Multi-Island Finite Automata and Their Even Computations

finite automata, graph-based decomposition, regulated computation, infinite hierarchies of language families

This paper discusses n-island finite automata whose transition graphs can be expressed as n-member sequences of islands i1, i2, ..., in, where there is a bridge leaving ij and entering i(j+1) for each 1 <= j <= n - 1. It concentrates its attention on even computation defined as any sequence of moves during which these automata make the same number of moves in each of the islands. Under the assumption that these automata work only in an evenly computational way, the paper proves its main result stating that n-island finite automata and Rosebrugh-Wood n-parallel right-linear grammars are equivalent. Then, making use of this main result, it demonstrates that under this assumption, the language family defined by n-island finite automata is properly contained in that defined by (n+1)-island finite automata for all n >= 1. The paper also points out that this infinite hierarchy occurs between the family of regular languages and that of context-sensitive languages. Open questions are formulated in the conclusion.