Showing papers on "Deterministic pushdown automaton published in 2014"

From LTL to Deterministic Automata: A Safraless Compositional Approach

Abstract: We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula i¾?. The automaton is the product of a master automaton and an array of slave automata, one for each G-subformula of i¾?. The slave automaton for G i¾? is in charge of recognizing whether FG i¾? holds. As opposed to standard determinization procedures, the states of all our automata have a clear logical structure, which allows for various optimizations. Our construction subsumes former algorithms for fragments of LTL. Experimental results show improvement in the sizes of the resulting automata compared to existing methods.

Complexity of input-driven pushdown automata

Abstract: In an input-driven pushdown automaton (IDPDA), the current input symbol determines whether the automaton performs a push operation, a pop operation, or does not touch the stack. Inputdriven pushdown automata, also known under alternative names of visibly pushdown automata and of nested word automata, have been intensively studied because of their desirable features: for instance, the model allows determinization, and the associated family of languages retains many of the strong closure and decidability properties of regular languages. This paper reports on various aspects of the complexity of input-driven pushdown automata, such as their descriptional complexity, the computational complexity of their membership problem and of other decision problems for input-driven languages. The research on IDPDAs has been associated to their complexity from the very beginning. When Mehlhorn [25] originally introduced the model, it was studied as a subclass of deterministic context-free languages with better space complexity. Further work on the model carried out in the 1980s [6, 12, 36] concentrated on improving the bounds on the complexity of the languages accepted by such automata, culminating in the proof of their containment in NC1. In 2004, the model was reintroduced by Alur and Madhusudan [2] under the name of visibly pushdown automata, and among their most important contributions were the first results on the descriptional complexity of the model, such as upper and lower bounds on the number of states in automata representing some operations on languages. Also, Alur and Madhusudan [2] established the computational complexity of several decision problems for the model. The paper by Alur and Madhusudan [2] sparked a renewed interest in IDPDAs, and inspired the research on various aspects of the model [1, 8, 10, 19, 39]. Alur and Madhusudan [3] also introduced an equivalent outlook on IDPDAs as automata operating on nested words, which provide a natural model for applications such as XML document processing, where data has a dual linear-hierarchical structure. Nested word automata have been studied in a number of recent papers [9, 11, 17, 35, 37]. Another equivalent outlook on IDPDAs is represented by pushdown forest automata [14], that are, roughly speaking, tree walking automata that traverse the tree in depth-first left-to-right order and are equipped with a synchronized pushdown.

Hyper-Ackermannian bounds for pushdown vector addition systems

Abstract: This paper studies the boundedness and termination problems for vector addition systems equipped with one stack. We introduce an algorithm, inspired by the Karp & Miller algorithm, that solves both problems for the larger class of well-structured pushdown systems. We show that the worst-case running time of this algorithm is hyper-Ackermannian for pushdown vector addition systems. For the upper bound, we introduce the notion of bad nested words over a well-quasi-ordered set, and we provide a general scheme of induction for bounding their lengths. We derive from this scheme a hyper-Ackermannian upper bound for the length of bad nested words over vectors of natural numbers. For the lower bound, we exhibit a family of pushdown vector addition systems with finite but large reachability sets (hyper-Ackermannian).

Equivalences of Pushdown Systems Are Hard

Abstract: Language equivalence of deterministic pushdown automata (DPDA) was shown to be decidable by Senizergues (1997, 2001); Stirling (2002) then showed that the problem is primitive recursive.

