We extend the propositional dynamic logic PDL of Fischer and Ladner with a restricted kind of recursive programs using the formalism of visibly pushdown automata (Alur, Madhusudan 2004). We show that the satisfiability problem for this extension remains decidable, generalising known decidability results for extensions of PDL by non-regular programs. Our decision procedure establishes a 2-ExpTime upper complexity bound, and we prove a matching lower bound that applies already to rather weak extensions of PDL with non-regular programs. Thus, we also show that such extensions tend to be more complex than standard PDL.
We first consider infinite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas  have presented a winning decidable condition that is Sigma_3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary finite Borel complexity. From this family, we deduce a family of decidable winning conditions of arbitrary finite Borel complexity for games played on finite graphs. The problem of deciding the winner for these conditions is shown to be non-elementary.
Determining for a given deterministic complete automaton the sequence of visited states while reading a given word is the core of important problems with automata-based solutions, such as approximate string matching. The main difficulty is to do this computation efficiently. Considering words as vectors and working on them using vectorial operations allows to solve the problem faster than using local operations. In this paper, we show first that the set of vectorial operations needed by an algorithm representing a given automaton depends on the language accepted by the automaton. We give precise characterizations for star-free, solvable and regular languages using vectorial algorithms. We also study classes of languages associated with restricted sets of vectorial operations and relate them with languages defined by fragments of linear temporal logic. Finally, we consider the converse problem of constructing an automaton from a given vectorial algorithm. As a byproduct, we show that the satisfiability problem for some extensions of LTL characterizing solvable and regular languages is PSPACE-complete.
We consider infinite two-player games on pushdown graphs. For parity winning conditions, we show that the set of winning positions of each player is regular and we give an effective construction of an alternating automaton recognizing it. This provides a DEXPTIME procedure to decide whether a position is winning for a given player. Finally, using the same methods, we show, for any É-regular winning condition, that the set of winning positions for a given player is regular and effective.
Communications avec actes
Christopher Broadbent, Arnaud Carayol, Matthew Hague et Olivier Serre. A Saturation Method for Collapsible Pushdown Systems, in Proceedings of Automata, Languages and Programming, 39th International Colloquium, ICALP 2012, Springer, Lecture Notes in Computer Science, pp.33 pages, 2012.
We introduce a natural extension of collapsible pushdown systems called annotated pushdown systems that replaces collapse links with stack annotations. We believe this new model has many advantages. We present a saturation method for global backwards reachability analysis of these models that can also be used to analyse collapsible pushdown systems. Beginning with an automaton representing a set of configurations, we build an automaton accepting all configurations that can reach this set. We also improve upon previous saturation techniques for higher-order pushdown systems by significantly reducing the size of the automaton constructed and simplifying the algorithm and proofs.
Higher-order recursion schemes are rewriting systems for simply typed terms and they are known to be equi-expressive with collapsible pushdown automata (CPDA) for generating trees. We argue that CPDA are an essential model when working with recursion schemes. First, we give a new proof of the translation of schemes into CPDA that does not appeal to game semantics. Second, we show that this translation permits to revisit the safety constraint and allows CPDA to be seen as Krivine machines. Finally, we show that CPDA permit one to prove the effective MSO selection property for schemes, subsuming all known decidability results for MSO on schemes.
Arnaud Carayol, Axel Haddad et Olivier Serre. Qualitative Tree Languages, in Twenty-Sixth Annual IEEE Symposium on Logic in Computer Science (LICS 2011), pp.13-22, 2011.
We study finite automata running over infinite binary trees and we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of non-accepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, called the qualitative tree languages that enjoys many properties. Then, we replace the existential quantification --a tree is accepted if there exists some accepting run over the input tree-- by a probabilistic quantification --a tree is accepted if almost every run over the input tree is accepting. Together with the qualitative acceptance and the Büchi condition, we obtain a class of probabilistic tree automata with a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees.
Christopher Broadbent, Arnaud Carayol, Luke Ong et Olivier Serre. Recursion Schemes and Logical Reflection, in Twenty-Fifth Annual IEEE Symposium on Logic in Computer Science (LICS 2010), pp.120-129, 2010.
Let 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.
We study a model of games that combines concurrency, imperfect information and stochastic aspects. Those are finite states games in which, at each round, the two players choose, simultaneously and independently, an action. Then a successor state is chosen accordingly to some fixed probability distribution depending on the previous state and on the pair of actions chosen by the players. Imperfect information is modeled as follows: both players have an equivalence relation over states and, instead of observing the exact state, they only know to which equivalence class it belongs. Therefore, if two partial plays are indistinguishable by some player, he should behave the same in both of them. We consider reachability (does the play eventually visit a final state?) and Büchi objective (does the play visit infinitely often a final state?). Our main contribution is to prove that the following problem is complete for 2-ExpTime: decide whether the first player has a strategy that ensures her to almost-surely win against any possible strategy of her oponent. We also characterise those strategies needed by the first player to almost-surely win.
