Strict Local Testability with Consensus Equals Regularity

Abstract

A recent language definition device named consensual is based on agreement between similar words. Considering, say, a regular set of words over a bipartite alphabet made by pairs of unmarked/marked letters, the match relation specifies when such words agree. Therefore a regular set (the “base”) over the bipartite alphabet specifies another language over the unmarked alphabet, called the consensual language. A word is in the consensual language if a set of corresponding matching words is in the base. From previous results, the family of consensual languages based on regular sets have an NLOGSPACE word problem, include non-semilinear languages, and are incomparable with the context-free (CF) ones; moreover the size of a consensual specification can be in a logarithmic ratio with respect to a NFA for the same language. We study the consensual languages that are produced by other language families: the Strictly Locally Testable of McNaughton and Papert and the context-free/sensitive ones. Using a recent generalization of Medvedev’s homomorphic characterization of regular languages, we prove that regular languages are exactly the consensual languages based on strictly locally testable sets, a result that hints at a novel parallel decomposition of finite automata into locally testable components. The consensual family based on context-free sets strictly includes the CF family, while the consensual and the base families collapse together if the context-sensitive languages are chosen instead of the CF.