Wojciech Czerwiński

Efficient separability of regular languages by subsequences and suffixes

When can two regular word languages K and L be separated by a simple language? We investigate this question and consider separation by piecewise- and suffix-testable languages and variants thereof. We give characterizations of when two languages can be separated and present an overview of when these problems can be decided in polynomial time if K and L are given by nondeterministic automata.

Fast equivalence-checking for normed context-free processes

Bisimulation equivalence is decidable in polynomial time over normed graphs generated by a context-free grammar. We present a new algorithm, working in timeO(n 5 ), thus improving the previously known complexityO(n 8 polylog(n)). It also improves the previously known complexity O(n 6 polylog(n)) of the equality problem for simple grammars.

A Note on Decidable Separability by Piecewise Testable Languages

The separability problem for word languages of a class \(\mathcal {C}\) by languages of a class \(\mathcal {S}\) asks, for two given languages I and E from \(\mathcal {C}\), whether there exists a language S from \(\mathcal {S}\) that includes I and excludes E, that is, \(I \subseteq S\) and \(S\cap E = \emptyset \). It is known that separability for context-free languages by any class containing all definite languages (such as regular languages) is undecidable. We show that separability of context-free languages by piecewise testable languages is decidable. This contrasts with the fact that testing if a context-free language is piecewise testable is undecidable. We generalize this decidability result by showing that, for every full trio (a class of languages that is closed under rather weak operations) which has decidable diagonal problem, separability with respect to piecewise testable languages is decidable. Examples of such classes are the languages defined by labeled vector addition systems and the languages accepted by higher order pushdown automata of order two. The proof goes through a result which is of independent interest and shows that, for any kind of languages I and E, separability can be decided by testing the existence of common patterns in I and E.

The (Almost) Complete Guide to Tree Pattern Containment

Tree pattern queries are being investigated in database theory for more than a decade. They are a fundamental and flexible query mechanism and have been considered in the context of querying tree structured as well as graph structured data. We revisit their containment, validity, and satisfiability problem, both with and without schema information. We present a comprehensive overview of what is known about the complexity of containment and develop new techniques which allow us to obtain tractability- and hardness results for cases that have been open since the early work on tree pattern containment. For the tree pattern queries we consider in this paper, it is known that the containment problem does not depend on whether patterns are evaluated on trees or on graphs. This means that our results also shed new light on tree pattern queries on graphs.

Reachability problem for weak multi-pushdown automata

This paper is about reachability analysis in a restricted subclass of multi-pushdown automata: we assume that the control states of an automaton are partially ordered, and all transitions of an automaton go downwards with respect to the order. We prove decidability of the reachability problem, and computability of the backward reachability set. As the main contribution, we identify relevant subclasses where the reachability problem becomes NP-complete. This matches the complexity of the same problem for communication-free vector addition systems (known also as commutative context-free graphs), a special case of stateless multi-pushdown automata.

Partially-Commutative Context-Free Processes

Bisimulation equivalence is decidable in polynomial time for both sequential and commutative normed context-free processes, known as BPA and BPP, respectively. Despite apparent similarity between the two classes, different techniques were used in each case. We provide one polynomial-time algorithm that works in a superclass of both normed BPA and BPP. It is derived in the setting of partially-commutative context-free processes , a new process class introduced in the paper. It subsumes both BPA and BPP and seems to be of independent interest.

Minimization of Tree Pattern Queries

We investigate minimization of tree pattern queries that use the child relation, descendant relation, node labels, and wildcards. We prove that minimization for such tree patterns is Sigma2P-complete and thus solve a problem first attacked by Flesca, Furfaro, and Masciari in 2003. We first provide an example that shows that tree patterns cannot be minimized by deleting nodes. This example shows that the M-NR conjecture, which states that minimality of tree patterns is equivalent to their nonredundancy, is false. We then show how the example can be turned into a gadget that allows us to prove Sigma2P-completeness.

Partially-commutative context-free languages

The paper is about a class of languages that extends context-free languages (CFL) and is stable under shuffle. Specifically, we investigate the class of partially-commutative context-free languages (PCCFL), where non-terminal symbols are commutative according to a binary independence relation, very much like in trace theory. The class has been recently proposed as a robust class subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We identify a natural corresponding automaton model: stateless multi-pushdown automata. We show stability of the class under natural operations, including homomorphic images and shuffle. Finally, we relate expressiveness of PCCFL to two other relevant classes: CFL extended with shuffle and trace-closures of CFL. Among technical contributions of the paper are pumping lemmas, as an elegant completion of known pumping properties of regular languages, CFL and commutative CFL.

Decidability of branching bisimulation on normed commutative context-free processes

We investigate normed commutative context-free processes (Basic Parallel Processes). We show that branching bisimilarity admits the small response property: in the Bisimulation Game, Duplicator always has a response leading to a process of size linearly bounded with respect to the Spoilers process. The linear bound is effective, which leads to decidability of branching bisimilarity. For weak bisimilarity, we are able merely to show existence of some linear bound, which is not sufficient for decidability. We conjecture however that the same effective bound holds for weak bisimilarity as well. We believe that further elaboration of novel techniques developed in this paper may be sufficient to demonstrate decidability.

Regular separability of one counter automata

The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSPACE-completeness, of the regular separability problem for languages of one counter automata without zero tests (also known as one counter nets). This contrasts with undecidability of the regularity problem for one counter nets, and with undecidability of the regular separability problem for one counter automata, which is our second result.