Polynomial time algorithm for left [right] local testability

TL;DR

This paper introduces polynomial time algorithms to determine if a language is locally testable from automaton transition graphs, providing necessary and sufficient conditions for right and left local testability and for automata with locally idempotent semi groups.

Contribution

It presents new polynomial time algorithms and characterizations for checking local testability of languages via automaton transition graphs and semi groups.

Findings

Polynomial time algorithms for right and left local testability.

Necessary and sufficient conditions for transition graphs and semi groups.

Verification method for automata with locally idempotent semi groups.

Abstract

A right [left] locally testable language S is a language with the property that for some non negative integer k two words u and v in alphabet S are equal in the semi group if (1) the prefix and suffix of the words of length k coincide, (2) the set of segments of length k of the words as well as 3) the order of the first appearance of these segments in prefixes [suffixes] coincide. We present necessary and sufficient condition for graph [semi group] to be transition graph [semi group] of the deterministic finite automaton that accepts right [left] locally testable language and necessary and sufficient condition for transition graph of the deterministic finite automaton with locally idempotent semi group. We introduced polynomial time algorithms for the right [left] local testable problem for transition semi group and transition graph of the deterministic finite automaton based on these conditions. Polynomial time algorithm verifies transition graph of automaton with locally idempotent transition semi group.