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MATHEMATICS
Yu. I. SMIRNOV
COMPUTATION OF THE MATHEMATICAL EXPECTATION OF A QUASI-ADDITIVE FUNCTION OF A PATH ON A GRAPH
(Presented by Academician A. A. Dorodnitsyn, 3 VII 1963)
We shall consider a multigraph \(G\) satisfying conditions (1) and (2) of paper \((^1)\). Let all inputs of each vertex \(\alpha\) of the graph \(G\) be numbered by the numbers from \(1\) to \(i_\alpha\), and all outputs by the numbers from \(1\) to \(k_\alpha\). By \({}^{i}a_k\) we shall denote the pair: the \(i\)-th input and the \(k\)-th output of the vertex \(\alpha\). A sequence
\[ {}^{i}s_{kn} = {}^{i}\alpha_{k_1}^{j_1}\, {}^{i_2}\alpha_{k_2}^{j_2}\ldots {}^{i_n}\alpha_{k_n}^{j_n} \]
such that, for \(l=1,2,\ldots,n-1\), \(\bigl(\alpha_{k_l}^{j_l}\,{}^{i_{l+1}}\alpha^{j_{l+1}}\bigr)\) is an edge of the graph \(G\), will be called a path of the graph \(G\). If
\[ {}^{i}s_k^{1} = {}^{i}\alpha_{k_1}^{j_1}\, {}^{i_2}\alpha_{k_2}^{j_2}\ldots {}^{i_n}\alpha_k^{j_n}, \qquad {}^{l}s_m^{2} = {}^{l}\alpha_{m_1}^{q_1}\, {}^{l_2}\alpha_{m_2}^{q_2}\ldots {}^{l_2}\alpha_m^{q_2} \]
and \(\bigl(\alpha_k^{j_n}\,{}^{l}\alpha^{q_1}\bigr)\) is an edge of the graph \(G\), then we shall say that the path
\[ {}^{i}s_m = {}^{i}\alpha_{k_1}^{j_1}\, {}^{i_2}\alpha_{k_2}^{j_2}\ldots {}^{i_n}\alpha_n^{j_n}\, {}^{l}\alpha_{m_1}^{q_1}\, {}^{l_2}\alpha_{m_2}^{q_2}\ldots {}^{l_2}\alpha_m^{q_2} \]
is composed of the paths \({}^{i}s_k^{1}\) and \({}^{l}s_m^{2}\), and we shall write
\({}^{i}s_m = {}^{i}s_k^{1}\,{}^{l}s_m^{2}\). A function \(f(s)\) of a path of the graph will be called quasi-additive if there exists a path function \(\varphi(s)\) such that, for any \(s^1\) and \(s^2\) such that \(s=s^1s^2\), the conditions
\[ \varphi(s)=\varphi(s^1)\cdot\varphi(s^2); \tag{1} \]
\[ f(s)=f(s^1)+\varphi(s^1)f(s^2). \tag{2} \]
are satisfied.
