V. V. VISHIN

IDENTICAL TRANSFORMATIONS IN FOUR-VALUED LOGIC

(Presented by Academician P. S. Novikov on 8 I 1963)

The problem of identical transformations of formulas of \(k\)-valued logics reduces to finding complete systems of identities for closed classes of functions having a finite basis. R. C. Lyndon \((^1)\) showed that for every closed class of two-valued logic there exists a finite complete system of identities. He also \((^2)\) obtained a negative result, consisting in the fact that for \(k = 7\) a closed class was constructed that has no finite system of identities. The question naturally arises: what is the least value of closed classes having no finite complete system of identities? In the present paper an example is constructed of a 4-valued closed class with this property, and therefore, for a complete solution of the question, it remains only to consider the case \(k = 3\).

Let us consider the closed class of functions generated by the function \(xy\), defined by Table 1. We shall denote this class by \(M\). The function \((xx)\), identically equal to zero, we shall denote by \(0\).

Table 1

The notation \(f = g\) will mean that the functions \(f\) and \(g\) are identically equal (i.e., have identical tables). Such notations we shall call identities, and in this case the formulas \(f\) and \(g\) will be called equivalent.

Lemma 1. In the class \(M\) the following identities hold:

\[ A_1:\ (x0)=0. \]

\[ A_2:\ (0x)=0. \]

\[ A_3:\ ((x_1x_2)x_2)=(x_1x_2). \]

\[ A_4:\ (x_1(x_2(x_3x_4)))=0. \]

\[ B_n:\ ((\ldots(((x_1x_2)x_3)x_4)\ldots x_n)x_2) = ((\ldots(((x_1x_3)x_2)x_4)\ldots x_n)x_3). \]

\[ C_{n,k}:\ ((((\ldots(((x_1x_2)x_3)x_4)\ldots x_k)x_{k+1})\ldots x_n)x_2) \]
\[ = ((((\ldots(((x_1x_2)x_3)\ldots x_{k+1})x_k)\ldots x_n)x_2) \quad (k=4,5,\ldots,n). \]

\[ D_n:\ ((\ldots((x_1x_2)x_3)\ldots x_n)x_1)=0. \]

The validity of this lemma is easily established by direct verification.

Formulas in which all left parentheses stand to the left of all occurrences of variables we shall call formulas of left association and denote by \(\varphi\); the remaining formulas we shall call formulas of right association and denote by \(\overline{\varphi}\).

Lemma 2. If \(\widehat{\varphi}_1 \ne 0\) and \(\overline{\varphi}_2 \ne 0\), then in \(M\) it cannot be the case that \(\widehat{\varphi}_1=\overline{\varphi}_2\).

Proof. 1. We shall show that if \(\widehat{\varphi}_1=\overline{\varphi}_2\), then all variables occurring in the formula \(\overline{\varphi}_2\) occur in \(\widehat{\varphi}_1\). Suppose this is not so. Let \(\widehat{\varphi}_1\) not depend on some \(x_k\), while \(\overline{\varphi}_2\) does depend on it; then set \(x_k=0\). By virtue of \(A_1\) and \(A_2\), \(\overline{\varphi}_2=0\) for any values of the remaining variables, i.e. \(\widehat{\varphi}_1=0\), which contradicts the assumption.

1. Suppose that \(\hat{\varphi}_1=\overline{\varphi}_2\). Let \(\hat{\varphi}_1\) have the form
   \[ ((\cdots((x_{i_1}x_{i_2})x_{i_3})\cdots)x_{i_k}). \]
   Since \(\hat{\varphi}_1\ne 0\), by virtue of \(D_n\) and \(A_2\), \(i_1\ne i_m\) \((m=2,3,\ldots,k)\). Let \(\overline{\varphi}_2\) have the form \((x_{j_1}\ldots(y_1y_2)\ldots)\), where \(y_2\) is either a variable \(x_{i_r}\) \((1\le r\le k)\), or some formula. Put \(x_{i_1}=3\), and the remaining \(x_s=1\); then \(\hat{\varphi}_1=3\), and since \(y_3\ne 2\) and, consequently, \((y_1y_2)\ne 1\) and \((y_1y_2)\ne 2\), we have \(\overline{\varphi}_2=0\), which contradicts the supposition.

Let a system of identities \(\varphi_1=\psi_1,\ \varphi_2=\psi_2,\ldots,\varphi_m=\psi_m\) be given. By an identical transformation of a formula with respect to this system we shall mean the replacement of one of its subformulas \(\varphi_i'\) by the subformula \(\psi_i'\), or conversely, where the identity \(\varphi_i'=\psi_i'\) is obtained from \(\varphi_i=\psi_i\) by substituting, in place of variables, arbitrary functions of the class under consideration or other variables. An identity \(g_1=g_2\) is derivable from the given system of identities \(\varphi_1=\psi_1,\ldots,\varphi_m=\psi_m\) if, by identical transformations with respect to this system, \(g_1\) can be obtained from \(g_2\), or conversely, \(g_2\) from \(g_1\). A system of identities is called complete if every identity holding in the class of functions under consideration is derivable from it. Let us single out from \(M\) the subclass \(M'\) of all left-associative functions not identically equal to zero.

