Showing papers on "Pushdown automaton published in 2017"

Collapsible Pushdown Automata and Recursion Schemes

Abstract: We consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed deterministic term rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this article, we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it, and contracts its silent transitions, which leads to an infinite tree, which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving effective transformations in both directions.

A new framework for the verification of service trust behaviors

Abstract: We propose in this paper a model checking framework for service trust behaviors. We devise a new trust behavior model, which is a deterministic PushDown Automaton (PDA) based trust behavior model. This model is built based on the observations' sequences, which are derived from the interactions with services. Furthermore, we express the regular and non-regular trust behavior properties using Fixed point Logic with Chop (FLC). The model checking of service trust behaviors with respect to trust properties is performed using a symbolic FLC model checking algorithm. Finally, we present some experiments to assess the efficiency of the proposed algorithm.

Model checking parameterized asynchronous shared-memory systems

Abstract: We characterize the complexity of liveness verification for parameterized systems consisting of a leader process and arbitrarily many anonymous and identical contributor processes. Processes communicate through a shared, bounded-value register. While each operation on the register is atomic, there is no synchronization primitive to execute a sequence of operations atomically. We analyze the case in which processes are modeled by finite-state machines or pushdown machines and the property is given by a Buchi automaton over the alphabet of read and write actions of the leader. We show that the problem is decidable, and has a surprisingly low complexity: it is NP-complete when all processes are finite-state machines, and is in NEXPTIME (and PSPACE-hard) when they are pushdown machines. This complexity is lower than for the non-parameterized case: liveness verification of finitely many finite-state machines is PSPACE-complete, and undecidable for two pushdown machines. For finite-state machines, our proofs characterize infinite behaviors using existential abstraction and semilinear constraints. For pushdown machines, we show how contributor computations of high stack height can be simulated by computations of many contributors, each with low stack height. Together, our results characterize the complexity of verification for parameterized systems under the assumptions of anonymity and asynchrony.

On Büchi One-Counter Automata

Abstract: Equivalence of deterministic pushdown automata is a famous problem in theoretical computer science whose decidability has been shown by Senizergues. Our first result shows that decidability no longer holds when moving from finite words to infinite words. This solves an open problem that has recently been raised by Loding. In fact, we show that already the equivalence problem for deterministic Buchi one-counter automata is undecidable. Hence, the decidability border is rather tight when taking into account a recent result by Loding and Repke that equivalence of deterministic weak parity pushdown automata (a subclass of deterministic Buchi pushdown automata) is decidable. Another known result on finite words is that the universality problem for vector addition systems is decidable. We show undecidability when moving to infinite words. In fact, we prove that already the universality problem for nondeterministic Buchi one-counter nets (or equivalently vector addition systems with one unbounded dimension) is undecidable.

Deterministic Stack Transducers

Abstract: We introduce and investigate stack transducers, which are one-way stack automata with an output tape. A one-way stack automaton is a classical pushdown automaton with the additional ability to move ...

The Complexity of Model Checking Multi-Stack Systems

Abstract: We study the linear-time 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 (18). The unifying feature of these temporal logics is that their modalities are defined in monadic second-order (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 24). While the MSO model checking problem in this case is non-elementary, our main result states that the model checking (and satisfiability) problem for all MSO-definable temporal logics is decidable in elementary time. More precisely, it is solvable in time exponential in the formula and (n+2)-fold exponential in the number of phases where n is the maximal level of the MSO modalities in the monadic quantifier alternation hierarchy (which is a vast improvement over the conference version of this paper from LICS 2013 where the space was also (n+2)-fold exponential in the size of the temporal formula). We complement this result and provide, for each level n, a temporal logic whose model checking problem is n-EXPSPACE-hard.

Timed pushdown automata and branching vector addition systems

Abstract: We prove that non-emptiness of timed register pushdown automata is decidable in doubly exponential time. This is a very expressive class of automata, whose transitions may involve state and top-of-stack clocks with unbounded differences. It strictly subsumes pushdown timed automata of Bouajjani et al., dense-timed pushdown automata of Abdulla et al., and orbit-finite timed register pushdown automata of Clemente and Lasota. Along the way, we prove two further decidability results of independent interest: for non-emptiness of least solutions to systems of equations over sets of integers with addition, union and intersections with N and -N, and for reachability in one-dimensional branching vector addition systems with states and subtraction, both in exponential time.

Edit Distance Neighbourhoods of Input-Driven Pushdown Automata

Abstract: Edit distance \(\ell \)-neighbourhood of a formal language is the set of all strings that can be transformed to one of the strings in this language by at most \(\ell \) insertions and deletions. Both the regular and the context-free languages are known to be closed under this operation, whereas the deterministic pushdown automata are not. This paper establishes the closure of the family of input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the edit distance neighbourhood operation. A construction of automata representing the result of the operation is given, and close lower bounds on the size of any such automata are presented.

Minimization of Visibly Pushdown Automata Using Partial Max-SAT

Abstract: We consider the problem of state-space reduction for nondeterministic weakly-hierarchical visibly pushdown automata (Vpa). Vpa recognize a robust and algorithmically tractable fragment of context-free languages that is natural for modeling programs.

