) of general automata, as defined by Arbib (1966). 4.2. Let A = ( X , Y, Q, 8, A) be an (M,R)-automaton, and let 0 = (a,/3, y) be a morphism from A to an automaton A' = (X', Y', Q', 8% A') with state-space Q' c_ Q~ × Q~ and output set Y' c_ Y~ × Q~. Denote by ~r~, ~r~ respectively the projections of Q' onto its first and second factors. In order that a' satisfy the first condition of Theorem 2 we need ~r~)t'(x', q~, q;) independent of q~ and ~r~A'(x', q~, q~) = q~. If a,/3 are onto then {x', q'l,q~) = (a(x), /3(ql, q2)). But A'(t~(x), /3(ql, q2)) = -fit(x, ql, q2) = T(A(x, ql), q2). Let ~r~/3(ql, q2)=/31(q~, q2), ~r~fl(q~, q2)--/32(q2), and let y(A(x, q0, q2) = (4/,~(x, ql), /32(q2)), where ~: Y~ ~ Y~. Then the conditions on A' are fulfilled.
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4.3. With this notation we can state the following proposition. THEOREM 3. /f A is an (M, R)-automaton and th = (a, fl, 3') is a morphism from A to an automaton A', then the following conditions are su~cient for A' to be an (M, R)-automaton. (i) There exist Q'1, Q~, Y[ such that Q' c_ Q'1 × Q~, Y' c_ Y'~ x Q~. (ii) 7r~fl(ql, q2) = fiE(q2) depends on q2 only. (iii) 3' = 4/x/32:I11 x Q2--~ Y~ x Q~. (iv) a,/3 are surjective. (v) "r = 4/ x /32 is bijective. Proof. Conditions (ii) and (iii) ensure the behaviour of A', while (iv) and (v) as in 2.4 give us X': Y'~ x Q ~ Q~ x Q~. Define X' = fiX(4/ 1,/321). Then
X'(A'(a(x), fl(q,, q2))) = flX(4/-',flJ')(4/;(x, ql),/3(q2)) = fiX(d( x, ql), q2) = fiB(x, ql, q2) =3'(a(x), fl(ql, q2))4.4. Note that ~r',/3 = fll(q~, q2) c a n d e p e n d on q~ and q2, although ~r~M(a(x), fll(q~, q2), /32(q2))= 4/A(x, ql) must be independent of q2. This affords some measure of incorporation of the metabolic and genetic structure of A into the metabolic structure of A'. ^
5. Comparison of Categories. 5.1. It will be noted that the categories MI(~), M2(~) have been defined formally in the context of automata theory and independently of their representations of (M, R)-systems. The class s¢2 of objects of ME is contained in the class sG of objects of M~, since products of sets (e.g. QI × Q2) are themselves sets. An object of ~2 ((M, R)-automaton), (X, Y x Q2, Q1 x Q2, 3, )t), with )(: Y, x Q2---~ QI x Q2 also has restriction (i) of Theorem 2 imposed on A. By T h e o r e m s 2 and 3 the morphisms of ME fulfil the requirement to be morphisms of M1. Thus ME is a full subcategory of M~. 5.2. But all sets may also be thought of as products by the device of letting the second factor (say) be a fixed one-point set. In our case take Q2 = {42} and identify Q1 x Q2 with Q1, by the c o r r e s p o n d e n c e (ql, 42) =ql. Similarly Y~ x Q2 is identified with Y~. With these identifications the objects of M~ form a subclass of ~2. The conditions of T h e o r e m 3 with
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
667
flz as the identity 1:42 ''') 42 then identify with those of Theorem 2, so that M1 is identified with a full subcategory of M2. 5.3. This identification of the categories M1 and Mz is not, however, natural with respect to their representations of (M,R)-systems. The argument of the preceding section does pass over naturally to the representation since 42 ~ Q2 = H(B, H(A, B)) represents the fixed genome • of a simple system. But, in the discussion of 5.1, although an object of M2 is indeed an object of M1, this object represents entirely different (M, R)-systems. For instance, if A = (X, Y1 x Qz, Q~ × Qz, ,~, h) represents J = (A, B, f, qb, g) with variable genome, then B = Y~, while if A represents a simple system, B - - Y~ x Q2. Let S be the class of simple (M,R)-systems and T the class of (M,R)-systems. Then there is no " c o m m u t a t i v e diagram" incorporating the two representations R1, R2 and the inclusion of Mz in M~. The top side of the square is missing. S
T
RII
J
Mt q
i
R~
M~
This illustrates Rosen's (1971) remark that every (M, R)-system can be regarded as simple if A, B , . . . are made complicated enough. Here J can be regarded as a simple (M, R)-system by taking (f, ~) as initial state and observing that (f, ~) ~ H(A, B) x H(B, H(A, B)) defines an element of H(A, B x H(B, H(A, B))), so that the variable genome qb E H(B, H(A, B)) has been replaced by X ~ H(B × H(B, H(A, B)), H(A, B ) x H(B, H(A, B))) now representing a fixed genome.
