A Note on Decidable Separability by Piecewise Testable Languages

Abstract

Piecewise testable languages form the first level of the Straubing-Thérien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. The question has not yet been addressed for NFAs. We fill in this gap by showing that it is PSpace-complete. The main result is then the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separa-bility by piecewise testable languages is known to be decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even for minimal DFAs.

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

“…Proof. This is an immediate consequence of a result of Czerwiński et al [11] who show that for any class of languages effectively closed under rational transductions, the problem reduces to solving the diagonal problem.…”

“…The diagonal problem is a decision problem with a number of interesting algorithmic consequences. It is a central subproblem for computing the downward closure of languages of words [27], as well as for the problem of separability by piecewise-testable languages [11]. It is used in deciding reachability of a certain type of parameterized concurrent systems [25].…”

“…In this context, a result by Zetzsche [27] entails computability of the downward closure of languages of words recognized by higher-order schemes. Moreover, a recent result by Czerwiński, Martens, van Rooijen, and Zeitoun [10] entails that the separability by piecewise testable languages is decidable for languages recognized by higher-order schemes. Finally, a third example comes from La Torre, Muscholl, and Walukiewicz [25] showing how to use downward closures to decide reachability in parameterized asynchronous shared-memory concurrent systems where every process is a higher-order scheme.…”

Intersection Types for Unboundedness Problems

“…ϕ, says that ϕ holds for arbitrarily large finite sets X . Let us also remark that decidability of SUP implies that given a language defined by a nondeterministic recursion scheme, it is possible to compute its downward closure [30], and given two such languages, it is possible to decide whether they can be separated by a piecewise testable language [10].…”

Separability by piecewise testable languages is PTime-complete

“…Separability by piecewise testable languages is of interest also outside regular languages. Although separability of context-free languages by regular languages is undecidable [17], separability by piecewise testable languages is decidable (even for some non-context-free languages) [9]. Piecewise testable languages are further investigated in natural language processing [11,33], cognitive and sub-regular complexity [34], and learning theory [12,22].…”