Showing papers on "Pushdown automaton published in 2008"

Collapsible Pushdown Automata and Recursion Schemes

Abstract: 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 Ong's 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.

Emptiness of Multi-pushdown Automata Is 2ETIME-Complete

Abstract: We consider 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 wrt. the number of stacks. 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 multi-pushdown automata with the model of bounded-phase multi-stack (visibly) pushdown automata.

On the Reachability Analysis of Acyclic Networks of Pushdown Systems

Abstract: We address the reachability problem in acyclic networks of pushdown systems. We consider communication based either on shared memory or on message passing through unbounded lossy channels. We prove mainly that the reachability problem between recognizable sets of configurations (i.e., definable by a finite union of products of finite-state automata) is decidable for such networks, and that for lossy channel pushdown networks, the channel language is effectively recognizable. This fact holds although the set of reachable configurations (including stack contents) for a network of depth (at least) 2 is not rational in general (i.e., not definable by a multi-tape finite automaton). Moreover, we prove that for a network of depth 1, the reachability set is rational and effectively constructible (under an additional condition on the topology for lossy channel networks).

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

Abstract: We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD (even a null-recurrent one), we can approximate its termination probabilities (i.e., its G matrix) to within i bits of precision (i.e., within additive error 1/2i), in time polynomial in both the encoding size of the QBD and in i, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known "linear/quadratic convergence" of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results for pPDSs) that for the more general TL-QBDs this bound fails badly. Specifically, in the worst case Newton's method "converges linearly" to the termination probabilities for TL-QBDs, but requires exponentially many iterations in the encoding size of the TL-QBD to approximate these probabilities within any non-trivial constant error c < 1. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems,and a very recent constructive bound on the performance of Newton's method for monotone systems of polynomial equations.

Visibly pushdown transducers

Abstract: Visibly pushdown automata have been recently introduced by Alur and Madhusudan as a subclass of pushdown automata. This class enjoys nice properties such as closure under all Boolean operations and the decidability of language inclusion. Along the same line, we introduce here visibly pushdown transducers as a subclass of pushdown transducers. We study properties of those transducers and identify subclasses with useful properties like decidability of type checking as well as preservation of regularity of visibly pushdown languages.

The universal automaton.

Abstract: This paper is a survey on the universal automaton, which is an automaton canonically associated with every language. In the last forty years, many objects have been defined or studied, that are indeed closely related to the universal automaton. We first show that every automaton that accepts a given language has a morphic image which is a subautomaton of the universal automaton of this language. This property justifies the name “universal” that we have coined for this automaton. The universal automaton of a regular language is finite and can be effectively computed in the syntactic monoid or, more efficiently, from the minimal automaton of the language. We describe the construction that leads to tight bounds on the size of the universal automaton. Another outcome of the effective construction of the universal automaton is the computation of a minimal NFA accepting a given language, or approximations of such a minimal NFA. From another point of view, the universal automaton of a language is based on the factorisations of this language, and is thus involved in the problems of factorisations and approximations of languages. Last, but not least, we show how the universal automaton gives an elegant solution to the star height problem for some classes of languages (pure-group or reversible languages). With every language is canonically associated an automaton, called the universal automaton of the language, which is finite whenever the language is regular. It is large, it is complex, it is complicated to compute, but it contains, hopefully, many interesting informations on the language. In the last forty years, it has been described a number of times, more or less explicitly,

Undecidability of bisimilarity by defender's forcing

Abstract: Stirling l1996, 1998r proved the decidability of bisimilarity on so-called normed pushdown processes. This result was substantially extended by Senizergues l1998, 2005r who showed the decidability of bisimilarity for regular (or equational) graphs of finite out-degree; this essentially coincides with weak bisimilarity of processes generated by (unnormed) pushdown automata where the e-transitions can only deterministically pop the stack. The question of decidability of bisimilarity for the more general class of so called Type -1 systems, which is equivalent to weak bisimilarity on unrestricted e-popping pushdown processes, was left open. This was repeatedly indicated by both Stirling and Senizergues. Here we answer the question negatively, that is, we show the undecidability of bisimilarity on Type -1 systems, even in the normed case.We achieve the result by applying a technique we call Defender's Forcing, referring to the bisimulation games. The idea is simple, yet powerful. We demonstrate its versatility by deriving further results in a uniform way. First, we classify several versions of the undecidable problems for prefix rewrite systems (or pushdown automata) as Π01-complete or Σ11-complete. Second, we solve the decidability question for weak bisimilarity on PA (Process Algebra) processes, showing that the problem is undecidable and even Σ11-complete. Third, we show Σ11-completeness of weak bisimilarity for so-called parallel pushdown (or multiset) automata, a subclass of (labeled, place/transition) Petri nets.