Benjamin Aminof, Axel Legay, Aniello Murano et Olivier Serre. μ-Calculus Pushdown Module Checking with Imperfect State Information, in Proceedings of the Fifth IFIP International Conference On Theoretical Computer Science - TCS 2008, IFIP 20th World Computer Congress, TC1, Foundations of Computer Science 273, Springer, IFIP, pp.333-348, 2008.
The model checking problem for open systems (module checking ) has recently been the sub ject of extensive study. The prob- lem was ﬁrst studied by Kupferman, Vardi, and Wolper for ﬁnite-state systems and properties expressed in the branching time logics CTL and CTL∗ . Further study continued mainly in two directions: considering systems equipped with a pushdown store, and considering environments with imperfect information about the system. A recent paper combined the two directions and considered the CTL pushdown module checking problem in the imperfect information set- ting, i.e., in the case where the environment has only a partial view of the system control states and pushdown store content. It has been shown that this problem is undecidable when the environment has imperfect information about the pushdown store content, while it is decidable and 2Exptime-complete when the imperfect information only concerns con- trol states. It was left open whether the latter remains decidable also for more expressive logics. In this paper, we answer this question in the aﬃrmative, showing that the pushdown module checking problem with imperfect information about the control states is decidable and 2Exptime-complete for the propositional and the graded µ-calculus, and 3Exptime-complete for CTL∗ .
Arnaud Carayol, Matthew Hague, Antoine Meyer, Luke Ong et Olivier Serre. Winning regions of higher-order pushdown games, in Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, IEEE Computer Society, pp.193-204, 2008.
In 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.
Blaise Genest, Anca Muscholl, Olivier Serre et Marc Zeitoun. Tree Pattern Rewriting Systems, in Proceedings ATVA'08 5311, Lecture Notes in Computer Science, pp.332-346, 2008.
Collapsible pushdown automata (CPDA) are a new kind of higher-order pushdown automata in which every symbol in the stack has a link to a stack situated somewhere below it. In addition to the higher-order stack operations pushi and pop_i , CPDA have an important operation called c ollapse, whose effect is to “collapse” a stack s to the preﬁx as indicated by the link from the t op 1 -symbol of s. Our ﬁrst result is that CPDA are equi-expressive with recursion schemes as generators of node-labelled ranked trees. In one direction, we give a simple algorithm that transforms an order-n CPDA to an order-n recursion scheme that generates the same tree, uniformly for all n ≥ 0. In the other direction, using ideas from game semantics, we give an effective transformation of order-n recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) to order-n CPDA that compute traversals over a certain ﬁnite graph determined by the scheme, and hence paths in the tree generated by the scheme. Our equi-expressivity result is the ﬁrst such automata-theoretic characterization of (general) recursion schemes. An important consequence of the equi-expressivity result is that it allows us to translate decision problems on trees generated by recursion schemes to equivalent problems on CPDA and vice versa. For example, since the Modal Mu-Calculus Model-Checking Problem for trees generated by order-n recursion schemes is n-EXPTIME complete, we show that it follows that the same decidability result holds for the problem of solving a parity game played on an order-n collapsible pushdown graph i.e. the conﬁguration graph of a corresponding (order-n) collapsible pushdown system; the latter subsumes several well-known results about the solvability of games over (higher-order) pushdown graphs by (respectively) Walukiewicz, Cachat, and Knapik et al. Moreover our approach yields techniques that are radically different from standard methods for solving model-checking problems on inﬁnite graphs generated by ﬁnite machines. This transfer of techniques goes both ways. Another innovation of our work is a self-contained proof of the solvability of parity games on collapsible pushdown graphs by generalizing standard techniques in the ﬁeld. By appealing to our equi-expressivity result, we obtain a new proof of the decidability (and optimal complexity) of the Modal Mu-Calculus Model-Checking Problem of trees generated by recursion schemes. In contrast to higher-order pushdown graphs, we show that Monadic Second-Order (MSO) theories of collapsible pushdown graphs are undecidable. Hence collapsible pushdown graphs are, to our knowledge, the ﬁrst example of a natural class of ﬁnitely-presentable graphs that have undecidable MSO theories while enjoying decidable modal mu-calculus theories.
