Abstract
Canonical positive linear recurrence numeration systems admit multiple coefficient presentations generating the same legality language. Paper III showed that every such presentation belongs to a unique equivalence class determined by an intrinsic periodic boundary sequence and that each class possesses a distinguished canonical reduction of minimal order.
In this paper we give an automata-theoretic interpretation of canonical reduction. To every coefficient presentation we associate a deterministic threshold-reset automaton accepting exactly its legality language. We prove that the natural automaton of every presentation minimizes to the natural automaton of the canonical reduction.
Consequently the minimal deterministic automaton of the legality language is uniquely determined by the canonical reduction, and the canonical order is exactly the number of live Myhill–Nerode classes of the legality language.
Thus canonical reduction is identified with automaton minimization, providing an intrinsic automata-theoretic characterization independent of recurrence order.