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Some algebraic properties of (M, R)-systems
BULLETIN OF MATHE~ATICAL B I O L O G Y
VOLII~IE 35, 1973
SOME ALGEBlZAIC PROPEI~TIES OF (M, R)-SYSTEMS
[] I. B~IA_~v* Faculty of Physics, 14 Academiei St., Bucharest
On the basis of Rosen's representation of (M, R)-systems as sequential machines (Rosen, Bull. Math. Biophys:, 26, 103-111, 1964), the existence of projective limits in categories of general (M, R)-systems is proved.
1. Introduction. In this note some algebraic aspects of categories of general (M, R)-systems will be presented, and their biological significance will be briefly discussed. The theory of (M, R)-systems was created by Robert Rosen (1958a, b), and it was further developed by him (l~osen 1959, 1961, 1962; 1963a, b; 1964a, b; 1966; 1968a, b; 1971), as well as by others (Arbib, 1966; Demetrius, 1966; Foster, 1966; B~ianu, 1971; B~ianu and M~rinescu, 1973). Let us recall first some abstract, algebraic aspects of this theory. It was shown by l=Losen that the simplest (M, R)-system [f, (I)s] with f e H(A, B) and (I)s e H(B, H(A, B)}, can be represented as a sequential machine A = (S, M, 2V, 3, ~) via the following identifications. S = H(A, B) is the set of "states" of the sequential machine; M --= A is the set of "inputs" of the machine; N ~ B is the set of "outputs" of the machine; * Present address: Physics Department, Queen Elizabeth College, Campden Hill Road, London W.8 7AH, England. 213
214
I. B ~ I A N U
8: A x H(A, B)-* H(A, B) =- 8(@,f) = Of{f(a)} is the next-state function or the "transition function"; 4: A x H ( A , B) --->B =- 2(a,f) = f(a) is the output function (l~osen, 1964).
A general (M,R)-system {fl, f2,f3 . . . . . f~; • h, 0 i ~ , . . . , Oil}, with f i e H(A~, Bi) and Os, e H{i--[ j Bj, H(A~, B~)}, was then represented as a sequential machine A = (S, M, N, ~, 4) by setting S - ~
H(A~, B~)
M - I-I A ~ t,
hr -- I--I B~
(1)
~:s x M-->S ~ : S x M--->N,
where 8 is defined by the assignments 8{(f~,..., f~), (a~ . . . . . a~)} "= [ ¢ h { f l ( a l ) ,
f2(a2) .....
fn(an)} .....
(SI)fl{fl(al), f2(a2) . . . .
,
and with h being defined b y the equality ~(fl, fa . . . . , fn) = {fl(al), f2(a~.) . . . . , f~(a~)}.
In this definition, all (I)r~ are factorizable through subproducts of their domains (loc. cit., pp. 107-108). The mappingsfi: Ai --> Bt exhibit metabolic activities, while the mappings q)i~: ]-[J Bj---~H(A~, Bi) represent the "genome". Subsequently, it was shown that the totality of sequential machines over a closed cartesian category s~ is in itself a category M ( ~ ) . ~ is a category of sets which is closed under the cartesian product functor P, that is, if A, B are two objects in ~', then P ( A , B) = A × B is also in ~ , and if f, g are mappings in ~ then so is f x g: A × B --> A' × B', for f: A --> A' and g: B --* B'. The state, input and output sets of sequential machines were defined as objects in ~ . Also, the next-state function and the output function were defined as mappings in ~ . I f A and A' are two sequential machines then H ( A , A') is taken to be the set of all homomorphisms ¢: A---> A'. For further Mgebraic details the reader is referred to Rosen (1966). In the sequel, the term automaton--instead of "sequential machine"--will be used. The category of automata was independently introduced later than Rosen b y C~z£neseu (1967), and he has proved that any projective system of the c~tegory of automata has a projective limit.
