Separability by piecewise testable languages and downward closures beyond subwords

Abstract

Indexed languages are a classical notion in formal language theory. As the language equivalent of second-order pushdown automata, they have received considerable attention in higher-order model checking. Unfortunately, counting properties are notoriously difficult to decide for indexed languages: So far, all results about non-regular counting properties show undecidability.In this paper, we initiate the study of slice closures of (Parikh images of) indexed languages. A slice is a set of vectors of natural numbers such that membership of 𝒖, 𝒖 +𝒗, 𝒖 +𝒘 implies membership of 𝒖 + 𝒗 + 𝒘. Our main result is that given an indexed language 𝐿, one can compute a semilinear representation of the smallest slice containing 𝐿's Parikh image.We present two applications. First, one can compute the set of all affine relations satisfied by the Parikh image of an indexed language. In particular, this answers affirmatively a question by Kobayashi: Is it decidable whether in a given indexed language, every word has the same number of 𝑎's as 𝑏's.As a second application, we show decidability of (systems of) word equations with rational constraints and a class of counting constraints: These allow us to look for solutions where a counting function (defined by an automaton) is not zero. For example, one can decide whether a word equation with rational constraints has a solution where the number of occurrences of 𝑎 differs between variables 𝑋 and 𝑌 .

An Approach to Regular Separability in Vector Addition Systems

“…In the case of non-regular languages as input languages, it was shown early that regular separability is undecidable for context-free languages [39,18]. Moreover, aside from the above mentioned results on regular separability, infinitestate systems have also been studied with respect to separability by bounded regular languages [6] and piecewise testable languages [10] and generalizations thereof [41].…”

The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings

“…In this chapter we are concerned with the issue of reasoning about, and computing with, downwards-closed and upwards-closed subsets of a WQO. These sets appear in program verification (prominently in model-checking of well structured systems [6], in verification of Petri nets [29], in separability problems [27,61], but also as an effective abstraction tool [5,60]). The question of how to handle downwards-closed subsets of WQOs in a generic way was first raised by Geeraerts et al: in [22] the authors postulated the existence of an adequate domain of limits satisfying some representation conditions.…”

The height of piecewise-testable languages and the complexity of the logic of subwords

“…They also constitute an important class of simple regular languages with applications in learning theory [KCM08], databases [BSS12], linguistics [RHF `13], etc. The concept of PT languages has been extended to richer notions of "subwords" [Zet18], to trees [BSS12], infinite words [PP04,CP18], pictures [Mat98], or any combinatorial wellquasi-order [GS16].…”