6. General Remarks. 6.1. The automaton-model approach discussed above is of course only one method of attacking the problem of representing (M, R)-systems by formal mathematical systems. Rosen (1971, 1973) discussed various methods of realising (M, R)-systems, moving on to a consideration of the dynamics by passing from the discrete automaton-model to a continuous dynamical system. At an intermediary stage the tolerance structure introduced on finite automata (Warner, 1981) could throw further light on, say, homogeneity in (M, R)-systems (Warner, 1980). 6.2. Full use is made of the categorical approach by Baianu and Marinescu (1974) and Baianu (1980). The latter presents a representation of variable genetic maps as natural transformations of functors. Another d e v e l o p m e n t has been that of Leguizam6n (1975, 1977), who
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has linked energy with the systems yielding the category of biDenvironmental systems. 6.3. In this paper we have restricted ourselves essentially to an adaption of Arbib's (1966) representation in order to give a partial answer in his own terminology to his question of finding morphisms allowing incorporation of metabolic into genetic structure. The relevance of the automaton representation to relational biology is more fully treated in the papers cited above due to Baianu and Rosen.
LITERATURE Arbib, M. A. 1966. "Categories of (M, R)-systems." Bull. Math. Biophys. 28, 511-517. Baianu, I. C. 1973, "Some algebraic properties of (M, R)-systems." Bull. Math. Biol. 35, 213-217• • 1980. "Natural transformations of organismic structures•" Bull. Math. Biol. 42, 431--446. and M. Marinescu. 1974. "On a functorial construction of (M, R)-systems." Rev. Roum. Math. Pures Appl. 19, 389-391. Chz~nescu, V. 1967. "On the category of abstract sequential automata." Ann. Univ. Bucharest, Math. and Mechanics Series 16, 31-37• Demetrius, L, A. 1966. "Abstract biological systems as sequential machines: behavioural reversibility." Bull. Math. Biophys. 28, 153-60. Glushkov, V. M. 1961. "The abstract theory of automata." Russ. Math. Surv. 16, 1-53. Leguizam6n, C. A. 1975. "Concept of energy in biological systems•" Bull. Math. Biol. 35, 565-572. 1977. "Transfers between biological and environmental systems•" Bull. Math. Biol. 39, 397--406. Rosen, R. 1964. "Abstract biological systems as sequential machines I." Bull. Math. Biophys. 26, 103-111. . 1966. "Abstract biological systems as sequentizl machines III." Bull. Math. Biophys. 28, 141-148. 1971. "Some realisations of (M, R)-systems and their interpretation." Bull. Math. Biophys. 33, 303-319• • 1973. "On the dynamical realizations of (M, R)-systems." Bull. Math. Biol. 35, 1-9. Warner, M. W. 1980. "Semi-group, group quotient and homogeneous automata." Information and Control 47, 59--66. • 1981. "On tolerating automata." Kybernetes 10, 173-178.
RECEIVED 5-6-81 REVISED 10-27-81
0092-8240/82/050661-08503.00/0 Pergamon Press Ltd. © I982 Society for Mathematical Biology
R E P R E S E N T A T I O N S OF (M, R ) - S Y S T E M S BY C A T E G O R I E S OF A U T O M A T A • M. W. WARNER Department of Mathematics, The City University, Northampton Square, London EC1, U.K. Arbib in a paper entitled 'Categories of (M, R)-Systems' represents both simple (M, R)systems and those with varying genome as subcategories of the category of automata. An alternative characterisation of general (M, R)-systems as automata is proposed and two theorems on (M,R)-automata are proved. The two categories of automata, namely Arbib in a paper entitled 'Categories of (M,R)-Systems' represents both simple (M, R)systems with variable genetic structure, are compared.
