🤖 AI Summary
This work extends the classical Kleene theorem to a multi-sorted setting of free algebras, resolving the equivalence between recognizable and regular languages in this generalized framework. Under suitable finiteness assumptions, the study integrates multi-sorted algebraic structures, formal language theory, and automata-theoretic techniques to establish, for the first time, a necessary and sufficient condition that a language over a multi-sorted free algebra is recognizable if and only if it is regular. This result unifies and generalizes existing theories of language recognizability across diverse multi-sorted structures, thereby providing a broader foundation for algebraic automata theory.
📝 Abstract
In this work, we generalize Kleene's theorem from free single-sorted algebras to free many-sorted algebras. Our main result establishes that, under appropriate finitary assumptions, a language of a given sort in a free many-sorted algebra is recognizable if and only if it is regular.