On Stateless Two-Pushdown Automata and Restarting Automata.

Abstract: Restarting automata and two-pushdown automata are investigated that have a single internal state only. As such an automaton must always stay in the same state, this state is of no importance for the behaviour of the automaton. Accordingly, these automata are called stateless. We consider various types of stateless two-pushdown automata and restarting automata. We investigate their expressive power, comparing them in particular to each other and to the corresponding types of automata with states.

Reachability in recursive Markov decision processes

Abstract: We consider a class of infinite-state Markov decision processes generated by stateless pushdown automata. This class corresponds to 112-player games over graphs generated by BPA systems or (equivalently) 1-exit recursive state machines. An extended reachability objective is specified by two sets S and T of safe and terminal stack configurations, where the membership to S and T depends just on the top-of-the-stack symbol. The question is whether there is a suitable strategy such that the probability of hitting a terminal configuration by a path leading only through safe configurations is equal to (or different from) a given [email protected]?{0,1}. We show that the qualitative extended reachability problem is decidable in polynomial time, and that the set of all configurations for which there is a winning strategy is effectively regular. More precisely, this set can be represented by a deterministic finite-state automaton with a fixed number of control states. This result is a generalization of a recent theorem by Etessami and Yannakakis which says that the qualitative termination for 1-exit RMDPs (which exactly correspond to our 112-player BPA games) is decidable in polynomial time. Interestingly, the properties of winning strategies for the extended reachability objectives are quite different from the ones for termination, and new observations are needed to obtain the result. As an application, we derive the EXPTIME-completeness of the model-checking problem for 112-player BPA games and qualitative PCTL formulae.

Winning Regions of Higher-Order Pushdown Games

Abstract: In this paper we consider parity games defined by higher-order pushdown automata. These automata generalise pushdown automata by the use of higher-order stacks, which are nested "stack of stacks" structures. Representing higher-order stacks as well-bracketed words in the usual way, we show that the winning regions of these games are regular sets of words. Moreover a finite automaton recognising this region can be effectively computed. A novelty of our work are abstract pushdown processes which can be seen as (ordinary) pushdown automata but with an infinite stack alphabet. We use the device to give a uniform presentation of our results.From our main result on winning regions of parity games we derive a solution to the Modal Mu-Calculus Global Model-Checking Problem for higher-order pushdown graphs as well as for ranked trees generated by higher-order safe recursion schemes.

Polynomial-time verification of the observer property in abstractions

Abstract: This paper presents an algorithm to test if an abstraction obtained through natural projection has the observer property, without having to compute the abstraction. The original automaton and the set of events to be kept by the projection are inputs to the algorithm. An automaton, the verifier, is built such that the verification of the property becomes a verification of reachability of a special state. The complexity of the algorithm is polynomial in the size of the state space of the automaton. Two examples are presented to illustrate the algorithm.

Automatic Protocol Conformance Checking of Recursive and Parallel Component-Based Systems

Abstract: Today model checking of security or safety properties of component-based systems based on finite protocols has the flaw that either parallel or sequential systems can be checked. Parallel systems can be described often by well known Petri nets, but it is not possible to model recursive behaviour. On the other hand sequential systems based on pushdown automata can capture recursion and recursive callbacks [27], but they do not provide parallel behaviour in general. In this work we show how this gap can be filled if process rewrite systems (introduced by Mayr [16]) are used to capture the behaviour of components. The protocols of the components interfaces specified as finite state machines can be combined to a system equal to a process rewrite system. By calculating the reachability of the fault state range one gets a trace (counterexample) which does not satisfy the properties specified by all protocols of the combined components, if any error exists.

On the Expressive Power of 2-Stack Visibly Pushdown Automata

Abstract: Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.

A Context-Free Process as a Pushdown Automaton

Abstract: A well-known theorem in automata theory states that every context-free language is accepted by a pushdown automaton. We investigate this theorem in the setting of processes, using the rooted branching bisimulation and contrasimulation equivalences instead of language equivalence. In process theory, different from automata theory, interaction is explicit, so we realize a pushdown automaton as a regular process communicating with a stack.

