Showing papers on "Pushdown automaton published in 2012"

Dense-Timed Pushdown Automata

Abstract: We propose a model that captures the behavior of real-time recursive systems. To that end, we introduce dense-timed pushdown automata that extend the classical models of pushdown automata and timed automata, in the sense that the automaton operates on a finite set of real-valued clocks, and each symbol in the stack is equipped with a real-valued clock representing its "age". The model induces a transition system that is infinite in two dimensions, namely it gives rise to a stack with an unbounded number of symbols each of which with a real-valued clock. The main contribution of the paper is an EXPTIME-complete algorithm for solving the reachability problem for dense-timed pushdown automata.

A saturation method for collapsible pushdown systems

Abstract: 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.

Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

Abstract: An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved eciently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by uniform constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0 -completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1 . Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC 0 , which encapsulates most of the technical diculties surrounding earlier results connecting tree automata and NC 1 . 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes

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

Abstract: 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.

Introspective pushdown analysis of higher-order programs

Abstract: In the static analysis of functional programs, pushdown flow analysis and abstract garbage collection skirt just inside the boundaries of soundness and decidability. Alone, each method reduces analysis times and boosts precision by orders of magnitude. This work illuminates and conquers the theoretical challenges that stand in the way of combining the power of these techniques. The challenge in marrying these techniques is not subtle: computing the reachable control states of a pushdown system relies on limiting access during transition to the top of the stack; abstract garbage collection, on the other hand, needs full access to the entire stack to compute a root set, just as concrete collection does. Introspective pushdown systems resolve this conflict. Introspective pushdown systems provide enough access to the stack to allow abstract garbage collection, but they remain restricted enough to compute control-state reachability, thereby enabling the sound and precise product of pushdown analysis and abstract garbage collection. Experiments reveal synergistic interplay between the techniques, and the fusion demonstrates "better-than-both-worlds" precision.

On the Significance of the Collapse Operation

Abstract: We show that deterministic collapsible pushdown automata of second level can recognize a language which is not recognizable by any deterministic higher order pushdown automaton (without collapse) of any level. This implies that there exists a tree generated by a second level collapsible pushdown system (equivalently: by a recursion scheme of second level), which is not generated by any deterministic higher order pushdown system (without collapse) of any level (equivalently: by any safe recursion scheme of any level). As a side effect, we present a pumping lemma for deterministic higher order pushdown automata, which potentially can be useful for other applications.

Decidability of DPDA Language Equivalence via First-Order Grammars

Abstract: Decidability of language equivalence of deterministic pushdown automata (DPDA) was established by G. Senizergues (1997), who thus solved a famous long-standing open problem. A simplified proof, also providing a primitive recursive complexity upper bound, was given by C. Stirling (2002). In this paper, the decidability is re-proved in the framework of first-order terms and grammars (given by finite sets of root-rewriting rules). The proof is based on the abstract ideas used in the previous proofs, but the chosen framework seems to be more natural for the problem and allows a short presentation which should be transparent for a general computer science audience.

An Improved Construction of Deterministic Omega-automaton Using Derivatives

Abstract: In an earlier paper, the author used derivatives to construct a deterministic automaton recognizing the language defined by an ω-regular expression. The construction was related to a determinization method invented by Safra. This paper describes a new construction, inspired by Piterman's improvement to Safra's method. It produces an automaton with fewer states. In addition, the presentation and proofs are simplified by going via a nondeterministic automaton having derivatives as states.

Model-Checking of Ordered Multi-Pushdown Automata

Abstract: We address the verification problem of ordered multi-pushdown automata: A multi-stack extension of pushdown automata that comes with a constraint on stack transitions such that a pop can only be pe ...

The minimal cost reachability problem in priced timed pushdown systems

Abstract: This paper introduces the model of priced timed pushdown systems as an extension of discrete-timed pushdown systems with a cost model that assigns multidimensional costs to both transitions and stack symbols. For this model, we consider the minimal cost reachability problem: i.e., given a priced timed pushdown system and a target set of configurations, determine the minimal possible cost of any run from the initial to a target configuration. We solve the problem by reducing it to the reachability problem in standard pushdown systems.