The Complexity of Mean-Payoff Pushdown Games

Abstract: Two-player games on graphs are central in many problems in formal verification and program analysis, such as synthesis and verification of open systems. In this work, we consider solving recursive game graphs (or pushdown game graphs) that model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives—such as reachability and ω-regular objectives—in this work, we study for the first time such games with the most well-studied quantitative objective, the mean-payoff objective. In pushdown games, two types of strategies are relevant: (1) global strategies, which depend on the entire global history; and (2) modular strategies, which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module. Our main results are as follows: (1) One-player pushdown games with mean-payoff objectives under global strategies are decidable in polynomial time. (2) Two-player pushdown games with mean-payoff objectives under global strategies are undecidable. (3) One-player pushdown games with mean-payoff objectives under modular strategies are NP-hard. (4) Two-player pushdown games with mean-payoff objectives under modular strategies can be solved in NP (i.e., both one-player and two-player pushdown games with mean-payoff objectives under modular strategies are NP-complete). We also establish the optimal strategy complexity by showing that global strategies for mean-payoff objectives require infinite memory even in one-player pushdown games and memoryless modular strategies are sufficient in two-player pushdown games. Finally, we also show that all the problems have the same complexity if the stack boundedness condition is added, where along with the mean-payoff objective the player must also ensure that the stack height is bounded.

Minimization of Visibly Pushdown Automata Using Partial Max-SAT

Abstract: We consider the problem of state-space reduction for nondeterministic weakly-hierarchical visibly pushdown automata (VPA). VPA recognize a robust and algorithmically tractable fragment of context-free languages that is natural for modeling programs. We define an equivalence relation that is sufficient for language-preserving quotienting of VPA. Our definition allows to merge states that have different behavior, as long as they show the same behavior for reachable equivalent stacks. We encode the existence of such a relation as a Boolean partial maximum satisfiability (PMax-SAT) problem and present an algorithm that quickly finds satisfying assignments. These assignments are sub-optimal solutions to the PMax-SAT problem but can still lead to a significant reduction of states. We integrated our method in the automata-based software verifier Ultimate Automizer and show performance improvements on benchmarks from the software verification competition SV-COMP.

Emptiness of Ordered Multi-Pushdown Automata is 2ETIME-Complete

Abstract: We consider ordered multi-pushdown automata, a multi-stack extension of pushdown automata that comes with a constraint on stack operations: a pop can only be performed on the first non-empty stack (which implies that we assume a linear ordering on the collection of stacks). We show that the emptiness problem for multi-pushdown automata is 2ETIME-complete. Containment in 2ETIME is shown by translating an automaton into a grammar for which we can check if the generated language is empty. The lower bound is established by simulating the behavior of an alternating Turing machine working in exponential space. We also compare ordered multi-pushdown automata with the model of bounded-phase (visibly) multi-stack pushdown automata, which do not impose an ordering on stacks, but restrict the number of alternations of pop operations on different stacks.

Bounded Pushdown Dimension vs Lempel Ziv Information Density

Abstract: In this paper we introduce a variant of pushdown dimension called bounded pushdown (BPD) dimension, that measures the density of information contained in a sequence, relative to a BPD automata, ie a finite state machine equipped with an extra infinite memory stack, with the additional requirement that every input symbol only allows a bounded number of stack movements BPD automata are a natural real-time restriction of pushdown automata We show that BPD dimension is a robust notion by giving an equivalent characterization of BPD dimension in terms of BPD compressors We then study the relationships between BPD compression, and the standard Lempel-Ziv (LZ) compression algorithm, and show that in contrast to the finite-state compressor case, LZ is not universal for bounded pushdown compressors in a strong sense: we construct a sequence that LZ fails to compress significantly, but that is compressed by at least a factor 2 by a BPD compressor As a corollary we obtain a strong separation between finite-state and BPD dimension

The Triple-Pair Construction for Weighted $\omega$-Pushdown Automata

Abstract: Let S be a complete star-omega semiring and Sigma be an alphabet. For a weighted omega-pushdown automaton P with stateset 1...n, n greater or equal to 1, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair ((S >)^nxn, (S >)^n) such that the behavior ||P|| of P is a component of a solution of this system. In case the basic semiring is the Boolean semiring or the semiring of natural numbers (augmented with infinity), we show that there exists a mixed context-free grammar that generates ||P||. The construction of the mixed context-free grammar from P is a generalization of the well known triple construction and is called now triple-pair construction for omega-pushdown automata.

A Kleene Theorem for Weighted ω-Pushdown Automata

Abstract: Weighted ω-pushdown automata were introduced as generalization of the classical pushdown automata accepting infinite words by Büchi acceptance. The main result in the proof of the Kleene Theorem is the construction of a weighted ω-pushdown automaton for the ω-algebraic closure of subsets of a continuous star-omega semiring.

