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

On the Complexity of Multi-Pushdown Games

We study the influence of parameters like the number of contexts, phases, and stacks on the complexity of solving parity games over concurrent recursive programs. Our first result shows that k-context games are b-EXPTIME-complete, where b = max{k-2, 1}. This means up to three contexts do not increase the complexity over an analysis for the sequential case. Our second result shows that for ordered k-stack as well as k-phase games the complexity jumps to k-EXPTIME-complete

Module Checking of Pushdown Multi-agent Systems

In this paper, we investigate the module-checking problem of pushdown multi-agent systems (PMS) against ATL and ATL* specifications. We establish that for ATL, module checking of PMS is 2EXPTIME-complete, which is the same complexity as pushdown module-checking for CTL. On the other hand, we show that ATL* module-checking of PMS turns out to be 4EXPTIME-complete, hence exponentially harder than both CTL* pushdown module-checking and ATL* model-checking of PMS. Our result for ATL* provides a rare example of a natural decision problem that is elementary yet but with a complexity that is higher than triply exponential-time

Module checking of pushdown multi-agent systems

In this paper, we investigate the module-checking problem of pushdown multi-agent systems (PMS) against ATL and ATL* specifications. We establish that for ATL, module checking of PMS is 2EXPTIME-complete, which is the same complexity as pushdown module-checking for CTL. On the other hand, we show that ATL* module-checking of PMS turns out to be 4EXPTIME-complete, hence exponentially harder than both CTL* pushdown module-checking and ATL* model-checking of PMS. Our result for ATL* provides a rare example of a natural decision problem that is elementary yet but with a complexity that is higher than triply exponential-time.Comment: arXiv admin note: substantial text overlap with arXiv:1709.0210

Verifying pushdown multi-agent systems against strategy logics

In this paper, we investigate model checking algorithms for variants of strategy logic over pushdown multi-agent systems, modeled by pushdown game structures (PGSs). We consider various fragments of strategy logic, i.e., SL[CG], SL[DG], SL[1G] and BSIL. We show that the model checking problems on PGSs for SL[CG], SL[DG] and SL[1G] are 3EXTIME-complete, which are not harder than the problem for the subsumed logic ATL*. When BSIL is concerned, the model checking problem becomes 2EXPTIME-complete. Our algorithms are automata-theoretic and based on the saturation technique, which are amenable to implementations

Recursion Schemes and Logical Reflection

International audienceLet R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given r in R and phi in L, we say that r_phi is a phi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if u in t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of r that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal

Recursion Schemes and Logical Reflection

International audienceLet R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given r in R and phi in L, we say that r_phi is a phi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if u in t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of r that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal

Constrained Dynamic Tree Networks

We generalise Constrained Dynamic Pushdown Networks, introduced by Bouajjani\et al, to Constrained Dynamic Tree Networks.<br>In this model, we have trees of processes which may monitor their children.<br>We allow the processes to be defined by any computation model for which the alternating reachability problem is decidable.<br>We address the problem of symbolic reachability analysis for this model. More precisely, we consider the problem of computing an effective representation of their reachability<br>sets using finite state automata. <div>We show that backwards reachability sets starting from regular sets of configurations are always regular. </div><div>We provide an algorithm for computing backwards reachability sets using tree automata.<br><br></div

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field