CYBERNETICS AND THE THEORY OF REGULATION

N. I. GLEBOV

ON THE REPRESENTATION OF OPERATORS OVER MEMORY

(Presented by Academician A. I. Berg on 8 I 1962)

In programming one has to deal with the representation of some operators over memory in the form of a product of other operators.

For what follows we need several definitions given by A. A. Lyapunov in reports at a meeting of the Moscow Mathematical Society and at the Fourth All-Union Mathematical Congress. A memory is a set \(\Omega\), whose elements are called cells. Let \(G\) be a set of elements called the states of cells. A state of memory is a mapping \(f : \Omega \to G\). Let \(\Xi = \{f\}\) be the set of all possible states of memory. An operator over the memory \(\Omega\) (or over \(\Omega\)) is a mapping \(A : \Xi_1 \to \Xi_2\), where \(\Xi_1\) and \(\Xi_2 \subset \Xi\). If \(A : \Xi_1 \to \Xi_2\) and \(B : \Xi_2 \to \Xi_3\), then the product of the operators \(A\) and \(B\) is the operator \(C = AB : \Xi_1 \to \Xi_3\), defined as follows: \(C(f) = B(A(f))\). Operators over \(\Omega\) form a category \((^1)\). A tuple of \(n\) cells \(x^1,\ldots,x^n\), a memory state \(f(x)\), and an arbitrary function \(\psi(g_1,\ldots,g_n)\) of \(n\) states determine an \(n\)-function of special form: \(\varphi(x^1,\ldots,x^n / f(x)) = \psi(f(x^1),\ldots,f(x^n))\). An \(n\)-function of special form is called an \((n,m)\)-operation if \(\psi(g_1,\ldots,g_n): G^n \dot{\to} G^m\), i.e. \(\varphi(x^1,\ldots,x^n / f(x)) = (\bar g_1,\ldots,\bar g_m)\). An ordered collection of \(m\) distinct cells \(y^1,\ldots,y^m\) and an \((n,m)\)-operation determine an \((n,m)\)-operator of special form

\[ A(f(x))=f_1(x)= \begin{cases} \bar g_i, & \text{if } x=y^i,\\ f(x), & \text{if } x\ne y^i,\quad i=1,\ldots,m. \end{cases} \]

We consider a memory \(\Omega\) with an arbitrary set of cells, each of which is capable of being in a finite number of states, and we study the question of representability of operators over the memory \(\Omega\) in the form of a product of other operators.

Theorem 1. Every \((n,m)\)-operator over \(\Omega\) is representable in the form of a product of \((n,1)\)-operators over \(\Omega\).

Theorem 2. For \(m<n\) there exist \((n,1)\)-operators over \(\Omega\) that are not representable in the form of a product of \((m,1)\)-operators over \(\Omega\).

Let the memory \(\Omega'\) be obtained by adding to the memory \(\Omega\) one cell taking two states.

Theorem 3. Every \((n,m)\)-operator over \(\Omega\) is representable in the form of a product of \((2,1)\)-operators over \(\Omega\).

The study of more complicated cases requires auxiliary considerations. Let \(K\) be an arbitrary category, \(\widetilde K\) the set of all non-identity elements of the category \(K\), and \(\mathfrak N \subset \widetilde K\). A subcategory \(K_1 \subset K\) will be called regular if every identity element of the subcategory \(K_1\) is an identity element of the category \(K\). The smallest regular subcategory \(K(\mathfrak N)\) such that \(K(\mathfrak N) \supset \mathfrak N\) will be called the categorical closure of the set \(\mathfrak N\). Sets \(\mathfrak N_1\) and \(\mathfrak N_2 \subset \widetilde K\) are called equivalent if \(K(\mathfrak N_1)=K(\mathfrak N_2)\). The question of the possibility of representing an operator \(A\) in the form of a product of operators from the set \(\mathfrak N\) reduces to the question of the equivalence of the sets \(\mathfrak N\) and \(\mathfrak N \cup \{A\}\).

We describe a certain class of criteria for the equivalence of subsets of an arbitrary category \(K\). Let \(K_\alpha\) be some operator category defined on the set \(\Xi_\alpha\). On the set \(\Xi_\alpha\) define a subordination relation \(\stackrel{r}{<}\), satisfying the following axioms: 1) if \(\xi_1 \stackrel{r}{<} \xi_2\) and \(\xi_2 \stackrel{r}{<} \xi_3\), then \(\xi_1 \stackrel{r}{<} \xi_3\); 2) if \(\xi_1 \stackrel{r}{<} \xi_2\), then \(A^\alpha \xi_1 \stackrel{r}{<} A^\alpha \xi_2\) for every operator \(A^\alpha \in K_\alpha\) (provided that the operator \(A^\alpha\) is defined on \(\xi_1\) and \(\xi_2\)). By a quasi-invariant of a set \(\mathfrak M \subset K_\alpha\) we shall mean any element \(\xi \in \Xi_\alpha\) such that, for every \(A^\alpha\) from \(\mathfrak M\), one has \(A^\alpha \xi \stackrel{r}{<} \xi\). The totality of all quasi-invariants of the set \(\mathfrak M\) is denoted by \(S(\mathfrak M)\). Let \(\sigma:K \to K_\alpha\) be a homomorphism of the category \(K\) into the category \(K_\alpha\).