A Parameter-Free Learning Automaton Scheme

Abstract: For a learning automaton, a proper configuration of its learning parameters, which are crucial for the automaton's performance, is relatively difficult due to the necessity of a manual parameter tuning before real applications. To ensure a stable and reliable performance in stochastic environments, parameter tuning can be a time-consuming and interaction-costing procedure in the field of LA. Especially, it is a fatal limitation for LA-based applications where the interactions with environments are expensive. In this paper, we propose a parameter-free learning automaton scheme to avoid parameter tuning by a Bayesian inference method. In contrast to existing schemes where the parameters should be carefully tuned according to the environment, the performance of this scheme is not sensitive to external environments because a set of parameters can be consistently applied to various environments, which dramatically reduce the difficulty of applying a learning automaton to an unknown stochastic environment. A rigorous proof of $\epsilon$-optimality for the proposed scheme is provided and numeric experiments are carried out on benchmark environments to verify its effectiveness. The results show that, without any parameter tuning cost, the proposed parameter-free learning automaton (PFLA) can achieve a competitive performance compared with other well-tuned schemes and outperform untuned schemes on consistency of performance.

Complexity of a problem concerning reset words for Eulerian binary automata

Abstract: A word is called a reset word for a deterministic finite automaton if it maps all the states of the automaton to a unique state. Deciding about the existence of a reset word of a given length for a given automaton is known to be an NP-complete problem. We prove that it remains NP-complete even if restricted to Eulerian automata with binary alphabets as it has been conjectured by Martyugin (2011).

The Quotient Operation on Input-Driven Pushdown Automata

Abstract: The quotient of a formal language K by another language L is the set of all strings obtained by taking a string from K that ends with a suffix from L, and removing that suffix The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation This paper establishes the closure of the family of input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be \(m^2n + O(m)\), where m and n is the number of states in the automata recognizing K and L, respectively

Weighted Operator Precedence Languages

Abstract: In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides the original grammar family, in terms of an automata family and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a B\"uchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.

Watson-Crick pushdown automata

Abstract: A multi-head 1-way pushdown automaton with $k$ heads is a pushdown automaton with $k$ 1-way read heads on the input tape and a stack. It was previously shown that the deterministic variant of the model cannot accept all the context free languages. In this paper, we introduce a 2-tape, 2-head model namely Watson-Crick pushdown automata where the content of the second tape is determined using a complementarity relation, similar to Watson-Crick automata. We show computational powers of nondeterministic two-head pushdown automata and nondeterministic Watson-Crick pushdown automata are same. Moreover, deterministic Watson-Crick pushdown automata can accept all the context free languages.

Edit distance for pushdown automata

Abstract: The edit distance between two words $w_1, w_2$ is the minimal number of word operations (letter insertions, deletions, and substitutions) necessary to transform $w_1$ to $w_2$. The edit distance generalizes to languages $\mathcal{L}_1, \mathcal{L}_2$, where the edit distance from $\mathcal{L}_1$ to $\mathcal{L}_2$ is the minimal number $k$ such that for every word from $\mathcal{L}_1$ there exists a word in $\mathcal{L}_2$ with edit distance at most $k$. We study the edit distance computation problem between pushdown automata and their subclasses. The problem of computing edit distance to a pushdown automaton is undecidable, and in practice, the interesting question is to compute the edit distance from a pushdown automaton (the implementation, a standard model for programs with recursion) to a regular language (the specification). In this work, we present a complete picture of decidability and complexity for the following problems: (1)~deciding whether, for a given threshold $k$, the edit distance from a pushdown automaton to a finite automaton is at most $k$, and (2)~deciding whether the edit distance from a pushdown automaton to a finite automaton is finite.

Visibly pushdown modular games

Abstract: Recursive game graphs can be used to reason about the control flow of sequential programs with recursion. Here, the most natural notion of strategy is the modular one, i.e., a strategy that is local to a module and is oblivious to previous module invocations. In this work, we study for the first time modular strategies with respect to winning conditions expressed as languages of pushdown automata. We show that pushdown modular 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 winning conditions. We carefully characterize the computational complexity of this decision problem by also considering as winning conditions nondeterministic and universal Buchi or co-Buchi visibly pushdown automata, and CaRet or Nwtl temporal logic formulas. As a further contribution, we present a different synthesis algorithm that improves on known solutions for large specifications and many exits.

One-Way Bounded-Error Probabilistic Pushdown Automata and Kolmogorov Complexity - (Preliminary Report).

Abstract: One-way probabilistic pushdown automata (or ppda’s) are a simple model of randomized computation with last-in first-out memory device known as stacks and, when error probabilities are bounded away from 1 / 2, ppda’s can characterize a family of bounded-error probabilistic context-free languages (BPCFL). We resolve a fundamental question raised by Hromkovic and Schnitger [Inf. Comput. 208 (2010) 982–995] concerning the limitation of the language recognition power of bounded-error ppda’s. More specifically, we prove that a well-known language—the set of palindromes—cannot be recognized by any bounded-error ppda; in other words, this language stays outside of BPCFL. Furthermore, we show that, with bounded-error probability, no ppda can determine whether the center bit of input string is 1 (one). For those impossibility results, we utilize a complexity measure of algorithmic information known as Kolmogorov complexity. In our proofs, we first transform ppda’s into an ideal shape and then lead to a key lemma by employing a Kolmogorov complexity argument.