P R O P E R T I E S OF (M, R)-SYSTEMS
215
Also, he has shown that any family of objects i t / t h e category of automata has a direct sum (coproduct), and that a cokernel exists in this category (loc. cir.). (M, R)-Systems over a category ~( form a subeategory of M(~/)--the category of automata over ~2 (Rosen, loc. cir.). Finally, it was shown that the totality of (M, R)-systems over a category is a subcategory of a more restricted category of automata 2~(~) (Rosen, 1966, Theorem 3 on p. 148). The objects of M ( ~ ) are the same as those of M(~). If A, A' are two automata in M(~), then/~(A, A') is i n / ~ ( ~ ) , and it consists of only those automata homomorphisms 1F = (¢1, ~2, ~ba) such that ~bl:S-->S' is onto, and ~b2:M-->M', ~b3:N-->/V' are one-one. In fact, Arbib (1966) has shown that M ( ~ ) must be enlarged since automata homomorphisms ~F = (~h~,¢2, ~3), with ~bl, ~b2 being onto, and ¢3 being 1-1, are still leading to automata A' which can represent simple (M, R)-systems (see Theorem 2 on p. 513, loc. cir.). Let us denote this morphic extension of M ( ~ ) by _M(~). Thus, the totality of (M, R)-systems over ~ is a subcategory of _M(~). Gathering these points together one could ask if _M(~) inherits the algebraic properties of M(~), and if the category M~ of all (M, R)-systems still has these properties. The answer to this last question is given in the next section.
2. Projective Limits of (M, R)-Systems. A direct answer to the above question can be given by examining first the proofs presented by C~z~nescu (1967) for some Mgebraie properties of the category of automata, and then, by applying these proofs to the particular case of (M, R)-systems. However, this implies rather complicated denotations which will be avoided here by providing a direct, different proof of the existence of projective limits (limits) in the category M~(~) of all (M, R)-systems over a closed cartesian category ~ .
Theorem 1. Any projective system in the category M~(~) of general (M, R)-systems, on a closed cartesian category ~ , has a projective limit (limit). Proof. The fact that ~ is a closed cartesian category is essential in proving the theorem. Let [ M ~ ] ~ be a projective system of general (M, R)-systems in Ma(~). An object M* of this projective system is defined as in (1) by (S~, M s, N~, 8~, A~). The general definition of a projective system requires that I be a partially ordered set, and that for any i, j, ]ce I, with i < j < k, the diagram
\
be commutative.
216
I. B~IANU
To prove that [M*]~, 1 has a projective limit is to construct this limit. (For a definition of projective limit see Mitchel, 1965 or B~ianu, 1970, on p. 557.) Consider the (M, R)-system defined by setting L = (S, M, N, 3, A), with
s=l-[s,, i
t
3:S x M-->S,
N=I--I:N,, i
:k:S x M---> N,
such that 3 = 81 x 32 x . . - x
3~ x . . . ,
A = A:t x A~ x . . . x
At x - . . ,
with 3~:S~ x Mt--->Si, ht:S~ x Mt--->Ni for any i e I . (The symbol 'T-i" denotes the cartesian product, and the symbol "× " between two mappings must be given the same meaning as in the definition of the closed cartesian category ~.) The automaton L = (S, M, N, 3, ~) is the projective limit, or limit, of [M*]t~i because this construction is valid for any projective system in categories of sets. Q.E.D.
Theorem 2. Any family of objects in the category Mn(~) has a direct sum, and a cokernel exists in Ms(~). The proofs are immediate and can be easily obtained as particular cases of the proofs given by Chz~nescu (1967). The biological interpretation of the first theorem would be that if a projective system of biological systems represented by [M~],~ I exists, then there also exists a biological system L which can be mapped (or projected) onto each member of the projective system. In a sense, it has the properties of any other biological system represented by the members of the projective system. Anyhow, the realizability problem for /5 remains unsolved since the realization of the projective system does not automatically imply the realizability of the projective limit. The problem could be eventually solved by taking into account the realizability of mappings in case of (M, R)systems (Rosen, 1964b). The second theorem has, in fact, two parts. The realizabflity problem for direct sums of biological systems is similar to that one encountered here, above. The existence of a cokernel in Mn(~) would imply the existence of a biological system in which some metabolic and genetic properties or components correspond to equivalence classes of metabolic and genetic components of "larger" biological organisms. Realizability problems of this kind will be discussed in a subsequent paper, on an experimental basis.
