Exact Local Annotations for Regular Languages

Abstract

A regular language is recognized by a finite monoid, but a locally checkable explanation of that recognition can have a nontrivial update geometry. We study exact bounded-arity annotations for regular word languages under one-symbol substitutions. The cost of an edit is the number of annotation cells that a canonical locally accepted representation must change, together with the corresponding bit movement and the number of local constraints that must be revalidated. For every morphism recognizing a regular language, the balanced product annotation gives constant locality, linear size, O(log n) edit stability, O(log n) revalidation, and constant access to the membership value. The matching lower bound proved here is restricted to product decompositions that expose an edit-active nontrivial group quotient as ordered product labels; in that setting one substitution changes every quotient label on an ancestor path. We also show that annotation-free bounded-window recognition is exactly strict locality, prove closure properties for a two-sided total decision variant, and formulate the remaining constant-stability boundary as a finite obstruction problem. The ancillary files include Lean, CP-SAT, and CUDA certificates, including a context-free interval-chart experiment.