CYBERNETICS AND CONTROL THEORY

V. A. ZYATITSKII

ON THE MINIMIZATION OF THE TEMPORAL MEASURE OF A MULTILAYER CYCLE

(Presented by Academician A. I. Berg on 23 V 1963)

1. Consider a certain polynomial

\[ \varphi=\sum_{j=0}^{n} b_{i_j}\prod_{l=0}^{j} a_{i_l}, \]

where \(p_{i_j}=(a_{i_j}, b_{i_j})\) are prescribed pairs of numbers, \(a>1\), \(b>0\), and \(i_j\in\{1,2,\ldots,n\}\); \(j=1,2,\ldots,n\); \(i_{j_1}\ne i_{j_2}\leftrightarrow j_1\ne j_2\). It is required to find such an arrangement of the pairs in \(\varphi\) that \(\varphi_0=\min\{\varphi\}\).

Theorem 1. The substitution

\[ \begin{pmatrix} 0, & 1, & \ldots, & n\\ i_0, & i_1, & \ldots, & i_n \end{pmatrix}, \]

defined by the chain of inequalities

\[ b_{i_n}+\frac{b_{i_n}}{a_{i_n}-1} \leq b_{i_{n-1}}+\frac{b_{i_{n-1}}}{a_{i_{n-1}}-1} \leq \cdots \leq b_{i_0}+\frac{b_{i_0}}{a_{i_0}-1}, \tag{1} \]

gives \(\min \varphi\,(p_{i_0}, p_{i_1}, \ldots, p_{i_n})\).

Lemma. If a transposition in \((i_0,i_1,\ldots,i_n)\) causes an inversion in (1), then the corresponding value of \(\varphi\) is not less than the preceding one.

Theorem 1 follows from the lemma.

2. For what follows we give some definitions.

By a cycle we shall mean a closed sequence of commands (or operators) \((^2)\). The predetermined number of realizations of a cycle will be called its multiplicity \((^2)\).

If \(C_1\) and \(C_2\) are two cycles, with \(C_2\subset C_1\), then the cycle \(C_1\) will be called the envelope of \(C_2\) \((^1)\), and \(C_1-C_2=C_1-C_1\cap C_2\) the envelope of the cycle \(C_2\) with respect to \(C_1\) \((^1)\). Otherwise, \(C_2\) is a two-layer cycle. If a sequence of cycles \(C_0,C_1,\ldots,C_n\) is given, with \(C_n\subset C_{n-1}\subset\cdots\subset C_1\subset C_0=C\), then each cycle \(C_{j-1}\) will be called the envelope of \(C_j\); \(C_{j-1}-C_j=C_{j-1}-C_j\cap C_{j-1}\) is the \((j-1)\)-st envelope of the cycle \(C\), and, correspondingly, the cycle \(C\) is an \((n+1)\)-layer cycle. Otherwise, \(C\) is a multilayer cycle.

Let some multilayer cycle \(C\) process a partially commutative direct product of tables \(F=f_1\times f_2\times\cdots\times f_m\), with the number of components of the tables equal to the multiplicities of the corresponding closures.

The factors in \(F\) are ordered, guided by the functional subordination of the parameters. In this case groups are distinguished in \(F\) within which the factors are commutative.

We shall be interested in finding such a structure in \(F\) that, all other conditions being equal, the realization time of the corresponding cycle \(C\) is minimal.

It is clear that under a permutation of tables in \(F\), the corresponding layers in \(C\) (we shall call them commutative) likewise, being modified, are moved.

Denote by \(t_{i_j}\) the time required for a single realization of the envelope of the \(i_j\)-th cycle, and by \(T(C)\) the time of a complete realization of the cycle \(C\), i.e., its temporal measure.

1. Idealized model 1. Suppose that the \(t_{ij}\) corresponding to commutative layers are invariant with respect to the arrangement of the layers in \(C\). Then the following holds.

Theorem 2. If the \(t\) of the commutative layers of the multilayer cycle \(C(F)\) are invariant with respect to the arrangement of the layers in \(C\), then, all other conditions being equal, minimization of \(T(C)\) is ensured by ordering the commutative layers in the commutativity groups \(F\) according to (1), where \(a_{ij} = k(\lfloor \_i \rfloor)\) is the multiplicity of the cycles of the corresponding commutative layers of group \(i\); \(b_{ij} = f_{ij}\).

Otherwise, (1) can be rewritten in the form

\[ t_{i_n} + \frac{t_{i_n}}{k(\lfloor \_ {i_n}\rfloor)-1} \leq t_{i_{n-1}} + \frac{t_{i_{n-1}}}{k(\lfloor \_ {i_{n-1}}\rfloor)-1} \leq \cdots \leq t_{i_0} + \frac{t_{i_0}}{k(\lfloor \_ {i_0}\rfloor)-1}. \tag{1*} \]

The proof follows from Theorem 1.

1. Idealized model 2. Let \(t_k = \delta t_{ij} + \Delta_k\), where \(\delta t_{ij} > 0\), \(\Delta_k \geq 0\) are fixed quantities, with \(\Delta_k\) invariant with respect to the ordinal number of the layer \(k=i_j\), and \(\delta t_{ij}\) invariant for the fixed layer \(i\).

Theorem 3. If, for (1), \(a_{i_n} \leq a_{i_{n_1}} \leq \cdots \leq a_{i_0}\), then, all other conditions being equal, (1) determines \(\min T(C)\), where \(b_{ij}=\delta t_{ij}\), \(a_{ij}=k(\lfloor \_i\rfloor)\).

1. Remark. In order for Theorems 2 and 3 to be applicable in constructing real multilayer cycles, it is necessary that either \(T(O_{ij}) \leq t_{i_{j-1}}\), or \(T(O_{ij}) \simeq T(O_{i_{j-1}})\).

2. If it is assumed that the changes in \(t_{ij}\) under permutation of layers are so small that relation (1) is not violated, then one can see that the limits of applicability of Theorems 2 and 3 are widened.

3. Thus, Theorems 2 and 3 indicate, in the models under consideration, in what order the commutative layers of the multilayer cycle \(C\) should be arranged so that the time measure \(T(C)\) is minimal.

Received
15 V 1963

CITED LITERATURE

1. Yu. I. Yanov, in: Problems of Cybernetics, 1, Moscow, 1958.
2. V. A. Zatitskii, Proceedings of the Seminar “Modern Digital Automation and Computer Technology”, 2, Moscow, 1962.