Recognizability of languages via deterministic finite automata with values on a monoid: General Myhill-Nerode Theorem

Abstract

This paper deals with the problem of recognizability of functions ℓ : Σ * → M that map words to values in the support set M of a monoid (M, •, 1). These functions are called M -languages. M -languages are studied from the aspect of their recognition by deterministic finite automata whose components take values on M (M -DFAs). The characterization of an M -language ℓ is based on providing a right congruence on Σ * that is defined through ℓ and a factorization on the set of all M -languages, L(Σ * , M ) (in short L). A factorization on L is a pair of functions (g, f ) such that, for each ℓ ∈ L, g(ℓ) • f (ℓ) = ℓ, where g(ℓ) ∈ M and f (ℓ) ∈ L. In essence, a factorization is a form of common factor extraction. In this way, a general Myhill-Nerode theorem, which is valid for any L(Σ * , M ), is provided. Basically, ℓ ∈ L is recognized by an M -DFA if and only if there exists a factorization on L, (g, f ), such that the right congruence on Σ * induced by the factorization (g, f ) and f (ℓ) ∈ L, has finite index. This paper shows that the existence of M -DFAs guarantees the existence of natural non-trivial factorizations on L without taking account any additional property on the monoid. In addition, the composition of factorizations is also a new factorization, and the composition of natural factorizations preserves the recognition capability of each individual natural factorization.