A Perfect Model for Bounded Verification

Abstract: A class of languages C is perfect if it is closed under Boolean operations and the emptiness problem is decidable. Perfect language classes are the basis for the automata-theoretic approach to model checking: a system is correct if the language generated by the system is disjoint from the language of bad traces. Regular languages are perfect, but because the disjointness problem for CFLs is undecidable, no class containing the CFLs can be perfect. In practice, verification problems for language classes that are not perfect are often under-approximated by checking if the property holds for all behaviors of the system belonging to a fixed subset. A general way to specify a subset of behaviors is by using bounded languages (languages of the form w1* ... wk* for fixed words w1,...,wk). A class of languages C is perfect modulo bounded languages if it is closed under Boolean operations relative to every bounded language, and if the emptiness problem is decidable relative to every bounded language. We consider finding perfect classes of languages modulo bounded languages. We show that the class of languages accepted by multi-head pushdown automata are perfect modulo bounded languages, and characterize the complexities of decision problems. We also show that bounded languages form a maximal class for which perfection is obtained. We show that computations of several known models of systems, such as recursive multi-threaded programs, recursive counter machines, and communicating finite-state machines can be encoded as multi-head pushdown automata, giving uniform and optimal under approximation algorithms modulo bounded languages.

Strictness of the collapsible pushdown hierarchy

Abstract: We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples that separate the levels of the collapsible pushdown graph hierarchy and of the collapsible pushdown tree hierarchy, i.e., the hierarchy of trees generated by higher-order recursion schemes. This confirms the open conjecture that higher orders allow one to generate more graphs and more trees. Full proofs can be found in the arXiv version[10] of this paper.

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 can model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives, such as reachability and parity objectives, in this work we study for the first time such games with the most well-studied quantitative objective, namely, mean payoff objectives. In pushdown games two types of strategies are relevant: (1) global strategies, that depend on the entire global history; and (2) modular strategies, that have only local memory and thus do not depend on the context of invocation, but 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 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 computational 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.

Superiority of one-way and realtime quantum machines∗∗∗

Abstract: In automata theory, quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to QFAs augmented with counters or stacks. In this paper, we focus on such generalizations of QFAs where the input head operates in one-way or realtime mode, and present some new results regarding their superiority over their classical counterparts. Our first result is about the nondeterministic acceptance mode: Each quantum model architecturally intermediate between realtime finite state automaton and one-way pushdown automaton (one-way finite automaton, realtime and one-way finite automata with one-counter, and realtime pushdown automaton) is superior to its classical counterpart. The second and third results are about bounded error language recognition: for any k > 0, QFAs with k blind counters outperform their deterministic counterparts; and, a one-way QFA with a single head recognizes an infinite family of languages, which can be recognized by one-way probabilistic finite automata with at least two heads. Lastly, we compare the nondeterminictic and deterministic acceptance modes for classical finite automata with k blind counter(s), and we show that for any k > 0, the nondeterministic models outperform the deterministic ones.

SPEDE: Probabilistic Edit Distance Metrics for MT Evaluation

Abstract: This paper describes Stanford University's submission to the Shared Evaluation Task of WMT 2012. Our proposed metric (SPEDE) computes probabilistic edit distance as predictions of translation quality. We learn weighted edit distance in a probabilistic finite state machine (pFSM) model, where state transitions correspond to edit operations. While standard edit distance models cannot capture long-distance word swapping or cross alignments, we rectify these shortcomings using a novel pushdown automaton extension of the pFSM model. Our models are trained in a regression framework, and can easily incorporate a rich set of linguistic features. Evaluated on two different prediction tasks across a diverse set of datasets, our methods achieve state-of-the-art correlation with human judgments.

Algorithmic games for full ground references

Abstract: We present a full classification of decidable and undecidable cases for contextual equivalence in a finitary ML-like language equipped with full ground storage (both integers and reference names can be stored). The simplest undecidable type is unit→unit→unit. At the technical level, our results marry game semantics with automata-theoretic techniques developed to handle infinite alphabets. On the automata-theoretic front, we show decidability of the emptiness problem for register pushdown automata extended with fresh-symbol generation.

Recursive schemes, krivine machines, and collapsible pushdown automata

Abstract: Higher-order recursive schemes are an interesting method of approximating program semantics. The semantics of a scheme is an infinite tree labeled with built-in constants. This tree represents the meaning of the program up to the meaning of built-in constants. It is much easier to reason about properties of such trees than properties of interpreted programs. Moreover some interesting properties of programs are already expressible on the level of these trees. Collapsible pushdown automata (CPDA) give another way of generating the same class of trees. We present a relatively simple translation from recursive schemes to CPDA using Krivine machines as an intermediate step. The later are general machines for describing computation of the weak head normal form in the lambda-calculus. They provide the notions of closure and environment that facilitate reasoning about computation.