Suppose that in \(M\) there exists a complete finite system of identities \(\varphi_1=\psi_1,\ldots,\varphi_m=\psi_m\); then, by Lemma 2, among them there are identities \(\hat{\varphi}_{i_1}=\hat{\psi}_{i_1},\ldots,\hat{\varphi}_{i_k}=\hat{\psi}_{i_k}\). As a consequence of Lemma 2, the system \(\hat{\varphi}_{i_1}=\hat{\psi}_{i_1},\ldots,\hat{\varphi}_{i_k}=\hat{\psi}_{i_k}\) must be complete in \(M'\), since an arbitrary \(\hat{\varphi}=\hat{\psi}\) can be derived only from the system \(\hat{\varphi}_{i_1}=\hat{\psi}_{i_1},\ldots,\hat{\varphi}_{i_k}=\hat{\psi}_{i_k}\).

Thus, if there exists a finite complete system of identities in \(M\), then there also exists a finite complete system of identities in \(M'\). We shall show that this necessary condition is not fulfilled; thereby it will be proved that in \(M\) there is no finite complete system of identities.

The system of identities \(A_3,\ B_n,\ C_{n,k}\) \((k=4,\ldots,n;\ n=1,2,\ldots,m)\) in the class \(M'\) will be denoted by \(E_m\). Since only left-associative formulas enter \(M'\), in writing them we shall omit all parentheses; in such formulas, subformulas of the form \(x_i x_{j_1}\ldots x_{j_k} x_i\), where \(k>0\), and also a single occurrence of a variable, we shall call intervals.

We shall call two intervals nonintersecting if they have no identical variables. If the variables entering an interval stand in increasing order of indices, except for its right end, then the interval will be called normal.

A collection of pairwise nonintersecting normal intervals will be called a canonical form of a formula.

Lemma 3. For every formula there exists at most one canonical formula equivalent to it.

Proof. Let there be two canonical formulas
\[ \varphi_1 \sim y_1^1\ldots y_{m_1}^1 \quad\text{and}\quad \varphi_2 \sim y_1^2\ldots y_{m_2}^2, \]
where \(y_i^1\ldots y_{m_i}^i\) are pairwise nonintersecting normal intervals, \(i=1,2\), and suppose that \(\varphi_1=\varphi_2\).

1) We shall show that \(y_1^1\sim y_1^2\). Indeed, in view of \(D_n\), the intervals \(y_1^1\) and \(y_1^2\) are variables; let, for example, \(y_1^1\sim x_i\). Putting \(x_i=3,\ x_j=1\) for \(j\ne i\), we obtain that \(\varphi_1=3\), and if \(y_1^2\) differs from \(x_i\), then \(\varphi_2=0\), which contradicts the supposition.

2) Suppose that \(y_s^1=y_s^2\) for \(s=1,\ldots,k\), \(k<\min(m_1,m_2)\), and show that then \(y_{k+1}^1\sim y_{k+1}^2\). Assume the contrary, i.e. that \(y_{k+1}^1\) is not \(\sim y_{k+1}^2\) (where one of these intervals may be empty). Then, because the intervals \(y_{k+1}^1\) and \(y_{k+1}^2\) are normal, in one of them, say in \(y_{k+1}^1\), there will be a variable \(x_j\) not entering the interval \(y_{k+1}^2\).

Since functions from \(M'\) depend essentially on all their variables, this means, in particular, that \(k + 1 < m_2\). Set the values of all variables from \(y_2^2 \ldots y_{k+1}^2\) equal to 1, and set the values of all remaining variables equal to 2. Then, evidently, \(\varphi_2 = 1\), while \(\varphi_1 = 0\), which contradicts the assumption that the functions \(\varphi_1\) and \(\varphi_2\) are equal.

Lemma 4. Every identity of formulas having no more than \(m\) occurrences of variables (of length \(m\)) is derivable from the system \(E_m\).

It is easy to see that, by means of the system \(E_m\), any function of length no more than \(m\) can be brought to canonical form; namely, by means of \(C_{n,k}\) and \(B_n\), intersecting intervals can be transformed into one, possibly longer, interval, and by means of \(A_3\) and \(C_{n,k}\) \((k = 4, \ldots, n;\ n = 1,\ldots,m)\), already nonintersecting intervals can be brought to normal form. As a consequence of Lemma 3, we obtain the validity of this lemma as well.

Theorem. There is no finite complete system of identities in \(M\).

As indicated above, a necessary condition for the existence of a finite complete system of identities in \(M\) is the existence of such a system in \(M'\).

Suppose that there exists a finite complete system of identities in \(M'\). Let all the identities of this system contain formulas of length no more than \(m\). Then, by Lemma 4, \(E_m\) is a complete system for \(M'\). We shall show that \(B_{m+1}\) is not derivable from \(E_m\), thereby proving the theorem.

Indeed, to either side of the identity \(B_{m+1}\), from the system \(E_m\) only the identity \(A_3\) is applicable. Likewise, only the identity \(A_3\) from the system \(E_m\) is applicable to the result, since this result is not any substitution instance of the remaining identities from \(E_m\). Thus, \(E_m\) is not complete. Consequently, the theorem is proved.

REFERENCES

¹ R. K. Lyndon, Cybernetics Collection, 1, IL, 1950, p. 246. ² R. K. Lyndon, Identities in finite algebras, ibid., p. 249.