1. Introduction. 1.1. Category theory has been used by Rosen (1964, 1966) Arbib (1966), Demetrius (1966), Baianu (1973, 1980) and Marinescu (1974) to represent biological systems. We consider the metabolismrepair systems of Rosen (1966) and representations of them as a subcategory of the category of automata (Rosen, 1964; Arbib, 1966; Baianu, 1973, 1980; Baianu and Marinescu, 1974). Arbib describes the subcategory M1 representing (M, R)-systems with fixed genome, then proceeds to a more general subcategory representing (M, R)-systems with variable genetic structure. We suggest an alternative description for this subcategory and present the corresponding representation theorem. 1.2. Arbib raised the question of finding a natural categorical characterisation for variable genome (M,R)-systems in which the morphisms allow some measure of incorporation of metabolic and genetic structure. This is achieved by our redefined subcategory M2. 1.3. Section 2 recapitulates the description of Arbib's representation of simple (M, R)-systems (with fixed genome) by the category M1 of simple (M,R)-automata. Sections 3 and 4 establish a category M2 of (M,R)automata representing (M, R)-systems of variable genetic structure. We end with a comparison of these categories, followed by a brief reference to related work on this topic and to Rosen's discussion on the general question of realizability of (M, R)-automata. 2. Representations of Simple (M, R)-Systems by Automata. 2.1. Definition 1. A category c~ is a class s¢ of objects A, B . . . . . together with a class of 2 / o f morphisms H~(A, B) . . . . , A, B E eg. Each 661
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He(A,B) is the set of morphisms from A to B. If /3 E He(B, C), a E He(A, B), then the composition of /3 by a is obtained from the function He(B, C ) x H e ( A , B ) to He(A, C), and is denoted by /3a. Whenever compositions make sense we have associativity, and He(A, A) contains an identity element, 1A. 2.2. Definition 2. A category ~g'= (M', At') is a subcategory of the category ~ if M' C M, and He,(A, B) C__He(A, B) for all (A, B) E A' x A'. The composition of morphisms in c~, is the same as in ~g, and the identity 1A is the same in ~' as in ~ for all A E M'. If, furthermore, He,(A, B) = He(A, B) for all A, B E M' x M', then rg, is a full subcategory of cC 2.3. Let rg be a category whose objects M are sets A, B , . . . and whose sets of morphisms H(A, B ) , . . . are subsets of the corresponding sets of all set-theoretic functions from A to B. Let M be closed under cartesian products. Definition 3. An automaton on the category cg is a sextuple A -- (X, Y, Q, ~, A), where X (the input set), Y (the output set), and Q (the state set) are objects of ~ , and 8: X x Q ~ Q , h: X x Q ~ Y , the next-state function and next-output function, respectively, are morphisms of At. We assume throughout that A is onto since there is no interest in outputs which are not achieved. Definition 4. A morphism ~b: A ~ A' from A to A' = (X', Y', Q',/Y, h') is a triple (a, /3, ~/), where a: X - ~ X ' , /3: Q ~ Q ' , y: Y-~ Y' and the diagram 8xX XxQ
• QxY
axB[
[Bxr • Q'x Y'
X'x O'
B'x X'
commutes. The identity morphism in H(A, A) has (a,/3, y) all as identity mappings, while the composition of ~b E H(A, A') with g,'E H(A', A") composes the corresponding triples (a,/3, y), (a',/3', 3") respectively to give ~ = (a'a, /3'/3, 3/30 E H(A, A") and the commutative diagram Xx 0
8xx
(a'a~#'#)[
"'Q x Y
I (#'#xr'y) V
X " x O"
- O"x Y" B" x X"
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
663
Here the double primes are taken to refer to the objects and morphisms of A". The class of automata thus forms a category (Glushkov, 1961; Rosen, 1964; C~z~nescu, 1967), denoted by M(C~). 2.4. Definition 5. A simple (M, R)-system on the category cg is a quadruple (A, B , / , @) where A, B are objects of ~/, H(A, B) is also an object of J , and f ~ H(A, B), dp E H(B, H(A, B)). A simple (M,R)-system may be represented by an automaton with Q = H(A, B), X = A and Y = B. Then if g E H(A, B), a E A, we have A: X x Q ~ Y given by A(a,g)=g(a), and 6: X x Q ~ Q by 6(a,g)= ¢}(g(a)) E H(A, B), since g(a) E B and @: B ~ H(A, B). We stipulate an initial state/o: A ~ B. Arbib (1966) proved that simple (M, R)-systems are represented by the objects of a subcategory MI(Cg) of M(~g) whose morphisms are triples (a, /3, y) with a,/3 surjective and 3' bijective. The proof followed from the representation of a simple (M, R)-system as an automaton with outputdependent state function, i.e. 6: X × Q ---> Q factorises through A: X × Q ---> Y by means of a function to: Y ---> Q which represents ~.
