Unified Analysis of Collapsible and Ordered Pushdown Automata via Term Rewriting

We model collapsible and ordered pushdown systems with term rewriting, by encoding higher-order stacks and multiple stacks into trees. We show a uniform inverse preservation of recognizability result for the resulting class of term rewriting systems, which is obtained by extending the classic saturation-based approach. This result subsumes and unifies similar analyses on collapsible and ordered pushdown systems. Despite the rich literature on inverse preservation of recognizability for term rewrite systems, our result does not seem to follow from any previous study.Comment: in Proc. of FRE

Scope-bounded multistack pushdown systems: fixed-point, sequentialization, and tree-width

We present a novel fixed-point algorithm to solve reachability of multi-stack pushdown systems restricted to runs of bounded-scope. The followed approach is compositional, in the sense that the runs of the system are summarized by bounded-size interfaces. Moreover, it is suitable for a direct implementation and can be exploited to prove two new results. We give a sequentialization for this class of systems, i.e., for each such multi-stack pushdown system we construct an equivalent single-stack pushdown system that faithfully simulates the behaviour of each thread. We prove that the behaviour graphs (multiply nested words) for these systems have bounded three-width, and thus a number of decidability results can be derived from Courcelle’s theorem

Reachability analysis of first-order definable pushdown systems

We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are first-order definable in a fixed countably-infinite structure. We show that the reachability analysis can be addressed with the well-known saturation technique for the wide class of oligomorphic structures. Moreover, for the more restrictive homogeneous structures, we are able to give concrete complexity upper bounds. We show ample applicability of our technique by presenting several concrete examples of homogeneous structures, subsuming, with optimal complexity, known results from the literature. We show that infinitely many such examples of homogeneous structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1

Model checking Branching-Time Properties of Multi-Pushdown Systems is Hard

We address the model checking problem for shared memory concurrent programs modeled as multi-pushdown systems. We consider here boolean programs with a finite number of threads and recursive procedures. It is well-known that the model checking problem is undecidable for this class of programs. In this paper, we investigate the decidability and the complexity of this problem under the assumption of bounded context-switching defined by Qadeer and Rehof, and of phase-boundedness proposed by La Torre et al. On the model checking of such systems against temporal logics and in particular branching time logics such as the modal $\mu$-calculus or CTL has received little attention. It is known that parity games, which are closely related to the modal $\mu$-calculus, are decidable for the class of bounded-phase systems (and hence for bounded-context switching as well), but with non-elementary complexity (Seth). A natural question is whether this high complexity is inevitable and what are the ways to get around it. This paper addresses these questions and unfortunately, and somewhat surprisingly, it shows that branching model checking for MPDSs is inherently an hard problem with no easy solution. We show that parity games on MPDS under phase-bounding restriction is non-elementary. Our main result shows that model checking a $k$ context bounded MPDS against a simple fragment of CTL, consisting of formulas that whose temporal operators come from the set {\EF, \EX}, has a non-elementary lower bound

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

A non-deterministic recursion scheme recognizes a language of finite trees. This very expressive model can simulate, among others, higher-order pushdown automata with collapse. We show decidability of the diagonal problem for schemes. This result has several interesting consequences. In particular, it gives an algorithm that computes the downward closure of languages of words recognized by schemes. In turn, this has immediate application to separability problems and reachability analysis of concurrent systems.Comment: technical report; to appear in LICS'1

Reachability in Concurrent Uninterpreted Programs

We study the safety verification (reachability problem) for concurrent programs with uninterpreted functions/relations. By extending the notion of coherence, recently identified for sequential programs, to concurrent programs, we show that reachability in coherent concurrent programs under various scheduling restrictions is decidable by a reduction to multistack pushdown automata, and establish precise complexity bounds for them. We also prove that the coherence restriction for these various scheduling restrictions is itself a decidable property

Higher-Order Model Checking Step by Step

We show a new simple algorithm that solves the model-checking problem for recursion schemes: check whether the tree generated by a given higher-order recursion scheme is accepted by a given alternating parity automaton. The algorithm amounts to a procedure that transforms a recursion scheme of order n to a recursion scheme of order n-1, preserving acceptance, and increasing the size only exponentially. After repeating the procedure n times, we obtain a recursion scheme of order 0, for which the problem boils down to solving a finite parity game. Since the size grows exponentially at each step, the overall complexity is n-EXPTIME, which is known to be optimal. More precisely, the transformation is linear in the size of the recursion scheme, assuming that the arity of employed nonterminals and the size of the automaton are bounded by a constant; this results in an FPT algorithm for the model-checking problem. Our transformation is a generalization of a previous transformation of the author (2020), working for reachability automata in place of parity automata. The step-by-step approach can be opposed to previous algorithms solving the considered problem "in one step", being compulsorily more complicated

Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

On the Separability Problem of String Constraints

We address the separability problem for straight-line string constraints. The separability problem for languages of a class C by a class S asks: given two languages A and B in C, does there exist a language I in S separating A and B (i.e., I is a superset of A and disjoint from B)? The separability of string constraints is the same as the fundamental problem of interpolation for string constraints. We first show that regular separability of straight line string constraints is undecidable. Our second result is the decidability of the separability problem for straight-line string constraints by piece-wise testable languages, though the precise complexity is open. In our third result, we consider the positive fragment of piece-wise testable languages as a separator, and obtain an ExpSpace algorithm for the separability of a useful class of straight-line string constraints, and a Pspace-hardness result