On the complexity of decision problems for some classes of machines and applications

We study the computational complexity of important decision problems — including general membership, fixed-machine membership, emptiness, disjointness, equivalence, containment, universe, and finiteness problems — for various restrictions and combinations of two-way nondeterministic reversal-bounded multicounter machines ($2\mathrm{NCM}$) and two-way pushdown automata. We show that the general membership problem (respectively fixed membership problem) for $2\mathrm{NCM}$ is $\mathrm{NP}$-complete (respectively in $P$).

We then give applications 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 if there is no non-empty string y in L that is an infix of more than k strings in $L-\left\{y\right\}$. Our general results imply the complexity of determining whether a given machine accepts a k-infix-free language, for one- and two-way deterministic and nondeterministic finite automata (answering an open question from the literature).