If \(\mathfrak N \subset \widetilde K\) and \(R=(K_\alpha,\stackrel{r}{<},\sigma)\), then by the \(R\)-characteristic of the set \(\mathfrak N\) we shall mean the set \(R(\mathfrak N)=S(\sigma\mathfrak N)\). The sets \(\mathfrak N_1\) and \(\mathfrak N_2 \subset \widetilde K\) are called \(R\)-equivalent if \(R(\mathfrak N_1)=R(\mathfrak N_2)\).

An \(R\)-criterion of equivalence. In order that the sets \(\mathfrak N_1\) and \(\mathfrak N_2 \subset \widetilde K\) be equivalent, it is necessary that they be \(R\)-equivalent.

Thus, to each triple \((K_\alpha,\stackrel{r}{<},\sigma)=R_\beta\) there corresponds an \(R_\beta\)-criterion of equivalence. We shall say that the \(R_1\)-criterion is weaker than the \(R_2\)-criterion (the \(R_2\)-criterion is stronger than the \(R_1\)-criterion) and write \(R_1 \preccurlyeq R_2\), if from the \(R_2\)-equivalence of arbitrary sets \(\mathfrak N_1\) and \(\mathfrak N_2 \subset \widetilde K\) there follows their \(R_1\)-equivalence. The \(R_1\)-criterion and the \(R_2\)-criterion are called equally strong if \(R_1 \preccurlyeq R_2\) and \(R_2 \preccurlyeq R_1\). Identifying equivalent \(R\)-criteria, we obtain a partially ordered set \(\mathfrak R(K)\), whose elements are classes of equally strong \(R\)-criteria of equivalence.

Theorem 4. The partially ordered set \(\mathfrak R(K)\) is a complete lattice.

Theorem 5. The unit element of the lattice \(\mathfrak R(K)\) is the class of sufficient \(R\)-criteria of equivalence.

A set \(H \subset \widetilde K\) is called \(\widetilde K\)-closed if \(\widetilde K \cap K(H) \subset H\). A system \(\mathscr E\) of nonempty \(\widetilde K\)-closed subsets of the set \(\widetilde K\) is called proper if: 1) \(\widetilde K \in \mathscr E\) and 2) together with the sets \(H_\alpha \in \mathscr E\) \((\alpha \in I)\), the system \(\mathscr E\) also contains their intersection \(\bigcap_{\alpha\in I} H_\alpha\), if it is nonempty.

If \(\mathscr E\) is a proper system and \(N \subset \widetilde K\), then by \(\mathscr E N\) we shall denote the smallest set \(H \in \mathscr E\) containing the set \(N\).

Every \(R\)-criterion of equivalence generates a certain partition \(D_R\) of the system of all subsets of the set \(\widetilde K\) into classes of \(R\)-equivalent sets. To each class \(Q_\alpha\) of the partition \(D_R\) we put in correspondence the set
\[ H_\alpha=\bigcup_{N\in Q_\alpha} N. \]
The totality of the sets \(H_\alpha\) corresponding to all possible classes of the partition \(D_R\) will be denoted by \(\mathscr E_R\). Thus, to each \(R\)-criterion of equivalence there corresponds a certain system of sets \(\mathscr E_R\).

Theorem 6. The system of sets \(\mathscr E_R\) corresponding to an \(R\)-criterion of equivalence is proper.

The question arises: does every proper system correspond to some \(R\)-criterion of equivalence?

Let \(\overline S\) be a semigroup with a system of generators \(K\) and a system of defining relations \(\mathfrak A\), consisting of all relations valid in the category \(K\). In what follows we shall regard the category \(K\) as a subset of the semigroup \(\overline S\), and by \(NM\), where \(N,M\subset \overline S\), we shall mean the product of sets in the semigroup \(\overline S\). If \(N\subset K\) and

\(\dot T(N)=\{a:aN\cap K\ne \Lambda,\ a\in K\}\), then by \(F(N)\) we denote the set \(\{a:T(N)a\cap K\ne \Lambda,\ a\in K\}\).

Let \(\mathscr E\) be a regular system and \(H\in\mathscr E\). A set \(H\) is called distinguishable in \(\mathscr E\) if there exist a set \(S\subset \overline S\) and a mapping \(f(a)=P_a\) \((a\in S,\ P_a\subset S)\) satisfying the following conditions:

1°. \(F(H)\subset S\).

2°. If \(ab\in S\) and \(b\in K\), then \(a(bK\cap K)\subset S\).

3°. If \(a\in F(H)\) and \(b\in H\), then \(S\cap H_a\subset P_a\) and \(S\cap P_ba\subset P_a\).

4°. If \(aN\subset P_a\) \((a\in S,\ N\subset \widetilde K)\), then \(a\mathscr E N\subset P_a\) and \(aF(\mathscr E N)\subset S\).

