Usefulness of Information and Unary Languages

Abstract

In this paper we continue the research on usefulness of information examining the effect of supplementary information on the complexity of solving a problem (see Rovan and Sádovský [7] for an overview). We use deterministic finite automata for a formal setting. Given a problem (a regular language) $L_{prob}$ we measure the complexity of its solution – a DFA $A_{prob}$ such that $L_{prob} = L(A_{prob})$ – using the state complexity. A supplementary information (advice) $L_{adv}$ given by $A_{adv}$ is useful if a simpler problem $L_{new}$ given by $A_{new}$ exists such that $L_{prob} = L_{new} \cap L_{adv}$ and both $L_{new}$ and $L_{adv}$ are simpler than $L_{prob}$. This is formalized via the notion of decomposability of finite automata (see [1] for DFA case and [7] for NFA case). We address the problem of decomposability of unary regular languages and give a characterization of $\lambda$-cyclic languages upon deterministic decomposability.

This research has been supported in part by the grant 1/0601/20 of the Slovak Scientific Grant Agency VEGA.