A variant of a recursively unsolvable problem. (English) Zbl 0063.06329

From the text: By a string on \(a,b\) we mean a row of \(a\)’s and \(b\)’s such as \(baabbbab\). It may involve only \(a\), or \(b\), or be null. If, for example, \(g_1, g_2, g_3\) represent strings \(bab\), \(aa\), \(b\) respectively, string \(g_2g_1g_1g_3g_2\) on \(g_1, g_2, g_3\) will represent, in obvious fashion, the string \(aababbabbaa\) on \(a,b\). By the correspondence decision problem we mean the problem of determining for an arbitrary finite set \((g_1, g'_1)\), \((g_2, g'_2), \ldots, (g_\mu, g'_\mu)\) of pairs of corresponding non-null strings on \(a, b\) whether there is a solution in \(n\), \(i_1, i_2, \ldots, i_n\) of equation
\[ g_{i_1}g_{i_2}\cdots g_{i_n}=g'_{i_1}g'_{i_2}\cdots g'_{i_n},\quad n\geq 1, i_j=1,2,\ldots,\mu.\tag{1} \]
That is, whether some non-null string on \(g_1,g_2, \ldots, g_n\), and corresponding string on \(g'_1,g'_2, \ldots, g'_n\) represent identical strings on \(a, b\).
In special cases, of course, the question posed by (1) may be answerable. Thus, if, with \(\mu=3\), \((g_1, g'_1)\), \((g_2, g'_2)\), \((g_3, g'_3)\) are \((bb, b)\), \((ab, ba)\), \((b, bb)\) respectively, \(g_1g_2g_2g_3 = bbababb=g'_1g'_2g'_2g'_3\), and (1) has a solution. Again, if each \(g_i\) is of greater length than the corresponding \(g'_i\), or if each \(g_i\) starts with a different letter than the corresponding \(g'_i\), (1) has no solution. We proceed to prove, on the other hand, that in its full generality the correspondence decision problem is recursively unsolvable, and hence, no doubt, unsolvable in the intuitive sense.


03D99 Computability and recursion theory
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