8 XxQ
"Q
Y
In the diagram, 6 = toA. Such an automaton is called a simple (M, R)automaton. Here d~: B ~ H(A, B) is the genome of the system with .fo: A ~ B as initial metabolic component (initial state), so that in the environment a E A the metabolic component g: A ~ B changes to ~ ( g ( a ) ) E H(A, B). 2.5. The dual of an output-dependent state function would be a statedependent output function (s.d.o.), i.e. A: X × Q ~ Y factorises through 6: X × Q ~ Q, and the next output depends on the next state at each stage through a function ~': Q ~ Y. In the diagram, A = if6, the direction of ~" being opposite to that of to in the previous diagram. The category of s.d.o, automata occurs more frequently in automata theory where the output is often considered to be functionally dependent on the next state. The morphisms (a,/3, 3') of this
664
M.W. WARNER
X×Q
. Q
category have a, 3' surjective a n d / 3 bijective. This follows f r o m the next proposition. THEOREM 1. If A is an s.d.o, automaton and x]¢ = (c~, /3, 3/) is a morphism from A to an automaton A', then A' is an s.d.o, automaton if a, /3 (and hence 3') are onto and if/3 is also 1 - 1. Proof. L e t ~"--- 3"ff/3 1. Then, if x E X, q ~ Q,
('(tS'(a(x),/3(q))) = 3"~'/3 '/3/~(x, q) = 3,(6(x, q) = 3"A(x, q) = 3"(a(x), fl(q)).
3. Variable Genetic Structure. 3.1. Variable genetic structures have been c o n s i d e r e d by several authors (e.g. Rosen, 1973; Baianu, 1973, 1980). In Arbib's paper a m e t h o d is introduced of r e p r e s e n t i n g an (M, R)-system with varying g e n o m e by an a u t o m a t o n with a fixed next-state function. He considers (f,D) E H ( A , B ) x H ( B , H ( A , B)) to be a state of this a u t o m a t o n so that both genetic and metabolic structure can be altered by an e n v i r o n m e n t A (input) acting on a given state (f, D). Definition 6. A n (M, R)-system on ~ is a quintuple J = (A, B, f, D, ~), where A , B , H ( A , B ) , H ( B , H ( A , B ) ) are objects of ~/, while f @ H ( A , B), D E H ( B , H ( A , B)), and g E H ( B x H ( B , H ( A , B)), H(B, H ( A , B))). T h e n if /Co and Do are the initial metabolic and genetic c o m p o n e n t s of the system in the e n v i r o n m e n t a E A, t h e y are c h a n g e d to (Do(fo(a)), g(fo(a), Do)). 3.2. This (M, R)-system m a y be replaced by an a u t o m a t o n A = (X, Y,
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
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Q, 8, A) by putting Q = H ( A , B) × H ( B , H ( A , B)) X=A Y = B x H ( B , H ( A , B))
A(a, (f, O)) = (f(a), O) 8(a, (f, O)) = O(f(a), g(f(a), 0)). Here we have differed from Arbib (1966) who takes Y = B as the output set and defines A(a, (/, O)) = f(a). 2. A n automation A on the category c~ represents an (M, R)-system if and only if the state-space Q can be written as a product Q~ × 02 and the output set Y as a product I11 × Q2 in such a way that 3.3. THEOREM
(i) A(x, ql, q2) = (2(x, ql), q2) for all x @ X, ql ~ Q1, q2 @ Q2, where 2: X x Q I ~ Yl, (ii) there exists X: Y1 x Q2--> QI x Q2 such that 6 = XA. Such an automaton is called an (M, R)-automaton. 3.4. Arbib's original theorem (Arbib, 1966, Theorem 3, p. 514) not only leaves the output set as YI but also introduces a function/3: I11 -> Q1 instead of ~rlX: I11 x Q2 ~ Q~, with 1r,: Q~ x Q2-~ Q1. However this does not provide an embedding of Q2 in H(Y~, QO, so it would seem that Arbib's automaton A does not provide an adequate representation for an (M, R)-system J defined as the quintuple (A, B, f, q~, 8). 4. The Subcategory of (M, R)-Automata. 4.1. By exhibiting the morphisms we show that (M, R)-automata form a subcategory M2(C~) of the category M(C~) of general automata, as defined by Arbib (1966). 