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24
MSO decidability of MultiPushdown Systems via SplitWidth
, 2012
"... Multithreaded programs with recursion are naturally modeled as multipushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW ca ..."
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Multithreaded programs with recursion are naturally modeled as multipushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW can be decomposed by two basic and natural operations: shuffle of two sequences of factors and merge of consecutive factors of a sequence. We say that the splitwidth of an MNW is k if it admits a decomposition where the number of factors in each sequence is at most k. The MSO theory of MNWs with splitwidth k is decidable. We introduce two very general classes of MNWs that strictly generalize known decidable classes and prove their MSO decidability via their splitwidth and obtain comparable or better bounds of treewidth of known classes.
Scopebounded Multistack Pushdown Systems: FixedPoint, Sequentialization, and TreeWidth
"... Abstract. Wepresentanovelfixedpointalgorithmtosolvereachability of multistack pushdown systems restricted to runs of boundedscope. The followed approach is compositional, in the sense that the runs of the system are summarized by boundedsize interfaces. Moreover, it is suitable for a direct impl ..."
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Abstract. Wepresentanovelfixedpointalgorithmtosolvereachability of multistack pushdown systems restricted to runs of boundedscope. The followed approach is compositional, in the sense that the runs of the system are summarized by boundedsize 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 multistack pushdown system we construct an equivalent singlestack pushdown system that faithfully simulates the behaviour of each thread. We prove that the behaviour graphs (multiply nested words) for these systems have bounded threewidth, and thus a number of decidability results can be derived from Courcelle’s theorem. 1
Temporal logics for concurrent recursive programs: Satisfiability and model checking
 In MFCS’11, volume 6907 of LNCS
, 2011
"... Abstract. We develop a general framework for the design of temporal logics for concurrent recursive programs. A program execution is modeled as a partial order with multiple nesting relations. To specify properties of executions, we consider any temporal logic whose modalities aredefinable in monadi ..."
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Cited by 9 (3 self)
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Abstract. We develop a general framework for the design of temporal logics for concurrent recursive programs. A program execution is modeled as a partial order with multiple nesting relations. To specify properties of executions, we consider any temporal logic whose modalities aredefinable in monadic secondorder logic and that, in addition, allows PDLlike path expressions. This captures, in a unifying framework, a wide range of logics defined for ranked and unranked trees, nested words, and Mazurkiewicz traces that have been studied separately. We show that satisfiability and model checking are decidable in EXPTIME and 2EXPTIME, depending on the precise path modalities. 1
Model Checking Languages of Data Words
"... We consider the modelchecking problem for data multipushdown automata (DMPA). DMPA generate data words, i.e, strings enriched with values from an infinite domain. The latter can be used to represent an unbounded number of process identifiers so that DMPA are suitable to model concurrent programs ..."
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We consider the modelchecking problem for data multipushdown automata (DMPA). DMPA generate data words, i.e, strings enriched with values from an infinite domain. The latter can be used to represent an unbounded number of process identifiers so that DMPA are suitable to model concurrent programs with dynamic process creation. To specify properties of data words, we use monadic secondorder (MSO) logic, which comes with a predicate to test two word positions for data equality. While satisfiability for MSO logic is undecidable (even for weaker fragments such as firstorder logic), our main result states that one can decide if all words generated by a DMPA satisfy a given formula from the full MSO logic.
Saturation of Concurrent Collapsible Pushdown Systems
"... Multistack pushdown systems are a wellstudied model of concurrent computation using threads with firstorder procedure calls. While, in general, reachability is undecidable, there are numerous restrictions on stack behaviour that lead to decidability. To model higherorder procedures calls, a gene ..."
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Cited by 5 (2 self)
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Multistack pushdown systems are a wellstudied model of concurrent computation using threads with firstorder procedure calls. While, in general, reachability is undecidable, there are numerous restrictions on stack behaviour that lead to decidability. To model higherorder procedures calls, a generalisation of pushdown stacks called collapsible pushdown stacks are required. Reachability problems for multistack collapsible pushdown systems have been little studied. Here, we study ordered, phasebounded and scopebounded multistack collapsible pushdown systems using saturation techniques, showing decidability of control state reachability and giving a regular representation of all configurations that can reach a given control state.
The complexity of modelchecking multistack systems
, 2012
"... Abstract—We consider the lineartime model checking problem for boolean concurrent programs with recursive procedure calls. While sequential recursive programs are usually modeled as pushdown automata, concurrent recursive programs involve several processes and can be naturally abstracted as pushdo ..."
