Olivier Serre

Collapsible Pushdown Automata and Recursion Schemes

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 push and pop operations, CPDA have an important operation called collapse, whose effect is to collapse a stack s to the prefix as indicated by the link from the topmost symbol of s. Our first result is that CPDA are equi-expressive with recursion schemes as generators of (possibly infinite) 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 Gt= 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 an abstract syntax graph of the scheme, and hence paths in the tree generated by the scheme. Our equi-expressivity result is the first automata-theoretic characterization of higher-order recursion schemes. Thus CPDA are also a characterization of the simply-typed lambda calculus with recursion (generated from uninterpreted 1st-order symbols) and of (pure) innocent strategies. An important consequence of the equi-expressivity result is that it allows us to reduce decision problems on trees generated by recursion schemes to equivalent problems on CPDA and vice versa. Thus we show, as a consequence of a recent result by Ong (modal mu-calculus model-checking of trees generated by recursion schemes is n-EXPTIME complete), that the problem of solving parity games over the configuration graphs of order-n CPDA is n-EXPTIME complete, subsuming several well-known results about the solvability of games over higher-order pushdown graphs by (respectively) Walukiewicz, Cachat, and Knapik et al. Another contribution of our work is a self-contained proof of the same solvability result by generalizing standard techniques in the field. By appealing to our equi-expressivity result, we obtain a new proof of Ongs result. In contrast to higher-order pushdown graphs, we show that the monadic second-order theories of the configuration graphs of CPDA are undecidable. It follows that -- as generators of graphs -- CPDA are strictly more expressive than higher-order pushdown automata.

A saturation method for collapsible pushdown systems

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.

Recursion Schemes and Logical Reflection

Let R be a class of generators of node-labelled infinite trees, and Lbe a logical language for describing correctness properties of the setrees. Given r in R and phi in L, we say that r_phi is aphi-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 uin 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.

Pushdown module checking with imperfect information

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 paper, 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 [emailxa0protected]?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 @m-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 @m-calculus, and 3Exptime-complete for CTL^@?.

Collapsible Pushdown Automata and Labeled Recursion Schemes: Equivalence, Safety and Effective Selection

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.

C-SHORe: a collapsible approach to higher-order verification

Higher-order recursion schemes (HORS) have recently received much attention as a useful abstraction of higher-order functional programs with a number of new verification techniques employing HORS model-checking as their centrepiece. This paper contributes to the ongoing quest for a truly scalable model-checker for HORS by offering a different, automata theoretic perspective. We introduce the first practical model-checking algorithm that acts on a generalisation of pushdown automata equi-expressive with HORS called collapsible pushdown systems (CPDS). At its core is a substantial modification of a recently studied saturation algorithm for CPDS. In particular it is able to use information gathered from an approximate forward reachability analysis to guide its backward search. Moreover, we introduce an algorithm that prunes the CPDS prior to model-checking and a method for extracting counter-examples in negative instances. We compare our tool with the state-of-the-art verification tools for HORS and obtain encouraging results. In contrast to some of the main competition tackling the same problem, our algorithm is fixed-parameter tractable, and we also offer significantly improved performance over the only previously published tool of which we are aware that also enjoys this property. The tool and additional material are available from http://cshore.cs.rhul.ac.uk.

Randomization in Automata on Infinite Trees

We study finite automata running over infinite binary trees. A run of such an automaton over an input tree is a tree labeled by control states of the automaton: the labeling is built in a top-down fashion and should be consistent with the transitions of the automaton. A branch in a run is accepting if the ω-word obtained by reading the states along the branch satisfies some acceptance condition (typically an ω-regular condition such as a Büchi or a parity condition). Finally, a tree is accepted by the automaton if there exists a run over this tree in which every branch is accepting.n In this article, we consider two relaxations of this definition, introducing a qualitative aspect. First, we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of nonaccepting 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, qualitative tree languages. This class enjoys many good properties: closure under union and intersection (but not under complement), and emptiness is decidable in polynomial time. A dual class, positive tree languages, is defined by requiring that an accepting run contains a non-negligeable set of branches.n The second relaxation is to replace the existential quantification (a tree is accepted if there exists some accepting run over the input tree) with a probabilistic quantification (a tree is accepted if almost every run over the input tree is accepting). For the run, we may use either classical acceptance or qualitative acceptance. In particular, for the latter, we exhibit a tight connection with partial observation Markov decision processes. Moreover, if we additionally restrict operation to the Büchi condition, we show that it leads to a class of probabilistic automata on infinite trees enjoying a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees.

Counting branches in trees using games

We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting. In this article, we relax the notion of accepting run by allowing a certain quantity of rejecting branches. More precisely we study the following criteria for a run to be accepting:(i)it contains at most finitely (resp. countably) many rejecting branches;(ii)it contains infinitely (resp. uncountably) many accepting branches;(iii)the set of accepting branches is topologically big. In all situations we provide a simple acceptance game that later permits to prove that the languages accepted by automata with cardinality constraints are always ω-regular. In the case (ii) where one counts accepting branches it leads to new proofs (without appealing to logic) of a result of Beauquier and Niwinski.

Parity games on undirected graphs

We examine the complexity of solving parity games in the special case when the underlying game graph is undirected. For strictly alternating games, that is, when the game graph is bipartite between the players, we observe that the solution can be computed in linear time. In contrast, when the assumption of strict alternation is dropped, we show that the problem is as hard in the undirected case as it is in the general, directed, case.

Pure Strategies in Imperfect Information Stochastic Games

We consider imperfect information stochastic games where we require the players to use pure (i.e. non randomised) strategies. We consider reachability, safety, Buchi and co-Buchi objectives, and investigate the existence of almost-sure/positively winning strategies for the first player when the second player is perfectly informed or more informed than the first player. We obtain decidability results for positive reachability and almost-sure Buchi with optimal algorithms to decide existence of a pure winning strategy and to compute one if exists. We complete the picture by showing that positive safety is undecidable when restricting to pure strategies even if the second player is perfectly informed.