Vince Bárány, Christof Loeding et Olivier Serre. Regularity Problems for Visibly Pushdown Languages, in Proceedings of 23rd International Symposium on Theoretical Aspects of Computer Science, STACS 2006, Marseille, France, February 23-25, 2006, Springer Verlag, LNCS, pp.420-431, 2006.
Visibly pushdown automata are special pushdown automata whose stack behavior is driven by the input symbols according to a partition of the alphabet. We show that it is decidable for a given visibly pushdown automaton whether it is equivalent to a visibly counter automaton, i.e. an automaton that uses its stack only as counter. In particular, this allows to decide whether a given visibly pushdown language is a regular restriction of the set of well-matched words, meaning that the language can be accepted by a finite automaton if only well-matched words are considered as input.
Christof Loeding et Olivier Serre. Propositional Dynamic Logic with Recursive Programs, in Proceedings of FOSSACS 2006: Foundations of Software Science and Computation Structures, Vienna, Austria, March 25 - 31, 2006, Springer Verlag, LNCS 3921, pp.292-306, 2006.
We extend the propositional dynamic logic PDL of Fischer and Ladner with a restricted kind of recursive programs using the formalism of visibly pushdown automata (Alur, Madhusudan 2004). We show that the satisfiability problem for this extension remains decidable, generalising known decidability results for extensions of PDL by non-regular programs.
We consider parity games played on special pushdown graphs, namely those generated by one-counter processes. For parity games on pushdown graphs, it is known from [Wal96] that deciding the winner is an Exptime-complete problem. An important corollary of this result is that the mu-calculus model checking problem for pushdown processes is Exptime-complete. As one-counter processes are special cases of pushdown processes, it follows that deciding the winner in a parity game played on the transition graph of a one-counter process can be achieved in Exptime. Nevertheless the proof for the Exptime-hardness lower bound of [Wal] cannot be adapted to that case. Therefore, a natural question is whether the Exptime upper bound can be improved in this special case. In this paper, we adapt techniques from [KV00,Cac02c] and provide a Pspace upper bound and a DP-hard lower bound for this problem. We also give two important consequences of this result. First, we improve the best upper bound known for model-checking one-counter processes against mu-calculus. Second, we show how these games can be used to solve pushdown games with winning conditions that are Boolean combinations of a parity condition on the control states with conditions on the stack height.
Christof Loeding, Parthasarathy Madhusudan et Olivier Serre. Visibly Pushdown Games, in Proceedings of FST TCS 2004: Foundations of Software Technology and Theoretical Computer Science, 24th Conference, Chennai, India, December 16-18 2004, Springer Verlag, Volume 3328 of Lecture Notes in Computer Sciences, pp.408-420, 2004.
The class of visibly pushdown languages has been recently defined as a subclass of context-free languages with desirable closure properties and tractable decision problems. We study visibly pushdown games, which are games played on visibly pushdown systems where the winning condition is given by a visibly pushdown language. We establish that, unlike pushdown games with pushdown winning conditions, visibly pushdown games are decidable and are 2Exptime-complete. We also show that pushdown games against Ltl specifications and Caret speciØcations are 3Exptime-complete. Finally, we establish the topological complexity of visibly pushdown languages by showing that they are a subclass of Boolean combinations of Sigma_3 sets. This leads to an alternative proof that visibly pushdown automata are not determinizable and also shows that visibly pushdown games are determined.
Olivier Serre. Games with Winning Conditions of High Borel Complexity, in Proceedings of Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004, Springer Verlag, Volume 3142 of Lecture Notes in Computer Sciences, pp.1150-1162, 2004.
We first consider in nite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas  have presented a winning decidable condition that is SIgma_3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary high finite Borel complexity. From this family, we deduce a family of decidable winning conditions of arbitrary finite Borel complexity for games played on finite graphs. The problem of deciding the winner for these winning conditions is shown to be non-elementary complete.
Alexis-Julien Bouquet, Olivier Serre et Igor Walukiewicz. Pushdown games with unboundedness and regular conditions, in Proceedings of FST TCS 2003: Foundations of Software Technology and Theoretical Computer Science, 23rd Conference, Mumbai, India, December 15-17 2003, Springer Verlag, Volume 2914 of Lecture Notes in Computer Sciences, pages 88-99, pp.88-99, 2003.
We consider infinitary two-player perfect information games defined over graphs of configurations of a pushdown automaton.We show how to solve such games when winning conditions are Boolean combinations of a Buchi condition and a new condition that we call unboundedness. An infinite play satisfies the unboundedness condition if there is no bound on the size of the stack during the play. We show that the problem of deciding a winner in such games is EXPTIME-complete.
