UDC 51.01:518.5

MATHEMATICS

Yu. V. MATIYASEVICH

SIMPLE EXAMPLES OF UNDECIDABLE ASSOCIATIVE CALCULI

(Presented by Academician P. S. Novikov on 21 VI 1966)

1. A method for constructing an associative calculus with an undecidable equivalence problem was first indicated in papers of A. A. Markov \((^1)\) and Post \((^2)\). The defining systems of calculi constructed by this method consist of a considerable number of relations. G. S. Tseitin, in paper \((^3)\), constructed an associative calculus with a defining system of 7 relations for which the equivalence problem is undecidable. A method for constructing an associative calculus with a defining system of 7 relations and an undecidable equivalence problem is also indicated by Scott \((^4)\).

In the present paper we construct an associative calculus with an undecidable equivalence problem for a fixed word, whose defining system consists of 5 relations. In addition, we indicate a method for constructing an associative calculus with a defining system of 3 relations for which the equivalence problem for some fixed word is also undecidable.

1. Let \(A\) denote the alphabet \(\{a, b, c, d, e\}\). Let \(i\) be a nonnegative integer. Denote by \(\mathfrak{C}_i\) the following associative calculus in the alphabet \(A\):

\[ \begin{gathered} ac \leftrightarrow ca,\qquad ad \leftrightarrow da,\\ bc \leftrightarrow cb,\qquad bd \leftrightarrow db,\\ ce \leftrightarrow eca,\qquad de \leftrightarrow edb,\\ cd^i ca \leftrightarrow cd^i cae. \end{gathered} \]

By \(\mathfrak{E}_i\) we denote the associative calculus in the alphabet \(A\) with defining system

\[ \begin{gathered} ac \leftrightarrow ca,\qquad ad \leftrightarrow da,\\ bc \leftrightarrow cb,\qquad bd \leftrightarrow db,\\ ce \leftrightarrow eca,\qquad de \leftrightarrow edb,\\ cd^i ca \leftrightarrow cd^i ce. \end{gathered} \]

G. S. Tseitin, in paper \((^3)\), proved that, whatever the nonnegative integer \(i\), for the calculus \(\mathfrak{C}_i\) the equivalence problem for some fixed word is undecidable. In an analogous way one can prove that, whatever the nonnegative integer \(i\), for the calculus \(\mathfrak{E}_i\) the equivalence problem for some fixed word is undecidable.

1. Denote by \(B\) the alphabet \(\{\alpha,\sigma\}\), and by \(\varphi\) the following normal algorithm in the alphabet \(A \cup B\):

\[ \begin{aligned} a &\to \alpha\alpha\sigma\alpha,\\ b &\to \alpha\sigma\alpha\alpha,\\ c &\to \alpha\sigma,\\ d &\to \sigma\sigma,\\ e &\to \alpha\alpha\alpha. \end{aligned} \]

Denote by \(\mathfrak H\) the following associative calculus in the alphabet \(B\):

\[ \begin{gathered} \alpha\sigma\alpha\alpha\sigma \leftrightarrow \sigma\sigma\alpha\alpha\sigma\alpha,\\ \alpha\alpha\sigma\alpha\sigma\alpha \leftrightarrow \sigma\sigma\alpha\alpha\alpha\sigma\alpha,\\ \alpha\sigma\alpha\alpha\alpha\sigma \leftrightarrow \alpha\sigma\sigma\alpha\sigma\alpha,\\ \sigma\sigma\sigma\alpha\alpha\sigma\sigma\alpha\alpha\sigma\alpha \leftrightarrow \sigma\sigma\sigma\alpha\sigma\sigma\alpha\alpha\alpha\alpha,\\ \alpha\alpha\alpha\alpha\sigma\sigma\alpha\alpha\sigma\alpha \leftrightarrow \sigma\sigma\alpha\alpha\alpha\alpha . \end{gathered} \]

Theorem 1. Whatever the words \(P\) and \(Q\) in the alphabet \(A\), in order that \(\mathfrak C_1 : P \parallel Q\), it is necessary and sufficient that \(\mathfrak H : \varphi(P) \parallel \varphi(Q)\).