4.2. Let A = ( X , Y, Q, 8, A) be an (M,R)-automaton, and let 0 = (a,/3, y) be a morphism from A to an automaton A' = (X', Y', Q', 8% A') with state-space Q' c_ Q~ × Q~ and output set Y' c_ Y~ × Q~. Denote by ~r~, ~r~ respectively the projections of Q' onto its first and second factors. In order that a' satisfy the first condition of Theorem 2 we need ~r~)t'(x', q~, q;) independent of q~ and ~r~A'(x', q~, q~) = q~. If a,/3 are onto then {x', q'l,q~) = (a(x), /3(ql, q2)). But A'(t~(x), /3(ql, q2)) = -fit(x, ql, q2) = T(A(x, ql), q2). Let ~r~/3(ql, q2)=/31(q~, q2), ~r~fl(q~, q2)--/32(q2), and let y(A(x, q0, q2) = (4/,~(x, ql), /32(q2)), where ~: Y~ ~ Y~. Then the conditions on A' are fulfilled.
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4.3. With this notation we can state the following proposition. THEOREM 3. /f A is an (M, R)-automaton and th = (a, fl, 3') is a morphism from A to an automaton A', then the following conditions are su~cient for A' to be an (M, R)-automaton. (i) There exist Q'1, Q~, Y[ such that Q' c_ Q'1 × Q~, Y' c_ Y'~ x Q~. (ii) 7r~fl(ql, q2) = fiE(q2) depends on q2 only. (iii) 3' = 4/x/32:I11 x Q2--~ Y~ x Q~. (iv) a,/3 are surjective. (v) "r = 4/ x /32 is bijective. Proof. Conditions (ii) and (iii) ensure the behaviour of A', while (iv) and (v) as in 2.4 give us X': Y'~ x Q ~ Q~ x Q~. Define X' = fiX(4/ 1,/321). Then
X'(A'(a(x), fl(q,, q2))) = flX(4/-',flJ')(4/;(x, ql),/3(q2)) = fiX(d( x, ql), q2) = fiB(x, ql, q2) =3'(a(x), fl(ql, q2))4.4. Note that ~r',/3 = fll(q~, q2) c a n d e p e n d on q~ and q2, although ~r~M(a(x), fll(q~, q2), /32(q2))= 4/A(x, ql) must be independent of q2. This affords some measure of incorporation of the metabolic and genetic structure of A into the metabolic structure of A'. ^
5. Comparison of Categories. 5.1. It will be noted that the categories MI(~), M2(~) have been defined formally in the context of automata theory and independently of their representations of (M, R)-systems. The class s¢2 of objects of ME is contained in the class sG of objects of M~, since products of sets (e.g. QI × Q2) are themselves sets. An object of ~2 ((M, R)-automaton), (X, Y x Q2, Q1 x Q2, 3, )t), with )(: Y, x Q2---~ QI x Q2 also has restriction (i) of Theorem 2 imposed on A. By T h e o r e m s 2 and 3 the morphisms of ME fulfil the requirement to be morphisms of M1. Thus ME is a full subcategory of M~. 5.2. But all sets may also be thought of as products by the device of letting the second factor (say) be a fixed one-point set. In our case take Q2 = {42} and identify Q1 x Q2 with Q1, by the c o r r e s p o n d e n c e (ql, 42) =ql. Similarly Y~ x Q2 is identified with Y~. With these identifications the objects of M~ form a subclass of ~2. The conditions of T h e o r e m 3 with
REPRESENTATIONS OF (M, R)-SYSTEMS BY CATEGORIES OF AUTOMATA
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flz as the identity 1:42 ''') 42 then identify with those of Theorem 2, so that M1 is identified with a full subcategory of M2. 5.3. This identification of the categories M1 and Mz is not, however, natural with respect to their representations of (M,R)-systems. The argument of the preceding section does pass over naturally to the representation since 42 ~ Q2 = H(B, H(A, B)) represents the fixed genome • of a simple system. But, in the discussion of 5.1, although an object of M2 is indeed an object of M1, this object represents entirely different (M, R)-systems. For instance, if A = (X, Y1 x Qz, Q~ × Qz, ,~, h) represents J = (A, B, f, qb, g) with variable genome, then B = Y~, while if A represents a simple system, B - - Y~ x Q2. Let S be the class of simple (M,R)-systems and T the class of (M,R)-systems. Then there is no " c o m m u t a t i v e diagram" incorporating the two representations R1, R2 and the inclusion of Mz in M~. The top side of the square is missing. S
