A Myhill-Nerode Theorem for Finite State Matrix Automata and Finite Matrix Languages

Abstract

We propose a deterministic version of finite state matrix automaton (DFSMA) which recognizes finite matrix languages (FML). Our main result is a generalization of the classical Myhill-Nerode theorem for DFSMA. Our generalization requires the use of two relations to capture the additional structure of DFSMA. Vertical equivalence captures that words sharing the same vertical location, horizontal equivalence captures that words sharing the same horizontal location. A finite matrix language is defined to be regular if relations and exist that satisfy certain conditions, in particular, they have finite index. We show that the language associated to a DFSMA is regular, and we construct, for each finite matrix language, a DFSMA that accepts this language. Our result provides a foundation for learning algorithms for DFSMA.