PI4OPERTIES OF (M, R)-SYSTEMS
217
LITEI~ATUlZE Arbib, M. 1966. "Categories of(M, R)-Systems." Bull. Math. Biophysics, 28, 511-517. B~ianu, I. 1970. "Organismic Supereategories: II. On Multistable Systems." Ibid., 32, 539-561. 1971. "Categories, Functors and Automata Theory." The Fourth Int. Congress L . M . P . S . , Bucharest, August-September 1971. - - - - - - . and M. Marincscu. 1973. "A Functorial Construction of (M, R)-Systems." Rev. Roum. Math. Puree et Appl. (in press). C~z~nescu, V. 1967. "On the Category of Abstract Sequential Automata" (paper in Romanian followed b y summary in French and Russian), Ann. Univ. Bucharest, Math. and Mechanics Series, 16, ~o. 1, 31-37. Demetrius, L. 1966. "Abstract Biological Systems as Sequential Machines: Behavioral Reversibility." Bull. Math, Biophysics, 28, 153-160. Foster, B . L . 1966. "Re-establishability in Abstract Biology." Ibid., 28, 371-374. Freyd, P. 1964. Abelian Categories. A n Introduction to the Theory of Functors." New York: Harper & Row. Mitchell, B. 1965. The Theory of Categories. New York and London: Academic Press. Rosen, Robert. 1958a. "A Relational Theory of Biological Systems." Bull. Math. Biophysics, 20, 245-260. 1958b. "The Representation of Biological Systems from the Standpoint of the Theory of Categories." Ibid., 20, 317-342. 1959. "A Relational Theory of Biological Systems I I . " Ibid., 21, 109-127. 1961. "A Relational Theory of Structural Changes Induced in Biological Systems b y Alterations in Environment." Ibid., 23, 165-171. 1962. "A Note on Abstract Relational Biologies." Ibid., 24, 31-38. 1963a, "On ~he l~cversibility of Environmentally Induced Alterations in Abstrac~ Biological Systems." Ibid., 25, 41-50. 1963b. "Some Results in Graph Theory and Their Application to Abstract Relational Biology." Ibid., 25, 231-241. 1964a. "Abstract Biological Systems as Sequential Machines." Ibid., 26, 103-111. 1964b. "Abstract Biological Systems as Sequential Machines I I : Strong Connectedness and Reversibility." Ibid., 26, 239-246. 1966a. "Abstract Biological Systems as Sequential Machines: I I I . Some Algebraic Aspects." Ibid., 28, 141-148. 1966b. "A Note on Replication in (M, R)-Systems." Ibid., 28, 149-151. 1968a. "On Analogous Systems." Ibid. 30, 481-492. 1968b. "Relational Biology and Cybernetics." I n Biokybernetil~, 49-56. Leipzig: Karl Marx Universit~t, Drischel, I-I. and Tiedt, N., eds. 1971. "Some Realizations of (M, R)-Systems and Their Interpretation." Bull. Math. Biophysics, 33, 303-319.
VOLII~IE 35, 1973
SOME ALGEBlZAIC PROPEI~TIES OF (M, R)-SYSTEMS
[] I. B~IA_~v* Faculty of Physics, 14 Academiei St., Bucharest
On the basis of Rosen's representation of (M, R)-systems as sequential machines (Rosen, Bull. Math. Biophys:, 26, 103-111, 1964), the existence of projective limits in categories of general (M, R)-systems is proved.
1. Introduction. In this note some algebraic aspects of categories of general (M, R)-systems will be presented, and their biological significance will be briefly discussed. The theory of (M, R)-systems was created by Robert Rosen (1958a, b), and it was further developed by him (l~osen 1959, 1961, 1962; 1963a, b; 1964a, b; 1966; 1968a, b; 1971), as well as by others (Arbib, 1966; Demetrius, 1966; Foster, 1966; B~ianu, 1971; B~ianu and M~rinescu, 1973). Let us recall first some abstract, algebraic aspects of this theory. It was shown by l=Losen that the simplest (M, R)-system [f, (I)s] with f e H(A, B) and (I)s e H(B, H(A, B)}, can be represented as a sequential machine A = (S, M, 2V, 3, ~) via the following identifications. S = H(A, B) is the set of "states" of the sequential machine; M --= A is the set of "inputs" of the machine; N ~ B is the set of "outputs" of the machine; * Present address: Physics Department, Queen Elizabeth College, Campden Hill Road, London W.8 7AH, England. 213
214
I. B ~ I A N U
8: A x H(A, B)-* H(A, B) =- 8(@,f) = Of{f(a)} is the next-state function or the "transition function"; 4: A x H ( A , B) --->B =- 2(a,f) = f(a) is the output function (l~osen, 1964).
A general (M,R)-system {fl, f2,f3 . . . . . f~; • h, 0 i ~ , . . . , Oil}, with f i e H(A~, Bi) and Os, e H{i--[ j Bj, H(A~, B~)}, was then represented as a sequential machine A = (S, M, N, ~, 4) by setting S - ~
H(A~, B~)
M - I-I A ~ t,
hr -- I--I B~
(1)
~:s x M-->S ~ : S x M--->N,
where 8 is defined by the assignments 8{(f~,..., f~), (a~ . . . . . a~)} "= [ ¢ h { f l ( a l ) ,
f2(a2) .....
fn(an)} .....