Fig. 1
Let, for each pair \({}^{i}a_k\), a number \(p({}^{i}a_k)\ge 0\) be given—the probability of exiting through the \(k\)-th output of the vertex \(\alpha\) under the condition that one entered through the \(i\)-th input of the vertex \(\alpha\). Obviously,
\[ \sum_{k=1}^{k_\alpha} p({}^{i}a_k)=1. \]
Define the probability \(p({}^{i_1}s_{kn})\) of the path
\[ {}^{i_1}s_{kn} = {}^{i_1}\alpha_{k_1}^{j_1}\, {}^{i_2}\alpha_{k_2}^{j_2}\ldots {}^{i_n}\alpha_{k_n}^{j_n}, \]
by putting
\[ p({}^{i_1}s_{kn})=\prod_{t=1}^{n} p({}^{i_t}\alpha_{k_t}^{j_t}). \]
Let the graph \(G\) have an input and an output (see \((^1)\)). We set ourselves the task of computing
\[ Mf(\bar{s})=\sum_{\bar{s}} p(\bar{s})f(\bar{s}). \]
Here \(\bar{s}\), as in paper \((^2)\), denotes a path going from the input to the output. Let \({}^{i}\alpha_{k_1}\) be a loop at the vertex \(\alpha\). Consider the paths
\[ {}^{i}_{-1}s_k = {}^{i}\alpha_k,\qquad {}^{i}_{0}s_k = {}^{i}\alpha_{k_1}\,{}^{i_1}\alpha_k,\qquad {}^{i}_{1}s_k = {}^{i}\alpha_{k_1}\,{}^{i_1}\alpha_{k_1}\,{}^{i_1}\alpha_k,\qquad {}^{i}_{2}s_k = {}^{i}\alpha_{k_1}\,{}^{i_1}\alpha_{k_1}\,{}^{i_1}\alpha_{k_1}\,{}^{i_1}\alpha_k,\ldots \]
\[ \ldots,\quad {}^{i}_{n}s_k = {}^{i}\alpha_{k_1}\, \underbrace{{}^{i_1}\alpha_{k_1}\,{}^{i_1}\alpha_{k_1}\ldots{}^{i_1}\alpha_{k_1}}_{n}\, {}^{i_1}\alpha_k. \]
Compute \(\varphi({}^{i}_{n}s_k)\). From (1) we have
\[ \varphi({}^{i}_{n}s_k) = \varphi({}^{i}\alpha_{k_1})\, \varphi^{\,n}({}^{i_1}\alpha_{k_1})\, \varphi({}^{i_1}\alpha_k) \quad \text{for } n\ge 0. \tag{3} \]
Compute \(f({}^{i}_{n}s_k)\). Introduce the notation
\[ {}^{i}_{n}\bar{s}_k = {}^{i}\alpha_{k_1}\, \underbrace{{}^{i_1}\alpha_{k_1}\ldots{}^{i_1}\alpha_{k_1}}_{n}, \qquad n\ge 0. \]
Obviously,
\[ f({}^{i}_{0}\bar{s}_k)=f({}^{i}\alpha_{k_1}),\qquad f({}^{i}_{1}\bar{s}_k) = f({}^{i}\alpha_{k_1}) + \varphi({}^{i}\alpha_{k_1})f({}^{i_1}\alpha_{k_1}). \]
From (2) and (3),
\[ f({}^{i}_{n}\bar{s}_{k_1}) = f({}^{i}\alpha_{k_1}) + \varphi({}^{i}\alpha_{k_1}) \left[ \sum_{t=1}^{n}\varphi^{\,t-1}({}^{i_1}\alpha_{k_1}) \right] f({}^{i_1}\alpha_{k_1}) \quad \text{for } n\ge 1. \]
Since \({}_n^i s_k = {}_n^i s_{k_1}\,{}^i\alpha_k\), from the quasiadditivity of \(f(s)\) we obtain
\[ f({}_n^i s_k)=f({}^i\alpha_{k_1})+\varphi({}^i\alpha_{k_1}) \left[\sum_{t=1}^{n}\varphi^{t-1}({}^{i_1}\alpha_{k_1})\right] f({}^{i_1}\alpha_{k_1}) +\varphi({}^i\alpha_{k_1})\varphi^n({}^{i_1}\alpha_{k_1})f({}^{i_1}\alpha_k). \tag{4} \]
If the element \(E-\varphi({}^{i_1}\alpha_{k_1})\) has an inverse, then
\[ f({}_n^i s_k)=f({}^i\alpha_{k_1}) +\varphi({}^i\alpha_{k_1}) \frac{E-\varphi^n({}^{i_1}\alpha_{k_1})}{E-\varphi({}^{i_1}\alpha_{k_1})} f({}^{i_1}\alpha_{k_1}) +\varphi({}^i\alpha_{k_1})\varphi^n({}^{i_1}\alpha_{k_1})f({}^{i_1}\alpha_k). \]
Lemma.
