5
$\begingroup$

The Post correspondence problem has the following version for finite monoids:

Input: a finite monoid $M$ and a finite list $(m_1,m_1'),\ldots, (m_n,m_n')$ of pairs of elements of $M$

Question: is there a natural number $k\geq 1$ and indices $i_1,\ldots, i_k\in \{ 1,\ldots, n\}$ such that $m_{i_1}\cdot\cdots \cdot m_{i_k} = m_{i_1}'\cdot\cdots \cdot m_{i_k}'$?

Is it known whether this problem is decidable?

$\endgroup$
9
$\begingroup$

Yes, it is decidable. Build a graph where each vertex is a pair $(r,s)$ of elements from $M$. Add all edges of the form $(r,s) \to (r m_i, s m'_i)$ for all $r,s,i$. Then, your question asks whether there exists a path in this graph from the vertex $(1,1)$ to any vertex of the form $(t,t)$. This can be answered using standard reachability algorithms (e.g., DFS). The running time is linear in the size of the graph (i.e., $O(|M|^2 n)$), so the problem is decidable.

$\endgroup$
  • $\begingroup$ So is DFS faster than BFS here? $\endgroup$ – Bjørn Kjos-Hanssen Aug 13 '18 at 22:50
  • 1
    $\begingroup$ @BjørnKjos-Hanssen, no, they both run in linear time (linear in the size of the graph) so their asymptotic worst-case running time is equivalent. $\endgroup$ – D.W. Aug 13 '18 at 23:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.