Necessity follows from the fact that

\[ \begin{array}{ll} \mathfrak H:\ \varphi(ac)\parallel \varphi(ca), & \mathfrak H:\ \varphi(ad)\parallel \varphi(da),\\ \mathfrak H:\ \varphi(bc)\parallel \varphi(cb), & \mathfrak H:\ \varphi(bd)\parallel \varphi(db),\\ \mathfrak H:\ \varphi(ce)\parallel \varphi(eca), & \mathfrak H:\ \varphi(de)\parallel \varphi(edb), \end{array} \]

\[ \mathfrak H:\ \varphi(cdca)\parallel \varphi(cdce). \]

Sufficiency is proved by induction on the length of the \(\mathfrak H\)-sequence connecting the words \(\varphi(P)\) and \(\varphi(Q)\).

The base of the induction is obvious.

The induction step is carried out on the basis of the following lemma.

Lemma 1. Whatever the word \(R\) in the alphabet \(A\) and the word \(V\) in the alphabet \(B\), if \(\mathfrak H:\varphi(R)\parallel V\), then one can construct a word \(S\) in the alphabet \(A\) such that \(\mathfrak C_1:R\parallel S\) and \(\varphi(S)=V\).

It follows immediately from Theorem 1 that for the calculus \(\mathfrak H\) the equivalence problem for a certain fixed word is undecidable.

1. Denote by \(B\) the alphabet \(A\cup\{f_1,\ldots,f_7\}\), and by \(\mathfrak K\) the following associative calculus in the alphabet \(B\):

\[ \begin{array}{ll} f_1\leftrightarrow ac, & f_1\leftrightarrow ca,\\ f_2\leftrightarrow ad, & f_2\leftrightarrow da,\\ f_3\leftrightarrow bc, & f_3\leftrightarrow cb,\\ f_4\leftrightarrow bd, & f_4\leftrightarrow db,\\ f_5\leftrightarrow ce, & f_5\leftrightarrow ef_1,\\ f_6\leftrightarrow de, & f_6\leftrightarrow ef_4,\\ f_7\leftrightarrow cf_1, & f_7\leftrightarrow f_7e,\\ f_7\leftrightarrow cf_1, & f_7\leftrightarrow f_7e \end{array} \]

(the last two relations are repeated because in what follows we shall need the fact that the number of relations of the calculus \(\mathfrak K\) is a power of the number \(2\)).

Lemma 2. Whatever the words \(P\) and \(Q\) in the alphabet \(A\), \(\mathfrak C_0:P\parallel Q\) if and only if \(\mathfrak K:P\parallel Q\).

Denote by \(\Gamma\) the alphabet \(\{\beta,\gamma\}\), and by \(\psi\) the following normal algorithm in the alphabet \(B\cup\Gamma\):

\[ f_1\to \beta\beta\gamma\beta\gamma^{12}, \]

\[ \ldots\ldots\ldots\ldots \]

\[ f_i\to \beta\beta\gamma^i\beta\gamma^{13-i}, \]

\[ \ldots\ldots\ldots\ldots \]

\[ f_{12}\to \beta\beta\gamma^{12}\beta\gamma, \]

where \(f_8,\ldots,f_{12}\) denote the letters \(a,b,c,d,e\), respectively.

Denote by \(A_1,\ldots,A_{16},B_1,\ldots,B_{16}\) words in the alphabet \(B\) such that the system \(A_i\leftrightarrow B_i\ (i=1,\ldots,16)\) is a defining system of the calcul-

calculus \(\mathfrak K\). Denote by \(C_i\) the word \(\psi(A_i)\), and by \(D_i\) the word \(\psi(B_i)\) \((i=1,\ldots,16)\). Denote by \(\mathfrak L\) the associative calculus in the alphabet \(\Gamma\) whose defining system is the system \(C_i \leftrightarrow D_i\) \((i=1,\ldots,16)\).

Lemma 3. Whatever the words \(P\) and \(Q\) in the alphabet \(B\), \(\mathfrak K: P \,\|\, Q\) if and only if \(\mathfrak L: \psi(P) \,\|\, \psi(Q)\).