\[ \sum_{n=-1}^{\infty} p({}^l s_m^1\,{}_n^i s_k\,{}^p s_q^3)\, f({}^l s_m^1\,{}_n^i s_k\,{}^p s_q^3) = p({}^l s_m^1\,{}^i\widetilde{s}_k\,{}^p s_q^3)\, f({}^l s_m^1\,{}^i\widetilde{s}_k\,{}^p s_q^3), \]
where
\[ p({}^i\widetilde{s}_k)=p({}^i\alpha_k)+\sum_{n=0}^{\infty} p({}^i\alpha_{k_1})p^n({}^{i_1}\alpha_{k_1})p({}^{i_1}\alpha_k), \]
\[ \varphi({}^i\widetilde{s}_k)= \frac{\displaystyle\sum_{n=-1}^{\infty}p({}_n^i s_k)\varphi({}_n^i s_k)} {p({}^i\widetilde{s}_k)},\qquad f({}^i\widetilde{s}_k)= \frac{\displaystyle\sum_{n=-1}^{\infty}p({}_n^i s_k)f({}_n^i s_k)} {p({}^i\widetilde{s}_k)}. \]
Proof. By virtue of the definition of \(p(s)\) and the quasiadditivity of \(f(s)\), we have
\[ \sum_{n=-1}^{\infty} p({}^l s_m^1\,{}_n^i s_k\,{}^p s_q^3) f({}^l s_m^1\,{}_n^i s_k\,{}^p s_q^3)= \]
\[ =\sum_{n=-1}^{\infty} p({}^l s_m^1)p({}_n^i s_k)p({}^p s_q^3) \left[f({}^l s_m^1\,{}_n^i s_k)+ \varphi({}^l s_m^1\,{}_n^i s_k)f({}^p s_q^3)\right]= \]
\[ =p({}^l s_m^1)p({}^p s_q^3) \sum_{n=-1}^{\infty}p({}_n^i s_k) \left[f({}^l s_m^1)+\varphi({}^l s_m^1)f({}_n^i s_k) +\varphi({}^l s_m^1)\varphi({}_n^i s_n)f({}^p s_q^3)\right]= \]
\[ =p({}^l s_m^1)p({}^i\widetilde{s}_k)p({}^p s_q^3) \left[ f({}^l s_m^1)+ \varphi({}^l s_m^1) \frac{\displaystyle\sum_{n=-1}^{\infty}p({}_n^i s_k)f({}_n^i s_k)} {p({}^i\widetilde{s}_k)} +\right. \]
\[ \left. +\varphi({}^l s_m^1) \frac{\displaystyle\sum_{n=-1}^{\infty}p({}_n^i s_k)\varphi({}_n^i s_k)} {p({}^i\widetilde{s}_k)} f({}^p s_q^3) \right]= \]
\[ =p({}^l s_m^1\,{}^i\widetilde{s}_k\,{}^p s_q^3) \left[f({}^l s_m^1)+\varphi({}^l s_m^1)f({}^i\widetilde{s}_k) +\varphi({}^l s_m^1)\varphi({}^i\widetilde{s}_k)f({}^p s_q^3)\right]= \]
\[ =p({}^l s_m^1\,{}^i\widetilde{s}_k\,{}^p s_q^3) f({}^l s_m^1\,{}^i\widetilde{s}_k\,{}^p s_q^3), \]
which was required to prove.
Remark.