Symbolic Visibly Pushdown Automata

Abstract: Nested words model data with both linear and hierarchical structure such as XML documents and program traces. A nested word is a sequence of positions together with a matching relation that connects open tags (calls) with the corresponding close tags (returns). Visibly Pushdown Automata are a restricted class of pushdown automata that process nested words, and have many appealing theoretical properties such as closure under Boolean operations and decidable equivalence. However, like any classical automata models, they are limited to finite alphabets. This limitation is restrictive for practical applications to both XML processing and program trace analysis, where values for individual symbols are usually drawn from an unbounded domain. With this motivation, we introduce Symbolic Visibly Pushdown Automata (SVPA) as an executable model for nested words over infinite alphabets. In this model, transitions are labeled with predicates over the input alphabet, analogous to symbolic automata processing strings over infinite alphabets. A key novelty of SVPAs is the use of binary predicates to model relations between open and close tags in a nested word. We show how SVPAs still enjoy the decidability and closure properties of Visibly Pushdown Automata. We use SVPAs to model XML validation policies and program properties that are not naturally expressible with previous formalisms and provide experimental results for our implementation.

Word-Mappings of Level 2

Abstract: A sequence of natural numbers is said to have level k, for some natural integer k, if it can be computed by a deterministic pushdown automaton of level k (Fratani and Senizergues in Ann Pure Appl. Log. 141:363–411, 2006). We show here that the sequences of level 2 are exactly the rational formal power series over one undeterminate. More generally, we study mappings from words to words and show that the following classes coincide: the mappings which are computable by deterministic pushdown automata of level 2 the mappings which are solution of a system of catenative recurrence equations the mappings which are definable as a Lindenmayer system of type HDT0L. We illustrate the usefulness of this characterization by proving three statements about formal power series, rational sets of homomorphisms and equations in words.

Removing nondeterminism in constant height pushdown automata

Abstract: We study the descriptional cost of removing nondeterminism in constant height pushdown automata-i.e., pushdown automata with a built-in constant limit on the height of the pushdown. We show a double-exponential size increase when converting a constant height nondeterministic pushdown automaton into an equivalent deterministic device. Moreover, we prove that such a double-exponential blow-up cannot be avoided by certifying its optimality. As a direct consequence, we get that eliminating nondeterminism in classical finite state automata is single-exponential even with the help of a constant height pushdown store.

Bisimulation Equivalence of First-Order Grammars

Abstract: A decidability proof for bisimulation equivalence of first-order grammars (finite sets of labelled rules for rewriting roots of first-order terms) is presented. The equivalence generalizes the DPDA (deterministic pushdown automata) equivalence, and the result corresponds to the result achieved by Senizergues (1998, 2005) in the framework of equational graphs, or of PDA with restricted epsilon-steps. The framework of classical first-order terms seems particularly useful for providing a proof that should be understandable for a wider audience. We also discuss an extension to branching bisimilarity, announced by Fu and Yin (2014).

Extended Two-Way Ordered Restarting Automata for Picture Languages

Abstract: We introduce a two-dimensional variant of the deterministic restarting automaton for processing rectangular pictures. Our device has a window of size three-by-three, in a rewrite step it can only replace the symbol in the central position of its window by a symbol that is smaller with respect to a fixed ordering on the tape alphabet, and it can only perform extended move-right and move-down steps. This automaton is strictly more expressive than the deterministic Sgraffito automaton, but its word problem can still be solved in polynomial time, and when restricted to one-dimensional input, it only accepts the regular languages.

Unary Pushdown Automata and Straight-Line Programs

Abstract: We consider decision problems for deterministic pushdown automata over the unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs—one for the prefix, one for the lasso—that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is Π2 P-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply Π2 P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

Degrees of Reversibility for DFA and DPDA

Abstract: The notion of k-reversibility is generalized to pushdown automata A pushdown automaton is said to be (k,l)-reversible if its predecessor configurations can uniquely be computed by a pushdown automaton with input lookahead of size k and stack lookahead of size l It turns out that there are problems which can be solved by (k + 1,1)-reversible pushdown automata, but not by (k,l)-reversible pushdown automata So, infinite hierarchies dependent on the degree of reversibility are shown On the other hand, any reversible pushdown automaton of degree (k,l + 1) can be simulated by a reversible pushdown automaton of degree (k,1) So, there are no hierarchies induced by the size of the stack lookahead These results complement the situation for finite automata which is also discussed and presented in our setting

Unary Pushdown Automata and Straight-Line Programs

Abstract: We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs---one for the prefix, one for the lasso---that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is $\Pi_2 \mathrm P$-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply $\Pi_2 \mathrm P$-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

Deterministic Set Automata

Abstract: We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As results the incomparability to all classes considered is obtained. In the second part of the paper, we examine the closure properties of the class of DSA languages under Boolean operations. Finally, we show that deterministic set automata may be an interesting model from a practical point of view by proving that their emptiness problem is decidable.

