Regular Separability and Intersection Emptiness Are Independent Problems

Ramanathan S. Thinniyam, Georg Zetzsche, Michael Wagner
2019 Foundations of Software Technology and Theoretical Computer Science  
The problem of regular separability asks, given two languages K and L, whether there exists a regular language S that includes K and is disjoint from L. This problem becomes interesting when the input languages K and L are drawn from language classes beyond the regular languages. For such classes, a mild and useful assumption is that they are full trios, i.e. closed under rational transductions. All the results on regular separability for full trios obtained so far exhibited a noteworthy
more » ... ondence with the intersection emptiness problem: In each case, regular separability is decidable if and only if intersection emptiness is decidable. This raises the question whether for full trios, regular separability can be reduced to intersection emptiness or vice-versa. We present counterexamples showing that neither of the two problems can be reduced to the other. More specifically, we describe full trios C1, D1, C2, D2 such that (i) intersection emptiness is decidable for C1 and D1, but regular separability is undecidable for C1 and D1 and (ii) regular separability is decidable for C2 and D2, but intersection emptiness is undecidable for C2 and D2. ACM Subject Classification Theory of computation → Models of computation; Theory of computation → Formal languages and automata theory later also concentrated on separability (e.g. [32, 33, 34, 35, 36, 37] ). Moreover, separability has been studied for regular tree languages, where separators are either piecewise testable tree languages [21] or languages of deterministic tree-walking automata [5] . For non-regular input languages, separability has been investigated with piecewise testable languages (PTL) [11] and generalizations thereof [42] as separators. Separability of subsets of trace monoids [7] and commutative monoids [9] by recognizable subsets has been studied as well. A natural choice for the separators is the class of regular languages. On the one hand, they have relatively high separation power and on the other hand, it is usually verifiable whether a given regular language is in fact a separator. For instance, they generalize piecewise testable languages but are less powerful than context-free languages (CFL). Since the intersection problem for CFL is undecidable, it is not easy to check if a given candidate CFL is a separator. This has motivated a recent research effort to understand for which language classes C, D regular separability is decidable [29, 9, 8 ]. An early result was that regular separability is undecidable for CFL (by this we mean that both input languages are context-free) [39, 25] . This was recently strengthened to undecidability already for visibly pushdown languages [28] and one-counter languages [29] . On the positive side, it was shown that regular separability is decidable for several subclasses of vector addition systems (VASS): for one-dimensional VASS [29] , for commutative VASS languages [9] , and for Parikh automata (equivalently, Z-VASS) [8] . Moreover, it is decidable for languages of well-structured transition systems [10] . Furthermore, decidability still holds in many of these cases if one of the inputs is a general VASS language [12] . However, if both inputs are VASS languages, decidability of regular separability remains a challenging open problem. Of course, if one of the input languages is regular, checking regular separability degenerates into checking intersection with a regular language. Thus, the problem becomes interesting when both input languages are non-regular. Many language classes beyond the regular languages constitute full trios, meaning that they are closed under rational transductions. This is typically the case for classes that originate from non-deterministic infinite-state systems [16] and from various types of grammars [16, 13] .
doi:10.4230/lipics.fsttcs.2019.51 dblp:conf/fsttcs/ThinniyamZ19 fatcat:rklvkbma6jcexd55xddgo4glsu