\[ p({}^i\widetilde{s}_k)=p({}^i\alpha_k)+ \frac{p({}^i\alpha_{k_1})p({}^{i_1}\alpha_k)} {1-p({}^{i_1}\alpha_{k_1})}. \]
If the element \(E-p({}^{i_1}\alpha_{k_1})\varphi({}^{i_1}\alpha_{k_1})\) has an inverse, then
\[ \varphi({}^i\widetilde{s}_k)= \frac{ p({}^i\alpha_k)\varphi({}^i\alpha_k) +p({}^i\alpha_{k_1})p({}^{i_1}\alpha_k)\varphi({}^i\alpha_{k_1}) \left(E-p({}^{i_1}\alpha_{k_1})\varphi({}^{i_1}\alpha_{k_1})\right)^{-1} \varphi({}^{i_1}\alpha_k) } {p({}^i\widetilde{s}_k)}. \]
If the element \(E-\varphi({}^{i_1}\alpha_{k_1})\) also has an inverse, then
\[ \begin{aligned} f({}^{i}\widetilde{s}_k)=\Bigg[& p({}^{i}\alpha_k)f({}^{i}\alpha_k)+p({}^{i}\alpha_{k_1})p({}^{i}\alpha_k) \Bigg\{ \frac{f({}^{i}\alpha_{k_1})}{1-p({}^{i}\alpha_{k_1})}+{}\\ &+\varphi({}^{i}\alpha_{k_1})\, \frac{E}{E-p({}^{i}\alpha_{k_1})\varphi({}^{i}\alpha_{k_1})}\, f({}^{i}\alpha_k) +\frac{2p({}^{i_1}\alpha_{k_1})}{1-p({}^{i_1}\alpha_{k_1})}\, \varphi({}^{i}\alpha_{k_1})f({}^{i}\alpha_{k_1})-{}\\ &-\varphi({}^{i}\alpha_{k_1}) \bigl(E-\varphi({}^{i_1}\alpha_{k_1})\bigr)^{-1} \bigl(E-p({}^{i_1}\alpha_{k_1})\varphi({}^{i_1}\alpha_{k_1})\bigr)^{-1} f({}^{i_1}\alpha_{k_1}) \Bigg\}\Bigg]/p({}^{i}\widetilde{s}_k). \end{aligned} \]
Transformation 1 of the graph \(G\). Construct, starting from the graph \(G\), a graph \(G'\) as follows: in the graph \(G\) remove the loop \({}^{i_1}\alpha_{k_1}\). We shall mark all elements of the graph \(G'\) and the functions specified on them by a prime. Put
\[ p'({}^{i}\beta'_k)= \begin{cases} p({}^{i}\beta_k), & \text{if } \beta\ne\alpha,\\ p({}^{i}\widetilde{s}_k), & \text{if } \beta=\alpha; \end{cases} \qquad \varphi'({}^{i}\beta'_k)= \begin{cases} \varphi({}^{i}\beta_k), & \text{if } \beta\ne\alpha,\\ \varphi({}^{i}\widetilde{s}_k), & \text{if } \beta=\alpha; \end{cases} \]
\[ f'({}^{i}\beta'_k)= \begin{cases} f({}^{i}\beta_k), & \text{if } \beta\ne\alpha,\\ f({}^{i}\widetilde{s}_k), & \text{if } \beta=\alpha. \end{cases} \]
A path \({}^{q_1}s'_{\nu_t}\) of the graph \(G'\) can be represented uniquely in the form
\[ {}^{q_1}s'_{\nu_t} = {}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}\alpha'_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\,{}^{m_2}\alpha'_{k_2}\ldots {}^{q_{t-1}}s^{t-1}_{\nu_{t-1}}\,{}^{m_{t-1}}\alpha'_{k_{t-1}}\, {}^{q_t}s^{t}_{\nu_t} \]
so that the paths \({}^{q_j}s^{j}_{\nu_j}\) \((j=1,2,\ldots,t)\) do not contain the vertex \(\alpha'\) (in this case some of the paths \({}^{q_j}s^{j}_{\nu_j}\) may be empty). To the path \({}^{q_1}s'_{\nu_t}\) of the graph \(G'\) let us put in correspondence all paths of the graph \(G\) having the form
\[ {}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}_{n_1}s_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\ldots {}^{m_{t-1}}_{n_{t-1}}s_{k_{t-1}}\, {}^{q_t}s^{t}_{\nu_t}, \]
\[ j=1,2,\ldots,t-1; \qquad n_j=-1,0,1,2,\ldots . \]
We denote this set of paths by \(s({}^{q_1}s'_{\nu_t})\). We shall compute
\[ \sum_{s({}^{q_1}s'_{\nu_t})} p(s)f(s)= \]
\[ =\sum_{j=1}^{t-1}\sum_{n_j=-1}^{\infty} p\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}_{n_1}s_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\ldots {}^{m_{t-1}}_{n_{t-1}}s_{k_{t-1}}\, {}^{q_t}s^{t}_{\nu_t}\bigr) f\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}_{n_1}s_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\ldots {}^{m_{t-1}}_{n_{t-1}}s_{k_{t-1}}\, {}^{q_t}s^{t}_{\nu_t}\bigr). \]
Theorem.