Bisimulation Equivalence of First-Order Grammars

Abstract: A decidability proof for bisimulation equivalence of first-order grammars (i.e., finite sets of labelled rules for rewriting roots of first-order terms) is presented. The result, generalizing the decidability of the DPDA (deterministic pushdown automata) equivalence, is equivalent to the result achieved by Senizergues (1998, 2005) in the framework of equational graphs, or of PDA with restricted e-steps, but the framework of classical first-order terms seems to be particularly useful for providing a concise proof that should be understandable for a wider audience.

Regular Strategies in Pushdown Reachability Games

Abstract: We show that positional winning strategies in pushdown reachability games can be implemented by deterministic finite state automata of exponential size. Such automata read the stack and control state of a given pushdown configuration and output the set of winning moves playable from that position.

Well-Structured Pushdown System: Case of Dense Timed Pushdown Automata

Abstract: This paper investigates a general framework of a pushdown system with well-quasi-ordered states and stack alphabet to show decidability of reachability, which is an extension of our earlier work (Well-structured Pushdown Systems, CONCUR 2013). As an instance, an alternative proof of the decidability of the reachability for dense-timed pushdown system (in P.A. Abdulla, M.F. Atig, F. Stenman, Dense-Timed Pushdown Automata, IEEE LICS 2012) is presented. Our proof would be more robust for extensions, e.g., regular valuations with time.

Enforcing Operational Properties including Blockfreeness for Deterministic Pushdown Automata.

Abstract: We present an algorithm which modifies a deterministic pushdown automaton (DPDA) such that (i) the marked language is preserved, (ii) lifelocks are removed, (iii) deadlocks are removed, (iv) all states and edges are accessible, and (v) operational blockfreeness is established (i.e., coaccessibility in the sense that every initial derivation can be continued to a marking configuration). This problem can be trivially solved for deterministic finite automata (DFA) but is not solvable for standard petri net classes. The algorithm is required for an operational extension of the supervisory control problem (SCP) to the situation where the specification in modeled by a DPDA.

Reducing an Operational Supervisory Control Problem by Decomposition for Deterministic Pushdown Automata

Abstract: The purpose of Supervisory Control Theory (SCT) is to synthesize a controller for a plant and a specification such that the desired closed-loop behavior is enforced. Effective solvers have been constructed in the past for the setting of plants and specifications modeled by Deterministic Finite Automata (DFA). We extend the domain of the specification to Deterministic Pushdown Automata (DPDA) and verify an effective solver (up to two basic building blocks which ensure controllability and blockfreeness, effectively solved for this setting in two companion papers). We verify the enforcement of desired operational criteria, which are, in contrast to the setting of DFA, partly oblivious to the (un)marked language of the closed loop. Our general approach trivially covers the setting of DFA and can be reused and adapted to develop effective solvers for other settings as the realizability of solutions to the supervisory control problem (SCP) is considered on an Abstract: level.

