Deterministic Logics for UL

The class of Unambiguous Star-Free Regular Languages (UL) has been widely studied and variously characterized by logics such as TL[X a ,Y a ],UITL,TL[F,P], FO 2[ < ], the variety DA and partially-ordered two-way DFA. However, explicit reductions from logics to automata are missing. In this paper, we introduce the concept of Deterministic Logics for UL. The formulas of deterministic logics uniquely parse a word in order to evaluate satisfaction. We consider three such deterministic logics with varied modalities, namely TL[X a ,Y a ],TL[Ũ,S] and UITL ±. Using effective reductions between them and to po2dfa, we show that they all characterize UL, and have NP-complete satisfiability. The reductions rely on features of deterministic logic such as unique parsability and ranker-directionality.