New Decidability Results Concerning Two-Way Counter Machines

Abstract

The authors study some decision questions concerning two-way counter machines and obtain the strongest decidable results to date concerning these machines. In particular, it is shown that the emptiness, containment, and equivalence (ECE, for short) problems are decidable for two-way counter machines whose counter is reversal-bounded (i.e., the counter alternates between increasing and decreasing modes at most a fixed number of times). This result is used to give a simpler proof of a recent result in which the ECE problems for two-way reversal-bounded pushdown automata accepting bounded languages (i.e., subsets of $w_1^*\ldots w_k^*$ for some nonnull words $w_1,\ldots,w_k$) are decidable. Other applications concern decision questions about simple programs. Finally, it is shown that nondeterministic two-way reversal-bounded multicounter machines are effectively equivalent to finite automata on unary languages, and hence their ECE problems are decidable also.