Conjunctive Grammars, Cellular Automata and Logic

Abstract

The expressive power of the class Conj of conjunctive languages, i.e. languages generated by the conjunctive grammars of Okhotin, is largely unknown, while its restriction LinConj to linear conjunctive grammars equals the class of languages recognized by real-time one-dimensional one-way cellular automata. We prove two weakened versions of the open question Conj ⊆? RealTime1CA, where RealTime1CA is the class of languages recognized by real-time one-dimensional two-way cellular automata: 1. it is true for unary languages; 2. Conj ⊆ RealTime2OCA, i.e. any conjunctive language is recognized by a real-time two-dimensional one-way cellular automaton. Interestingly, we express the rules of a conjunctive grammar in two Horn logics, which exactly characterize the complexity classes RealTime1CA and RealTime2OCA. 2012 ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Formal languages and automata theory