Olivier Serre. Vectorial Languages and Linear Temporal Logic, in Proceedings of the 2nd IFIP International Conference Theoretical Computer Science (TCS@2002), Aug. 2002, Montreal, Canada TCS'2002, Kluwer, volume 223 of IFIP Conference Proceedings, pp.576-587, 2002.
Determining for a given deterministic complete automaton the sequence of visited states while reading a given word is the core of important problems with automata-based solutions, such as approximate string matching. The main difficulty is to do this computation efficiently, especially when dealing with very large texts. Considering words as vectors and working on them using vectorial (parallel) operations allows to solve the problem faster than in linear time using sequential computations. In this paper, we show first that the set of vectorial operations needed by an algorithm representing a given automaton depends only on the language accepted by the automaton. We give precise characterizations of vectorial algorithms for star-free, solvable and regular languages in terms of the vectorial operations allowed. We also consider classes of languages associated with restricted sets of vectorial operations and relate them with languages defined by fragments of linear temporal logic. Finally, we consider the converse problem of constructing an automaton from a given vectorial algorithm. As a byproduct, we show that the satisfiability problem for some extensions of linear-time temporal logic characterizing solvable and regular languages is PSPACE-complete.
Les jeux à deux joueurs sur des graphes finis ou infinis permettent de modéliser de nombreux problèmes liés à la vérification des systèmes. Le système spécifié dépend de la nature du graphe de jeu considéré tandis que la propriété à vérifier est décrite par la condition de gain. Le premier joueur, Eve, représente un programme qui évolue dans un environnement hostile représenté par le second joueur, Adam. Dans ce formalisme, Eve possède une stratégie gagnante si et seulement si le programme peut être contrôlé de sorte à satisfaire la propriété spécifiée par la condition de gain. On souhaite alors décider si Eve possède une stratégie gagnante et si oui la déterminer, afin de synthétiser ensuite un contrôleur. Dans cette thèse, les graphes de jeu considérés sont des graphes de processus à pile qui offrent une représentation finie simple de systèmes infinis relativement complexes. Sur de tels graphes, on peut considérer des conditions de gain classiques (accessibilité, Büchi ou parité) mais aussi des conditions plus spécifiques au modèle comme celles portant sur le bornage de la pile. On peut également combiner ces dernières entre elles. Une première contribution a été de fournir une représentation des ensembles de positions gagnantes pour les jeux de parité ainsi qu'une nouvelle présentation des résultats connus pour ces derniers. On a alors pu étendre de façon naturelle les techniques de preuves à d'autres conditions de gain, notamment à celles portant sur le bornage de la pile. Une autre contribution a été la description d'une famille de conditions de gain de complexité borélienne arbitraire finie pour lesquelles les jeux (sur des graphes finis ou sur des graphes de processus à pile) restent décidables. L'étude des jeux sur les graphes de BPA et sur les graphes de processus à compteur a permis de proposer des techniques propres à ces modèles qui fournissent alors des bornes de complexité meilleures que celles obtenues dans le cas général des graphes de processus à pile. Enfin, une dernière contribution a été de proposer une solution pour les jeux sur des graphes de processus à pile munis de conditions combinant des conditions régulières et des conditions sur la hauteur de pile et pour des conditions décrites par des automates à pile avec visibilité.
We focus on the emptiness problem for alternating parity tree automata. The usual technique to tackle this problem first removes alternation, going to non-determinism, and then checks emptiness by reduction to a two-player perfect-information parity game. In this note, we give an alternative roadmap to this problem by providing a direct reduction to the emptiness problem to solving an imperfect-information two-player parity game.
Preprint, Working Paper, Document sans référence, etc.
The model checking problem for finite-state open systems (module checking) has been extensively studied in the literature, both in the context of environments with perfect and imperfect information about the system. Recently, the perfect information case has been extended to infinite-state systems (pushdown module checking). In this part, we extend pushdown module checking to the imperfect information setting; i.e., to the case where the environment has only a partial view of the system's control states and pushdown store content. We study the complexity of this problem with respect to the branching-time temporal logics CTL, CTL* and the propositional mu-calculus. We show that pushdown module checking, which is by itself harder than pushdown model checking, becomes undecidable when the environment has imperfect information. We also show that undecidability relies on hiding information about the pushdown store. Indeed, we prove that with imperfect information about the control states, but a visible pushdown store, the problem is decidable and its complexity is 2ExpTime-complete for CTL and the propositional mu-calculus, and 3ExpTime-complete for CTL*.