5°. If \(a\in P_b\), then \(P_a\subset P_b\).

6°. If \(a\in P_b,\ c\in K\) and \(ac,\ bc\in S\), then \(ac\in P_{bc}\).

7°. If \(a\in H\), then \(\widetilde K\cap P_a\subset H\).

Otherwise the set \(H\) is called indistinguishable in \(\mathscr E\).

Theorem 7. In order that there exist an \(R\)-equivalence criterion corresponding to the regular system \(\mathscr E\), it is necessary and sufficient that every set \(H\in\mathscr E\) be distinguishable in \(\mathscr E\).

Theorem 8. If there exist elements \(a,b\in K\) and a set \(N\subset \widetilde K\) such that \(ab\in H\), \(aN\subset Ha\cap K\), and \(a\mathscr E Nb\cap(\widetilde K/H)\ne \Lambda\), then the set \(H\in\mathscr E\) is indistinguishable in \(\mathscr E\).

The following theorem gives sufficient conditions for distinguishability under the assumption that the category \(K\) is a semigroup.

Theorem 9. If for arbitrary \(a\in K\) and \(N\subset \widetilde K\), from \(aN\subset Ha\cup a\) it follows that \(a\mathscr E N\subset Ha\cup a\), then the set \(H\in\mathscr E\) is distinguishable in \(\mathscr E\).

Let us apply the results obtained to groups. Let \(K\) be an arbitrary group, and let \(H\) be an arbitrary \(\widetilde K\)-closed set of nonidentity elements of the group \(K\).

Theorem 10. In order that the set \(H\) be distinguishable in \(\mathscr E_0=\{\widetilde K,H\}\), it is necessary and sufficient that for every \(a\in K\) the relation \(aH\subset Ha\) hold.

Theorem 11. In order that the subgroup \(K_1\) be a normal divisor of the group \(K\), it is necessary and sufficient that there exist an \(R_0\)-equivalence criterion corresponding to the system \(\mathscr E_0=\{\widetilde K,K_1\cap \widetilde K\}\).

Theorem 12. In order that to every regular system there correspond some \(R\)-equivalence criterion, it is necessary and sufficient that the group \(K\) be abelian or Hamiltonian.

As an illustration, let us give one example. Let the memory \(\Omega=\{x_1,x_2,x_3\}\) consist of three cells, each of which can be in two states (0 and 1). \(\Xi=\{f(x)\}\) is the set of all states of the memory \(\Omega\). The category \(K\) consists of all invertible operators \(A:\Xi\to\Xi\); \(\mathfrak N\) is the set of all \((2,1)\)-operators from \(K\). We construct an \(\widetilde R\)-criterion recognizing the equivalence of the sets \(\mathfrak N\) and \(\mathfrak N\cup\{A\}\), where \(A\) is an arbitrary operator from \(K\). Let \(E=\{e\}\) be the set whose elements are all four-element subsets \(e\subset\Xi\), and let \(\Xi_1=\{\xi\}\) be the set of all subsets \(\xi\subset E\) consisting of 14 elements of the set \(E\). The operator category \(K_1\) consists of all invertible operators \(A_1:\Xi_1\to\Xi_1\).

The subordination relation \(\overset{1}{<}\) in \(\Xi_1\) is defined as follows: \(\xi_1\overset{1}{<}\xi_2\) if and only if \(\xi_1=\xi_2\). As the homomorphism \(\sigma_1:K\to K_1\) we take the mapping which assigns to an operator \(A\in K\) the corresponding operator \(A^1\in K_1\) induced by the operator \(A\).

If \(\widetilde R=(K_1,\overset{1}{<},\sigma_1)\), then the \(\widetilde R\)-criterion completely solves the question of the equivalence of the sets \(\mathfrak N\) and \(\mathfrak N\cup\{A\}\): in order that the sets \(\mathfrak N\) and \(\mathfrak N\cup\{A\}\) be equivalent, it is necessary and sufficient that they be \(\widetilde R\)-equivalent. The \(\widetilde R\)-equivalence of \(\mathfrak N\) and \(\mathfrak N\cup\{A\}\) is equivalent to the relation \(\widetilde R(\mathfrak N)\subset \widetilde R(A)\). In the present case the \(\widetilde R\)-characteristic of the set

\(\mathfrak N\) consists of a single element \(\xi_0=\{e_1,\ldots,e_{14}\}\), where the sets \(e_i\subset \Xi\) \((i=1,\ldots,14)\) are characterized by the following property: \(\sum_{f\in e_i} f(x_j)=r_j^i\), \((i=1,\ldots,14;\ j=1,2,3)\), is an even number. Thus, in order that an operator \(A\in K\) be representable as a product of \((2,1)\)-operators over \(\Omega\), it is necessary and sufficient that \(\xi_0\in \widetilde R(A)\).

Received
26 XI 1961

References

1. A. G. Kurosh, A. Kh. Livshits, E. G. Shulgeifer, UMN, 15, no. 6 (96) (1960).