\[ \sum_{s({}^{q_1}s'_{\nu_t})} p(s)f(s) = p({}^{q_1}s'_{\nu_t})f({}^{q_1}s'_{\nu_t}). \]
Proof. By the lemma,
\[ \sum_{n_1=-1}^{\infty} p\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}_{n_1}s_{k_1}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr) f\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}_{n_1}s_{k_1}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr) = \]
\[ = p\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}\widetilde{s}_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\,{}^{m_2}_{n_2}s_{k_2}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr) f\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}\widetilde{s}_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr). \]
Suppose
\[ \sum_{s({}^{q_1}s'_{\nu_t})} p(s)f(s) = \sum_{j=r}^{t-1}\sum_{n_j=-1}^{\infty} p\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}\widetilde{s}_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\,{}^{m_2}\widetilde{s}_{k_2}\ldots \]
\[ \ldots\, {}^{q_{r-1}}s^{r-1}_{\nu_{r-1}}\, {}^{m_{r-1}}\widetilde{s}_{k_{r-1}}\, {}^{q_r}s^{r}_{\nu_r}\,{}^{m_r}_{n_r}s_{k_r}\, {}^{q_{r+1}}s^{r+1}_{\nu_{r+1}}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr) f\bigl({}^{q_1}s^{1}_{\nu_1}\,{}^{m_1}\widetilde{s}_{k_1}\, {}^{q_2}s^{2}_{\nu_2}\,{}^{m_2}\widetilde{s}_{k_2}\ldots \]
\[ \ldots\, {}^{q_{r-1}}s^{r-1}_{\nu_{r-1}}\, {}^{m_{r-1}}\widetilde{s}_{k_{r-1}}\, {}^{q_r}s^{r}_{\nu_r}\,{}^{m_r}_{n_r}s_{k_r}\, {}^{q_{r+1}}s^{r+1}_{\nu_{r+1}}\ldots{}^{q_t}s^{t}_{\nu_t}\bigr). \]
Since, by the lemma,
\[ \sum_{n_r=-1}^{\infty} p\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\,{}^{q_2}s_{\nu_2}^{2}\ldots{}^{m_{r-1}}\widetilde{s}_{k_{r-1}}\,{}^{q_r}s_{\nu_r}^{r}\,{}^{m_r}_{n_r}s_{k_r}\ldots{}^{q_t}s_{\nu_t}^{t}\right) f\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\,{}^{q_2}s_{\nu_2}^{2}\ldots{}^{m_{r-1}}\widetilde{s}_{k_{r-1}}\,{}^{q_r}s_{\nu_r}^{r}\,{}^{m_r}_{n_r}s_{k_r}\ldots{}^{q_t}s_{\nu_t}^{t}\right) \]
\[ = p\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\,{}^{q_2}s_{\nu_2}^{2}\ldots{}^{q_r}s_{\nu_r}^{r}\,{}^{m_r}\widetilde{s}_{k_r}\,{}^{q_{r+1}}s_{\nu_{r+1}}^{r+1}\,{}^{m_{r+1}}_{n_{r+1}}s_{k_{r+1}}\ldots{}^{q_t}s_{\nu_t}^{t}\right) \]
\[ \ldots{}^{q_t}s_{\nu_t}^{t}\, f\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\,{}^{q_2}s_{\nu_2}^{2}\ldots{}^{q_r}s_{\nu_r}^{r}\,{}^{m_r}\widetilde{s}_{k_r}\,{}^{q_{r+1}}s_{\nu_{r+1}}^{r+1}\,{}^{m_{r+1}}_{n_{r+1}}s_{k_{r+1}}\ldots{}^{q_t}s_{\nu_t}^{t}\right), \]
we have
\[ \sum_{s({}^{q_1}s_{\nu_t}^{t})} p(s)f(s) = \sum_{j=r+1}^{t-1}\sum_{n_j=-1}^{\infty} p\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\ldots{}^{m_r}\widetilde{s}_{k_r}\,{}^{q_{r+1}}s_{\nu_{r+1}}^{r+1}\,{}^{m_{r+1}}_{n_{r+1}}s_{k_{r+1}}\ldots\right) \]
\[ \ldots {}^{q_t}s_{\nu_t}^{t} f\left({}^{q_1}s_{\nu_1}^{1}\,{}^{m_1}\widetilde{s}_{k_1}\ldots{}^{m_r}\widetilde{s}_{k_r}\,{}^{q_{r+1}}s_{\nu_{r+1}}^{r+1}\,{}^{m_{r+1}}_{n_{r+1}}s_{k_{r+1}}\ldots{}^{q_t}s_{\nu_t}^{t}\right). \]
Hence, by induction, we obtain
\[ \sum_{s({}^{q_1}s_{\nu_t}^{t})} p(s)f(s) = p\left({}^{q_1}s_{\nu_t}^{t}\right)f\left({}^{q_1}s_{\nu_t}^{t}\right), \]
which was required to be proved.