Visibly pushdown transducers with look-ahead

Abstract: Visibly Pushdown Transducers (VPT) form a subclass of pushdown transducers. In this paper, we investigate the extension of VPT with visibly pushdown look-ahead (VPTla ). Their transitions are guarded by visibly pushdown automata that can check whether the well-nested subword starting at the current position belongs to the language they define. First, we show that VPTla are not more expressive than VPT, but are exponentially more succinct. Second, we show that the class of deterministic VPTla corresponds exactly to the class of functional VPT, yielding a simple characterization of functional VPT. Finally, we show that while VPTla are exponentially more succinct than VPT, checking equivalence of functional VPTla is, as for VPT, ExpT-C. As a consequence, we show that any functional VPT is equivalent to an unambiguous one.

OpenNWA: a nested-word automaton library

Abstract: Nested-word automata (NWAs) are a language formalism that helps bridge the gap between finite-state automata and pushdown automata. NWAs can express some context-free properties, such as parenthesis matching, yet retain all the desirable closure characteristics of finite-state automata. This paper describes OpenNWA, a C++ library for working with NWAs. The library provides the expected automata-theoretic operations, such as intersection, determinization, and complementation. It is packaged with WALi--the Weighted Automaton Library--and interoperates closely with the weighted pushdown system portions of WALi.

Evolutionary Automata

Abstract: Expressiveness and convergence of evolutionary computation (EC) is studied using the evolutionary automata model. It turns out that all standard classes of evolutionary automata are equally expressive when they operate in the terminal mode, i.e. in the terminal mode, evolutionary finite automata (EFA) are as expressive as evolutionary pushdown automata, evolutionary linearly bounded automata, evolutionary Turing machines or evolutionary inductive Turing machines. For example, the simplest class of evolutionary automata, EFA, can accept all recursively enumerable languages (i.e. EFA have power of Turing machines) and even more—they can accept languages that are not recursively enumerable. Due to utilization of evolutionary automata, we obtain also very simple sufficient conditions for convergence of EC.

A pushdown transducer extension for the openfst library

Abstract: Pushdown automata are devices that can efficiently represent context-free languages, have natural weighted versions, and combine naturally with finite automata. We describe a pushdown transducer extension to OpenFst, a weighted finite-state transducer library. We present several weighted pushdown algorithms, some with clear finite-state analogues, describe their library usage and give some applications of these methods to recognition, parsing and translation.

On Bisimilarity of Higher-Order Pushdown Automata: Undecidability at Order Two

Abstract: We show that bisimulation equivalence of order-two pushdown automata is undecidable. Moreover, we study the lower order problem of higher-order pushdown automata, which asks, given an order-k pushdown automaton and some k' = 2 even when the input k-PDA is deterministic and real-time.

Systems and methods for determining the N-best strings

Abstract: Systems and methods for identifying the N-best strings of a weighted automaton. A potential for each state of an input automaton to a set of destination states of the input automaton is first determined. Then, the N-best paths are found in the result of an on-the-fly determinization of the input automaton. Only the portion of the input automaton needed to identify the N-best paths is determinized. As the input automaton is determinized, a potential for each new state of the partially determinized automaton is determined and is used in identifying the N-best paths of the determinized automaton, which correspond exactly to the N-best strings of the input automaton.

Descriptional Complexity of Input-Driven Pushdown Automata

Abstract: It is known that a nondeterministic input-driven pushdown automaton (IDPDA) can be determinized. Alur and Madhusudan (“Adding nesting structure to words”, J.ACM 56(3), 2009) showed that a deterministic IDPDA simulating a nondeterministic IDPDA with n states and stack symbols may need, in the worst case, \(2^{\Omega(n^2)}\) states. In their construction, the equivalent deterministic IDPDA does, in fact, not need to use the stack. This paper considers the size blow-up of determinization in more detail, and gives a lower bound construction, that is tight within a multiplicative constant, with respect to the size of the nondeterministic automaton both for the number of states and the number of stack symbols. The paper also surveys the recent results on operational state complexity of IDPDAs, and on the cost of converting a nondeterministic automaton to an unambiguous one, and an unambiguous automaton to a deterministic one.

Arbology: Trees and pushdown automata

Abstract: Trees are (data) structures used in many areas of human activity. Tree as the formal notion has been introduced in the theory of graphs. Nevertheless, trees have been used a long time before the foundation of the graph theory. An example is the notion of a genealogical tree. The area of family relationships was an origin of some terminology in the area of the tree theory (parent, child, sibling, ...) in addition to the terms originating from the area of the dendrology (root, branch, leaf, ...).