Denote by \(p_{i,j}\) the \(i\)-th letter from the left of the word \(C_j\) \((i=1,\ldots,16^*,\ j=1,\ldots,16)\), and by \(q_{i,j}\) the \(i\)-th letter from the left of the word \(D_j\) \((i=1,\ldots,32^{**},\ j=1,\ldots,16)\). Denote by \(M\) the word

\[ p_{1,1}\ldots p_{1,16}p_{2,1}\ldots p_{2,16}\ldots p_{16,1}\ldots p_{16,16}, \]

and by \(N\) the word

\[ q_{1,1}\ldots q_{1,16}q_{2,1}\ldots q_{2,16}\ldots q_{32,1}\ldots q_{32,16}. \]

Denote by \(\Delta\) the alphabet \(\Gamma\cup\{\varepsilon\}\), and by \(\mathfrak M\) the following associative calculus in the alphabet \(\Delta\):

\[ \varepsilon\beta\beta \leftrightarrow \beta\varepsilon,\qquad \varepsilon\gamma\beta \leftrightarrow \beta\varepsilon, \]

\[ \varepsilon\beta\gamma \leftrightarrow \gamma\varepsilon,\qquad \varepsilon\gamma\gamma \leftrightarrow \gamma\varepsilon, \]

\[ M \leftrightarrow N. \]

Theorem 2. Whatever the words \(P\) and \(Q\) in the alphabet \(B\), \(\mathfrak K: P \,\|\, Q\) if and only if \(\mathfrak M: \psi(P)\gamma\varepsilon^4 \,\|\, \psi(Q)\gamma\varepsilon^4\).

Denote by \(\tau\) the following normal algorithm in the alphabet \(B\cup\Delta\):

\[ \beta \to \sigma\alpha, \]

\[ \gamma \to \sigma, \]

\[ \varepsilon \to \alpha\alpha. \]

Denote by \(\mathfrak N\) the associative calculus in the alphabet \(B\) whose defining system is the system

\[ \alpha\alpha\sigma\alpha\sigma \leftrightarrow \sigma\alpha\alpha,\qquad \alpha\alpha\sigma\sigma \leftrightarrow \sigma\alpha\alpha, \]

\[ \tau(M) \leftrightarrow \tau(N). \]

Theorem 3. Whatever the words \(P\) and \(Q\) in the alphabet \(B\), in order that \(\mathfrak K: P \,\|\, Q\), it is necessary and sufficient that \(\mathfrak N: \tau(\psi(P)\gamma\varepsilon^4) \,\|\, \tau(\psi(Q)\gamma\varepsilon^4)\).

Necessity follows immediately from Theorem 2 and the following easily verified assertions:

\[ \mathfrak N: \tau(\varepsilon\beta\beta) \,\|\, \tau(\beta\varepsilon),\qquad \mathfrak N: \tau(\varepsilon\gamma\beta) \,\|\, \tau(\beta\varepsilon), \]

\[ \mathfrak N: \tau(\varepsilon\beta\gamma) \,\|\, \tau(\gamma\varepsilon),\qquad \mathfrak N: \tau(\varepsilon\gamma\gamma) \,\|\, \tau(\gamma\varepsilon), \]

\[ \mathfrak N: \tau(M) \,\|\, \tau(N). \]

From Lemma 2 and Theorem 3 it follows immediately that for the calculus \(\mathfrak N\) the problem of equivalence to a certain fixed word is undecidable.

Remark. We have reduced the equivalence problem for the calculus \(\mathfrak C_0\) to the equivalence problem for the calculus \(\mathfrak N\). In an analogous way, the equivalence problem for an arbitrary associative calculus can be reduced to the equivalence problem for a certain associative calculus in a two-letter alphabet, whose defining system consists of 3 relations.

Leningrad State University
named after A. A. Zhdanov

Received
16 IV 1966

CITED LITERATURE

\(^1\) A. A. Markov, DAN, 55, No. 7, 587 (1947). \(^2\) E. L. Post, J. Symb. Logic, 12, 1 (1947). \(^3\) G. S. Tseitin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 52, 172 (1958). \(^4\) D. Scott, J. Symb. Logic, 21, 111 (1956).

\[ \begin{aligned} &{}^*\,16 \text{ is the length of the word } C_j.\\ &{}^{**}\,32 \text{ is the length of the word } D_j. \end{aligned} \]