Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow $\epsilon$-contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy.Comment: 12 pages Accepted for STACS 201

Reachability in Higher-Order-Counters

Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level $k$ \HOCS with $0$-test is complete for \mbox{$(k-2)$}-fold exponential space; leaving out the $0$-test leads to completeness for \mbox{$(k-2)$}-fold exponential time. Restricting \HOCS (without $0$-test) to level $2$, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level $2$ \HOCS without $0$-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201

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

Visibly Pushdown Modular Games

Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. In this work, we study for the first time modular strategies with respect to winning conditions that can be expressed by a pushdown automaton. We show that such games are undecidable in general, and become decidable for visibly pushdown automata specifications. Our solution relies on a reduction to modular games with finite-state automata winning conditions, which are known in the literature. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal Buchi or co Buchi visibly pushdown winning condition are EXPTIME-complete, and when the winning condition is given by a CARET or NWTL temporal logic formula the problem is 2EXPTIME-complete, and it remains 2EXPTIME-hard even for simple fragments of these logics. As a further contribution, we present a different solution for modular games with finite-state automata winning condition that runs faster than known solutions for large specifications and many exits.Comment: In Proceedings GandALF 2014, arXiv:1408.556

First-Order Model Checking on Generalisations of Pushdown Graphs

We study the first-order model checking problem on two generalisations of pushdown graphs. The first class is the class of nested pushdown trees. The other is the class of collapsible pushdown graphs. Our main results are the following. First-order logic with reachability is uniformly decidable on nested pushdown trees. Considering first-order logic without reachability, we prove decidability in doubly exponential alternating time with linearly many alternations. First-order logic with regular reachability predicates is uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested pushdown trees are first-order interpretable in collapsible pushdown graphs of level 2. This interpretation can be extended to an interpretation of the class of higher-order nested pushdown trees in the collapsible pushdown graph hierarchy. We prove that the second level of this new hierarchy of nested trees has decidable first-order model checking. Our decidability result for collapsible pushdown graph relies on the fact that level 2 collapsible pushdown graphs are uniform tree-automatic. Our last result concerns tree-automatic structures in general. We prove that first-order logic extended by Ramsey quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page

Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611

Winning regions of higher-order pushdown games

International audienceIn this paper we consider parity games defined by higher-order pushdown automata. These automata generalise pushdown automata by the use of higher-order stacks, which are nested ``stack of stacks'' structures. Representing higher-order stacks as well-bracketed words in the usual way, we show that the winning regions of these games are regular sets of words. Moreover a finite automaton recognising this region can be effectively computed. A novelty of our work are abstract pushdown processes which can be seen as (ordinary) pushdown automata but with an infinite stack alphabet. We use the device to give a uniform presentation of our results. From our main result on winning regions of parity games we derive a solution to the Modal Mu-Calculus Global Model-Checking Problem for higher-order pushdown graphs as well as for ranked trees generated by higher-order safe recursion schemes

The Complexity of Model Checking (Collapsible) Higher-Order Pushdown Systems

We study (collapsible) higher-order pushdown systems --- theoretically robust and well-studied models of higher-order programs --- along with their natural subclass called (collapsible) higher-order basic process algebras. We provide a comprehensive analysis of the model checking complexity of a range of both branching-time and linear-time temporal logics. We obtain tight bounds on data, expression, and combined-complexity for both (collapsible) higher-order pushdown systems and (collapsible) higher-order basic process algebra. At order-$k$, results range from polynomial to $(k+1)$-exponential time. Finally, we study (collapsible) higher-order basic process algebras as graph generators and show that they are almost as powerful as (collapsible) higher-order pushdown systems up to MSO interpretations