(SI)fl{fl(al), f2(a2) . . . .
,
and with h being defined b y the equality ~(fl, fa . . . . , fn) = {fl(al), f2(a~.) . . . . , f~(a~)}.
In this definition, all (I)r~ are factorizable through subproducts of their domains (loc. cit., pp. 107-108). The mappingsfi: Ai --> Bt exhibit metabolic activities, while the mappings q)i~: ]-[J Bj---~H(A~, Bi) represent the "genome". Subsequently, it was shown that the totality of sequential machines over a closed cartesian category s~ is in itself a category M ( ~ ) . ~ is a category of sets which is closed under the cartesian product functor P, that is, if A, B are two objects in ~', then P ( A , B) = A × B is also in ~ , and if f, g are mappings in ~ then so is f x g: A × B --> A' × B', for f: A --> A' and g: B --* B'. The state, input and output sets of sequential machines were defined as objects in ~ . Also, the next-state function and the output function were defined as mappings in ~ . I f A and A' are two sequential machines then H ( A , A') is taken to be the set of all homomorphisms ¢: A---> A'. For further Mgebraic details the reader is referred to Rosen (1966). In the sequel, the term automaton--instead of "sequential machine"--will be used. The category of automata was independently introduced later than Rosen b y C~z£neseu (1967), and he has proved that any projective system of the c~tegory of automata has a projective limit.
P R O P E R T I E S OF (M, R)-SYSTEMS
215
Also, he has shown that any family of objects i t / t h e category of automata has a direct sum (coproduct), and that a cokernel exists in this category (loc. cir.). (M, R)-Systems over a category ~( form a subeategory of M(~/)--the category of automata over ~2 (Rosen, loc. cir.). Finally, it was shown that the totality of (M, R)-systems over a category is a subcategory of a more restricted category of automata 2~(~) (Rosen, 1966, Theorem 3 on p. 148). The objects of M ( ~ ) are the same as those of M(~). If A, A' are two automata in M(~), then/~(A, A') is i n / ~ ( ~ ) , and it consists of only those automata homomorphisms 1F = (¢1, ~2, ~ba) such that ~bl:S-->S' is onto, and ~b2:M-->M', ~b3:N-->/V' are one-one. In fact, Arbib (1966) has shown that M ( ~ ) must be enlarged since automata homomorphisms ~F = (~h~,¢2, ~3), with ~bl, ~b2 being onto, and ¢3 being 1-1, are still leading to automata A' which can represent simple (M, R)-systems (see Theorem 2 on p. 513, loc. cir.). Let us denote this morphic extension of M ( ~ ) by _M(~). Thus, the totality of (M, R)-systems over ~ is a subcategory of _M(~). Gathering these points together one could ask if _M(~) inherits the algebraic properties of M(~), and if the category M~ of all (M, R)-systems still has these properties. The answer to this last question is given in the next section.
2. Projective Limits of (M, R)-Systems. A direct answer to the above question can be given by examining first the proofs presented by C~z~nescu (1967) for some Mgebraie properties of the category of automata, and then, by applying these proofs to the particular case of (M, R)-systems. However, this implies rather complicated denotations which will be avoided here by providing a direct, different proof of the existence of projective limits (limits) in the category M~(~) of all (M, R)-systems over a closed cartesian category ~ .
Theorem 1. Any projective system in the category M~(~) of general (M, R)-systems, on a closed cartesian category ~ , has a projective limit (limit). Proof. The fact that ~ is a closed cartesian category is essential in proving the theorem. Let [ M ~ ] ~ be a projective system of general (M, R)-systems in Ma(~). An object M* of this projective system is defined as in (1) by (S~, M s, N~, 8~, A~). The general definition of a projective system requires that I be a partially ordered set, and that for any i, j, ]ce I, with i < j < k, the diagram
\
be commutative.
216
I. B~IANU
To prove that [M*]~, 1 has a projective limit is to construct this limit. (For a definition of projective limit see Mitchel, 1965 or B~ianu, 1970, on p. 557.) Consider the (M, R)-system defined by setting L = (S, M, N, 3, A), with
s=l-[s,, i
t
3:S x M-->S,
N=I--I:N,, i
:k:S x M---> N,
such that 3 = 81 x 32 x . . - x
3~ x . . . ,
A = A:t x A~ x . . . x
At x - . . ,
with 3~:S~ x Mt--->Si, ht:S~ x Mt--->Ni for any i e I . (The symbol 'T-i" denotes the cartesian product, and the symbol "× " between two mappings must be given the same meaning as in the definition of the closed cartesian category ~.) The automaton L = (S, M, N, 3, ~) is the projective limit, or limit, of [M*]t~i because this construction is valid for any projective system in categories of sets. Q.E.D.