It follows from the theorem that transformation 1 of the graph does not change \(Mf(\bar{s})\).
Transformation 2 of the graph \(G\). Let \(a^1\) and \(a^2\) be adjacent vertices of the graph \(G\); \(l_1^1,l_2^1,\ldots,l_{n_1}^1\) are the inputs of vertex \(a^1\); \(l_1^2,l_2^2,\ldots,l_{n_2}^2\) are the inputs of vertex \(a^2\); \(k_1^1,k_2^1,\ldots,k_{m_1}^1\) are the outputs of vertex \(a^1\); \(k_1^2,k_2^2,\ldots,k_{m_2}^2\) are the outputs of vertex \(a^2\); \(k_1^1,k_2^1,\ldots,k_{i_1}^1\) are those outputs of \(a^1\) which lead to the inputs of vertex \(a^2\); \(k_1^2,k_2^2,\ldots,k_{i_2}^2\) are those outputs of \(a^2\) which lead to the inputs of \(a^1\).
Fig. 2
It is easy to see that, in computing \(Mf(\bar{s})\), the pair of vertices \(a^1\) and \(a^2\) may be replaced by a vertex \(a'\) with inputs \(l_1^1,l_2^1,\ldots,l_{n_1}^1,l_1^2,l_2^2,\ldots,l_{n_2}^2\) and outputs \(k_1^1,k_2^1,\ldots,k_{m_1}^1,k_1^2,k_2^2,\ldots,k_{m_2}^2\), putting
\[ p\left({}^{k}a_l'\right)= \begin{cases} p\left({}^{k_m^t}a_{l_q^t}^{t_\nu}\right), & \text{if } k=k_m^t,\ l=l_q^\nu \text{ and } t=\nu,\\ 0, & \text{otherwise;} \end{cases} \]
\[ \varphi\left({}^{k}a_l'\right)= \begin{cases} \varphi\left({}^{k_m^t}a_{l_q^t}^{t_\nu}\right), & \text{if } k=k_m^t,\ l=l_q^\nu \text{ and } t=\nu,\\ 0, & \text{otherwise.} \end{cases} \]
\[ f\left({}^{k}a_l'\right)= \begin{cases} f\left({}^{k_m^t}a_{l_q^t}^{t_\nu}\right), & \text{if } k=k_m^t,\ l=l_q^\nu \text{ and } t=\nu,\\ 0, & \text{otherwise.} \end{cases} \]
Transformation 2, without changing \(Mf(\bar{s})\), reduces the number of vertices of the graph; transformation 1, without changing \(Mf(\bar{s})\), removes loops at vertices of the graph. Thus, by these transformations the graph can be reduced to a single vertex \(\bar{a}\) with one input \(k_1\) and one output \(l_1\). Then \(Mf(\bar{s})=f({}^{k_1}a_{l_1})\).
Received
28 VI 1963
CITED LITERATURE
- A. I. Ershov, Problems of Cybernetics, vol. 3, Moscow, 1960, p. 5.
- Yu. I. Smirnov, Zh. Vychisl. Mat. i Mat. Fiz., 3, No. 3, 539 (1963).