Extending Supervisory Controller Synthesis to Deterministic Pushdown Automata— Enforcing Controllability Least Restrictively

Abstract: In this paper a step towards the generalization of supervisory control theory to situations where the specification is modeled by a deterministic pushdown automaton (DPDA) is provided In particular, this paper presents an algorithm to calculate the largest controllable marked sublanguage of a given deterministic context free language (DCFL) by least restrictively removing controllability problems in a DPDA realization of this DCFL It also provides a counterexample which shows that the algorithm by Griffin (2008) intended to solve the considered problem is not minimally restrictive

New Results on the Minimum Amount of Useful Space

Abstract: We present several new results on minimal space requirements to recognize a nonregular language: (i) realtime nondeterministic Turing machines can recognize a nonregular unary language within weak $\log\log n$ space, (ii) $\log\log n$ is a tight space lower bound for accepting general nonregular languages on weak realtime pushdown automata, (iii) there exist unary nonregular languages accepted by realtime alternating one-counter automata within weak $\log n$ space, (iv) there exist nonregular languages accepted by two-way deterministic pushdown automata within strong $\log\log n$ space, and, (v) there exist unary nonregular languages accepted by two-way one-counter automata using quantum and classical states with middle $\log n$ space and bounded error.

Visibly Pushdown Transducers with Well-Nested Outputs

Abstract: Visibly pushdown transducers (VPTs) are visibly pushdown automata extended with outputs. They have been introduced to model transformations of nested words, i.e. words with a call/return structure. When outputs are also structured and well nested words, VPTs are a natural formalism to express tree transformations evaluated in streaming. We prove the class of VPTs with well-nested outputs to be decidable in Ptime. Moreover, we show that this class is closed under composition and that its type-checking against visibly pushdown languages is decidable.

Computing Optimal Reachability Costs in Priced Dense-Timed Pushdown Automata

Abstract: We study priced dense-timed pushdown automata that are a generalization of the classic model of pushdown automata, in the sense that they operate on real-valued clocks, and that the stack symbols have real-valued ages. Furthermore, the model allows a cost function that assigns transition costs to transitions and storage costs to stack symbols. We show that the optimal cost, i.e., the infimum of the costs of the set of runs reaching a given control state, is computable.

Input-Driven Pushdown Automata with Limited Nondeterminism (Invited Paper)

Abstract: It is known that determinizing a nondeterministic input- driven pushdown automaton (NIDPDA) of size n results in the worst case in a machine of size 2 Θ(n 2 ) (R Alur, P Madhusudan, "Adding nest- ing structure to words", JACM 56(3), 2009) This paper considers the special case of k-path NIDPDAs, which have at most k computations on any input It is shown that the smallest deterministic IDPDA equivalent to a k-path NIDPDA of size n is of size Θ(n k ) The paper also gives an algorithm for deciding whether or not a given NIDPDA has the k-path property, for a given k ;i fk is fixed, the problem is P-complete

To Design a English Language Recognizer by using Nondeterministic Pushdown Automata (ELR-NPDA)

Abstract: Natural language recognization is a popular topic of research as it covers many areas such as computer science, artificial intelligence, theory of computation, and machine leaning etc Many of the techniques are used for natural language recognization by the researchers, parsing is one of them The aim to propose this paper is to implement nondeterministic pushdown automata (NPDA) for the English Language (ELR-NPDA) that can modernize Context Free Grammar (CFG) for English language and then refurbish into Nondeterministic Pushdown Automata (NPDA) This converting procedure can uncomplicatedly parse legitimate English language sentences Parsing can be organized by Nondeterministic Pushdown Automata (NPDA) that used push down stack and input tape for recognizing English language sentences To formulate this NPDA convertor we have to exchange Context Free Grammar into Chomsky Normal Form (CNF) The move toward this is more appropriate because it uses nondeterministic approach of PDA that can improve language recognizing capabilities as compare to other parsing approach

Visibly Pushdown Modular Games

Abstract: Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. In this work, we study for the first time modular strategies with respect to winning conditions that can be expressed by a pushdown automaton. We show that such 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 automata winning conditions, which are known in the literature. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal Buchi or co Buchi visibly pushdown winning condition are EXPTIME-complete, and when the winning condition is given by a CARET or NWTL temporal logic formula the problem is 2EXPTIME-complete, and it remains 2EXPTIME-hard even for simple fragments of these logics. As a further contribution, we present a different solution for modular games with finite-state automata winning condition that runs faster than known solutions for large specifications and many exits.