Theorem 2. Any family of objects in the category Mn(~) has a direct sum, and a cokernel exists in Ms(~). The proofs are immediate and can be easily obtained as particular cases of the proofs given by Chz~nescu (1967). The biological interpretation of the first theorem would be that if a projective system of biological systems represented by [M~],~ I exists, then there also exists a biological system L which can be mapped (or projected) onto each member of the projective system. In a sense, it has the properties of any other biological system represented by the members of the projective system. Anyhow, the realizability problem for /5 remains unsolved since the realization of the projective system does not automatically imply the realizability of the projective limit. The problem could be eventually solved by taking into account the realizability of mappings in case of (M, R)systems (Rosen, 1964b). The second theorem has, in fact, two parts. The realizabflity problem for direct sums of biological systems is similar to that one encountered here, above. The existence of a cokernel in Mn(~) would imply the existence of a biological system in which some metabolic and genetic properties or components correspond to equivalence classes of metabolic and genetic components of "larger" biological organisms. Realizability problems of this kind will be discussed in a subsequent paper, on an experimental basis.
PI4OPERTIES OF (M, R)-SYSTEMS
217
LITEI~ATUlZE Arbib, M. 1966. "Categories of(M, R)-Systems." Bull. Math. Biophysics, 28, 511-517. B~ianu, I. 1970. "Organismic Supereategories: II. On Multistable Systems." Ibid., 32, 539-561. 1971. "Categories, Functors and Automata Theory." The Fourth Int. Congress L . M . P . S . , Bucharest, August-September 1971. - - - - - - . and M. Marincscu. 1973. "A Functorial Construction of (M, R)-Systems." Rev. Roum. Math. Puree et Appl. (in press). C~z~nescu, V. 1967. "On the Category of Abstract Sequential Automata" (paper in Romanian followed b y summary in French and Russian), Ann. Univ. Bucharest, Math. and Mechanics Series, 16, ~o. 1, 31-37. Demetrius, L. 1966. "Abstract Biological Systems as Sequential Machines: Behavioral Reversibility." Bull. Math, Biophysics, 28, 153-160. Foster, B . L . 1966. "Re-establishability in Abstract Biology." Ibid., 28, 371-374. Freyd, P. 1964. Abelian Categories. A n Introduction to the Theory of Functors." New York: Harper & Row. Mitchell, B. 1965. The Theory of Categories. New York and London: Academic Press. Rosen, Robert. 1958a. "A Relational Theory of Biological Systems." Bull. Math. Biophysics, 20, 245-260. 1958b. "The Representation of Biological Systems from the Standpoint of the Theory of Categories." Ibid., 20, 317-342. 1959. "A Relational Theory of Biological Systems I I . " Ibid., 21, 109-127. 1961. "A Relational Theory of Structural Changes Induced in Biological Systems b y Alterations in Environment." Ibid., 23, 165-171. 1962. "A Note on Abstract Relational Biologies." Ibid., 24, 31-38. 1963a, "On ~he l~cversibility of Environmentally Induced Alterations in Abstrac~ Biological Systems." Ibid., 25, 41-50. 1963b. "Some Results in Graph Theory and Their Application to Abstract Relational Biology." Ibid., 25, 231-241. 1964a. "Abstract Biological Systems as Sequential Machines." Ibid., 26, 103-111. 1964b. "Abstract Biological Systems as Sequential Machines I I : Strong Connectedness and Reversibility." Ibid., 26, 239-246. 1966a. "Abstract Biological Systems as Sequential Machines: I I I . Some Algebraic Aspects." Ibid., 28, 141-148. 1966b. "A Note on Replication in (M, R)-Systems." Ibid., 28, 149-151. 1968a. "On Analogous Systems." Ibid. 30, 481-492. 1968b. "Relational Biology and Cybernetics." I n Biokybernetil~, 49-56. Leipzig: Karl Marx Universit~t, Drischel, I-I. and Tiedt, N., eds. 1971. "Some Realizations of (M, R)-Systems and Their Interpretation." Bull. Math. Biophysics, 33, 303-319.