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Cited by 4 (0 self)
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Abstract—We consider the lineartime model checking problem for boolean concurrent programs with recursive procedure calls. While sequential recursive programs are usually modeled as pushdown automata, concurrent recursive programs involve several processes and can be naturally abstracted as pushdown automata with multiple stacks. Their behavior can be understood as words with multiple nesting relations, each relation connecting a procedure call with its corresponding return. To reason about multiply nested words, we consider the class of all temporal logics as defined in the book by Gabbay, Hodkinson, and Reynolds (1994). The unifying feature of these temporal logics is that their modalities are defined in monadic secondorder (MSO) logic. In particular, this captures numerous temporal logics over concurrent and/or recursive programs that have been defined so far. Since the general model checking problem is undecidable, we restrict attention to phase bounded executions as proposed by La Torre, Madhusudan, and Parlato (LICS 2007). While the MSO model checking problem in this case is nonelementary, our main result states that the model checking (and satisfiability) problem for all these temporal logics is decidable in elementary time. More precisely, it is solvable in (n + 2)EXPTIME where n is the maximal level of the MSO modalities in the monadic quantifier alternation hierarchy. We complement this result and provide, for each level n, a temporal logic whose model checking problem is nEXPSPACEhard. I.
ScopeBounded Pushdown Languages
"... Abstract. We study the formal language theory of multistack pushdown automata (Mpa) restricted to computations where a symbol can be popped from a stack S only if it was pushed within a bounded number of contexts of S (scoped Mpa). We contribute to show that scoped Mpa are indeed a robust model of ..."
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Cited by 3 (1 self)
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Abstract. We study the formal language theory of multistack pushdown automata (Mpa) restricted to computations where a symbol can be popped from a stack S only if it was pushed within a bounded number of contexts of S (scoped Mpa). We contribute to show that scoped Mpa are indeed a robust model of computation, by focusing on the corresponding theory of visibly Mpa (Mvpa). We prove the equivalence of the deterministic and nondeterministic versions and show that scopebounded computations of an nstack Mvpa can be simulated, rearranging the input word, by using only one stack. These results have several interesting consequences, such as, the closure under complement, the decidability of universality, inclusion and equality, and a Parikh theorem. We also give a logical characterization and compare the expressiveness of the scopebounded restriction with Mvpa classes from the literature. 1
A Unifying Approach for Multistack Pushdown Automata (Track B)
, 2014
"... We give a general approach to show the closure under complement and decide the emptiness for many classes of multistack visibly pushdown automata (Mvpa). A central notion in our approach is the visibly pathtree, i.e., a stack tree with the encoding of a path that denotes a linear ordering of the no ..."
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We give a general approach to show the closure under complement and decide the emptiness for many classes of multistack visibly pushdown automata (Mvpa). A central notion in our approach is the visibly pathtree, i.e., a stack tree with the encoding of a path that denotes a linear ordering of the nodes. We show that the set of all such trees with a bounded size labeling is regular, and pathtrees allow us to design simple conversions between tree automata and Mvpa’s. As corollaries of our results we get the closure under complement of ordered Mvpa that was an open problem, and a better upper bound on the algorithm to check the emptiness of boundedphase Mvpa’s, that also shows that this problem is fixed parameter tractable in the number of phases.
Verifying Communicating Multipushdown Systems
"... Abstract. Communicating multipushdown systems model networks of multithreaded recursive programs communicating via reliable FIFO channels. Hence their verification problems are undecidable in general. The behaviours of these systems can be represented as directed graphs, which subsume both Message ..."
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Abstract. Communicating multipushdown systems model networks of multithreaded recursive programs communicating via reliable FIFO channels. Hence their verification problems are undecidable in general. The behaviours of these systems can be represented as directed graphs, which subsume both Message Sequence Charts and nested words. We extend the notion of splitwidth [8] to these graphs, defining a simple algebra to compose/decompose these behaviours using two natural operations: shuffle and merge. We obtain simple, uniform and optimal decision procedures for various verification problems parametrized by splitwidth, ranging from reachability to modelchecking against MSO, PDL and Temporal Logics. 1
WeaklySynchronized Ground Tree Rewriting (with applications to verifying multithreaded programs)
"... Abstract. Ground tree rewrite systems (GTRS) are a wellknown treeextension of prefixrewrite systems on words (a.k.a. pushdown systems), where subtrees (instead of word prefixes) are rewritten. GTRS can model programs withunboundedrecursion depthandthreadspawning,wherein the threads have a treesh ..."
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Abstract. Ground tree rewrite systems (GTRS) are a wellknown treeextension of prefixrewrite systems on words (a.k.a. pushdown systems), where subtrees (instead of word prefixes) are rewritten. GTRS can model programs withunboundedrecursion depthandthreadspawning,wherein the threads have a treeshaped dependency graph. We consider the extension of GTRS with a finite (global) control unit for synchronizing among the active threads, a.k.a. stateextended GTRS (sGTRS). Since sGTRS is Turingcomplete, we restrict the finite control unit to dags possibly with selfloops, a.k.a. weaklysynchronized GTRS (wGTRS). wGTRS can be regarded as a generalization of contextbounded analysis of multipushdown systems with dynamic thread spawning. We show that reachability, repeated reachability, and the complement of model checking deterministic LTL over weaklysynchronized GTRS (wGTRS) are NPcomplete by a polynomial reduction to checking existential Presburger formulas, for which highly optimized solvers are available. 1