Verification Tools for Checking some kinds of Testability

TL;DR

This paper discusses verification tools and procedures for determining various forms of local testability in regular languages, including automaton-based and algebraic methods, with implementation details and bounds analysis.

Contribution

It introduces new verification procedures for local testability properties of automata and languages, along with bounds on their order and algebraic characterizations.

Findings

Implemented procedures in the TESTAS package for local testability verification.

Derived bounds on the order of local testability for transition graphs and semigroups.

Presented new proofs and conditions for local testability of deterministic finite automata.

Abstract

A locally testable language L is a language with the property that for some non negative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k + 1 and (2) the set of intermediate substrings of length k of the word u. For given k the language is called k-testable. The local testability has a wide spectrum of generalizations. A set of procedures for deciding whether or not a language given by its minimal automaton or by its syntactic semigroup is locally testable, right or left locally testable, threshold locally testable, strictly locally testable, or piecewise testable was implemented in the package TESTAS written in C=C++. The bounds on order of local testability of transition graph and order of local testability of transition semigroup are also found. For given k, the k-testability of transition graph is verified. We consider some approaches to verify these procedures and use for this aim some auxiliary programs. The approaches are based on distinct forms of presentation of a given finite automaton and on algebraic properties of the presentation. New proof and fresh wording of necessary and sufficient conditions for local testability of deterministic finite automaton is presented.