T
RII
J
Mt q
i
R~
M~
This illustrates Rosen's (1971) remark that every (M, R)-system can be regarded as simple if A, B , . . . are made complicated enough. Here J can be regarded as a simple (M, R)-system by taking (f, ~) as initial state and observing that (f, ~) ~ H(A, B) x H(B, H(A, B)) defines an element of H(A, B x H(B, H(A, B))), so that the variable genome qb E H(B, H(A, B)) has been replaced by X ~ H(B × H(B, H(A, B)), H(A, B ) x H(B, H(A, B))) now representing a fixed genome.
6. General Remarks. 6.1. The automaton-model approach discussed above is of course only one method of attacking the problem of representing (M, R)-systems by formal mathematical systems. Rosen (1971, 1973) discussed various methods of realising (M, R)-systems, moving on to a consideration of the dynamics by passing from the discrete automaton-model to a continuous dynamical system. At an intermediary stage the tolerance structure introduced on finite automata (Warner, 1981) could throw further light on, say, homogeneity in (M, R)-systems (Warner, 1980). 6.2. Full use is made of the categorical approach by Baianu and Marinescu (1974) and Baianu (1980). The latter presents a representation of variable genetic maps as natural transformations of functors. Another d e v e l o p m e n t has been that of Leguizam6n (1975, 1977), who
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has linked energy with the systems yielding the category of biDenvironmental systems. 6.3. In this paper we have restricted ourselves essentially to an adaption of Arbib's (1966) representation in order to give a partial answer in his own terminology to his question of finding morphisms allowing incorporation of metabolic into genetic structure. The relevance of the automaton representation to relational biology is more fully treated in the papers cited above due to Baianu and Rosen.
LITERATURE Arbib, M. A. 1966. "Categories of (M, R)-systems." Bull. Math. Biophys. 28, 511-517. Baianu, I. C. 1973, "Some algebraic properties of (M, R)-systems." Bull. Math. Biol. 35, 213-217• • 1980. "Natural transformations of organismic structures•" Bull. Math. Biol. 42, 431--446. and M. Marinescu. 1974. "On a functorial construction of (M, R)-systems." Rev. Roum. Math. Pures Appl. 19, 389-391. Chz~nescu, V. 1967. "On the category of abstract sequential automata." Ann. Univ. Bucharest, Math. and Mechanics Series 16, 31-37• Demetrius, L, A. 1966. "Abstract biological systems as sequential machines: behavioural reversibility." Bull. Math. Biophys. 28, 153-60. Glushkov, V. M. 1961. "The abstract theory of automata." Russ. Math. Surv. 16, 1-53. Leguizam6n, C. A. 1975. "Concept of energy in biological systems•" Bull. Math. Biol. 35, 565-572. 1977. "Transfers between biological and environmental systems•" Bull. Math. Biol. 39, 397--406. Rosen, R. 1964. "Abstract biological systems as sequential machines I." Bull. Math. Biophys. 26, 103-111. . 1966. "Abstract biological systems as sequentizl machines III." Bull. Math. Biophys. 28, 141-148. 1971. "Some realisations of (M, R)-systems and their interpretation." Bull. Math. Biophys. 33, 303-319• • 1973. "On the dynamical realizations of (M, R)-systems." Bull. Math. Biol. 35, 1-9. Warner, M. W. 1980. "Semi-group, group quotient and homogeneous automata." Information and Control 47, 59--66. • 1981. "On tolerating automata." Kybernetes 10, 173-178.
RECEIVED 5-6-81 REVISED 10-27-81