On the Complexity of Decision Problems for Counter Machines with Applications to Coding Theory

Abstract

We study the computational complexity of several decision problems (including the emptiness, disjointness, finiteness, and containment problems) for various restrictions of two-way reversal-bounded multicounter machines ($2\textsf {NCM}$). We then apply the results to some problems in coding theory. We examine generalizations of various types of codes with marginal errors; for example, a language L is k-infix-free ($k \ge 0$) if there is no non-empty string y in L that is an infix of more than k strings in $L - \{y\}$. This allows for bounded error versus standard infix-free languages. We show that it is $\textsf {PSPACE}$-complete to decide, given k and a $2\textsf {NCM}$ M whose input is finite-crossing, whether L(M) is not k-infix-free. It follows that the problem is also $\textsf {PSPACE}$-complete for one-way nondeterministic and deterministic finite automata (even for the two-way models), answering an open question in [12]. We also look at the complexity of the problem for restricted models of $2\textsf {NCM}$ and for other types of codes, and improve/generalize some previous results.

The research of I. McQuillan was supported, in part, by Natural Sciences and Engineering Research Council of Canada.