An Approach to Regular Separability in Vector Addition Systems

Abstract

We study the problem of regular separability of languages of vector addition systems with states (VASS). It asks whether for two given VASS languages K and L, there exists a regular language R that includes K and is disjoint from L. While decidability of the problem in full generality remains an open question, there are several subclasses for which decidability has been shown: It is decidable for (i) one-dimensional VASS, (ii) VASS coverability languages, (iii) languages of integer VASS, and (iv) commutative VASS languages. We propose a general approach to deciding regular separability. We use it to decide regular separability of an arbitrary VASS language from any language in the classes (i), (ii), and (iii). This generalizes all previous results, including (iv).

Regular Separability and Intersection Emptiness are Independent Problems

“…For showing decidability of regular separability, we use the following well-known fact (please see Appendix D for a proof). The last ingredient for our decision procedure is the following simple but powerful observation from [14] (for the convenience of the reader, a proof can be found in Appendix E). Lemma 5.7.…”

“…Moreover, it is decidable for languages of well-structured transition systems [12]. Furthermore, decidability still holds in many of these cases if one of the inputs is a general VASS language [14]. However, if both inputs are VASS languages, decidability of regular separability remains a challenging open problem.…”

Regular Separability in Büchi VASS

“…If the input languages are themselves regular and S is a subclass [42,41,40,39,43,44,35,15], then separability generalizes the classical subclass membership problem. Moreover, separability for languages of infinite-state systems has received a significant amount of attention [17,16,14,13,10,9,12,1,51,48,11,8]. Let us point out two prominent cases.…”

“…Here, a run is accepting if it reaches a final state with all counters being zero. While there have been several decidability results for subclasses of the VASS languages [17,14,13,10,9], the general case remains open. Second, a surprising result is that if K and L are coverability languages of well-structured transition systems (WSTS), then K and L are separable by a regular language if and only if they are disjoint [14].…”

Deterministic and game separability for regular languages of infinite trees

“…Over finite words, variants of the pC, Dq-separability problem have been studied for classes C both more general than the regular languages, such as the context free languages [22,49] and higher-order languages [14] (later extended to safe schemes over finite trees [1]), and for classes D more restrictive than the regular languages, such as in [39,40]. The separability and membership problems have also been studied for several classes of infinite-state systems, such as vector addition systems [11,10,23], well-structured transition systems [21], one-counter automata [20], and timed automata [13,12]. Recent developments on efficient algorithms solving parity games are based on the ability to find a simple separator, yielding both upper bounds on the problem, and lower bounds for a wide family of algorithms [5,19,Chapter 3].…”