Piecewise Testable Languages and Nondeterministic Automata

A regular language is k-piecewise testable if it is a finite boolean combination of languages of the form Sigma^* a_1 Sigma^* ... Sigma^* a_n Sigma^*, where a_i in Sigma and 0 = 0, it is an NL-complete problem to decide whether the language L(A) is piecewise testable and, for k >= 4, it is coNP-complete to decide whether the language L(A) is k-piecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on k. Namely, if L(A) is piecewise testable, then it is k-piecewise testable for k equal to the depth of A. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on k than the minimal DFA. We provide an application of our result, discuss the relationship between k-piecewise testability and the depth of NFAs, and study the complexity of k-piecewise testability for ptNFAs

On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)

Eilenberg correspondence, based on the concept of syntactic monoids, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence related more general classes of regular languages with classes of more complex algebraic objects. Such generalized varieties also have natural counterparts formed by classes of finite automata equipped with a certain additional algebraic structure. In this survey, we overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527

Testing Simon’s congruence

Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon’s congruence ∼kis one of the most classical approaches. Two words are ∼k-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of ∼k-classes. For each equivalence class of ∼k, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in ∼k. We present an algorithm for computing the canonical representative of the ∼k-class of a given word w ∈ A∗of length n. The running time of our algorithm is in O(|A|n) even if k ≤ n is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case k > n is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for ∼kis possible in deterministic logarithmic space. One of the consequences of our algorithm is that one can check with the same complexity whether two words are ∼k-equivalent (with k being part of the input)