The separation problem for regular languages by piecewise testable languages

Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. We present and discuss the separation problem for regular languages. We then give a direct polynomial time algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a $B{\Sigma}1(<)$ sentence can witness that the languages are indeed disjoint. The proof is a reformulation and a refinement of an algebraic argument already given by Almeida and the second author

On external presentations of infinite graphs

The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems

How Much Lookahead is Needed to Win Infinite Games?

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. For $\omega$-regular winning conditions it is known that such games can be solved in doubly-exponential time and that doubly-exponential lookahead is sufficient. We improve upon both results by giving an exponential time algorithm and an exponential upper bound on the necessary lookahead. This is complemented by showing EXPTIME-hardness of the solution problem and tight exponential lower bounds on the lookahead. Both lower bounds already hold for safety conditions. Furthermore, solving delay games with reachability conditions is shown to be PSPACE-complete. This is a corrected version of the paper https://arxiv.org/abs/1412.3701v4 published originally on August 26, 2016

Concurrency Makes Simple Theories Hard

A standard way of building concurrent systems is by composing several individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like Hennessy-Milner (HM) logic and its extension with the reachability operator (EF-logic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EF-logic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by Feferman-Vaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems

Remarks on Parikh-recognizable omega-languages

Several variants of Parikh automata on infinite words were recently introduced by Guha et al. [FSTTCS, 2022]. We show that one of these variants coincides with blind counter machine as introduced by Fernau and Stiebe [Fundamenta Informaticae, 2008]. Fernau and Stiebe showed that every $\omega$-language recognized by a blind counter machine is of the form $\bigcup_iU_iV_i^\omega$ for Parikh recognizable languages $U_i, V_i$, but blind counter machines fall short of characterizing this class of $\omega$-languages. They posed as an open problem to find a suitable automata-based characterization. We introduce several additional variants of Parikh automata on infinite words that yield automata characterizations of classes of $\omega$-language of the form $\bigcup_iU_iV_i^\omega$ for all combinations of languages $U_i, V_i$ being regular or Parikh-recognizable. When both $U_i$ and $V_i$ are regular, this coincides with B\"uchi's classical theorem. We study the effect of $\varepsilon$-transitions in all variants of Parikh automata and show that almost all of them admit $\varepsilon$-elimination. Finally we study the classical decision problems with applications to model checking.Comment: arXiv admin note: text overlap with arXiv:2302.04087, arXiv:2301.0896

Solving Infinite Games in the Baire Space

Infinite games (in the form of Gale-Stewart games) are studied where a play is a sequence of natural numbers chosen by two players in alternation, the winning condition being a subset of the Baire space $\omega^\omega$. We consider such games defined by a natural kind of parity automata over the alphabet $\mathbb{N}$, called $\mathbb{N}$-MSO-automata, where transitions are specified by monadic second-order formulas over the successor structure of the natural numbers. We show that the classical B\"uchi-Landweber Theorem (for finite-state games in the Cantor space $2^\omega$) holds again for the present games: A game defined by a deterministic parity $\mathbb{N}$-MSO-automaton is determined, the winner can be computed, and an $\mathbb{N}$-MSO-transducer realizing a winning strategy for the winner can be constructed.Comment: Minor revision. 26 pages, 1 figur

Forward Analysis and Model Checking for Trace Bounded WSTS

We investigate a subclass of well-structured transition systems (WSTS), the bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth. Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all $\omega$-regular properties on the set of infinite traces of the system