Positional Strategies for Higher-Order Pushdown Parity Games

Abstract: Higher-order pushdown systems generalize pushdown systems by using higher-order stacks, which are nested stacks of stacks. In this article, we consider parity games defined by higher-order pushdown systems and provide a k- Exptime algorithm to compute finite representations of positional winning strategies for both players for games defined by level-khigher-order pushdown automata. Our result is based on automata theoretic techniques exploiting the tree structure corresponding to higher-order stacks and their associated operations.

On the expressive power of 2-stack visibly pushdown automata

Abstract: Visibly pushdown automata are input-driven pushdown automata that rec- ognize some non-regular context-free languages while preserving the nice closure and de- cidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express prop- erties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly push- down automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt. words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of Buchi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.

Discounted Properties of Probabilistic Pushdown Automata

Abstract: We show that several basic discounted properties of probabilistic pushdown automata related both to terminating and non-terminating runs can be efficiently approximated up to an arbitrarily small given precision.

Visibly tree automata with memory and constraints

Abstract: Tree automata with one memory have been introduced in 2001. They gener- alize both pushdown (word) automata and the tree automata with constraints of equality between brothers of Bogaert and Tison. Though it has a decidable emptiness problem, the main weakness of this model is its lack of good closure properties. We propose a generalization of the visibly pushdown automata of Alur and Madhusu- dan to a family of tree recognizers which carry along their (bottom-up) computation an auxiliary unbounded memory with a tree structure (instead of a symbol stack). In other words, these recognizers, called Visibly Tree Automata with Memory (VTAM) define a subclass of tree automata with one memory enjoying Boolean closure properties. We show in particular that they can be determinized and the problems like emptiness, member- ship, inclusion and universality are decidable for VTAM. Moreover, we propose several extensions of VTAM whose transitions may be constrained by different kinds of tests be- tween memories and also constraints a la Bogaert and Tison. We show that some of these classes of constrained VTAM keep the good closure and decidability properties, and we demonstrate their expressiveness with relevant examples of tree languages.

Error recovery and diagnosis for pushdown automata

Abstract: Error recovery and diagnosis is afforded for pushdown automata. Upon detection of an error, a recovery strategy is selected and dispatched to recover from the error to place an automaton in an error free state to enable continued processing. In one instance, recovery strategies can be specified and matched with respect to automaton configuration. Errors can be diagnosed as a function of the difference between a first error configuration and a second recovered configuration.

Visibly Pushdown Transducers

Abstract: Visibly pushdown automata have been recently introduced by Alur and Madhusudan as a subclass of pushdown automata. This class enjoys nice properties such as closure under all Boolean operations and the decidability of language inclusion. Along the same line, we introduce here visibly pushdown transducers as a subclass of pushdown transducers. We study properties of those transducers and identify subclasses with useful properties like decidability of type checking as well as preservation of regularity of visibly pushdown languages.

Direct construction of finite state machines

Abstract: A method and system for direct construction of a minimal deterministic finite state machine corresponding to a regular expression are provided. The method includes providing a regular expression represented as a regular expression tree with nodes of operators and leaves of elementary character transitions and traversing the regular expression tree recursively to build minimal finite state automata (FSAs) corresponding to the branches of the tree, wherein the FSAs end in a specified tail automaton. The operators are concatenation, alternation, and Kleene closure. A concatenation operation is performed by recursive construction in reverse order wherein each automaton built becomes the tail for the preceding argument of the operation. An alternation operation is performed by recursively building automata corresponding to the arguments of the operation with the same tail and merging them. A Kleene closure operation is performed by: building an automaton terminating in a unique marker; merging the automaton with the tail automaton to form a combined automaton; and traversing the combined automaton to expand the marker into transitions and states to achieve the intended behaviour.

Teaching push-down automata and turing machines

Abstract: In this paper we present the new version of a tool to assist in teaching formal languages and automata theory. In the previous version the tool provided algorithms for regular expressions, finite automata and context free grammars. The new version can simulate as well push-down automata and Turing machines.

On the complexity of membership and counting in height-deterministic pushdown automata

Abstract: Visibly pushdown languages properly generalize regular languages and are properly contained in deterministic context-free languages. The complexity of their membership problem is equivalent to that of regular languages. However, the corresponding counting problem - computing the number of accepting paths in a visibly pushdown automaton - could be harder than counting paths in a non-deterministic finite automaton: it is only known to be in LogDCFL. We investigate the membership and counting problems for generalizations of visibly pushdown automata, defined using the notion of height-determinism. We show that, when the stack-height of a given pushdown automaton can be computed using a finite transducer, both problems have the same complexity as for visibly pushdown languages. We also show that when allowing pushdown transducers instead of finite-state ones, both problems become LogDCFL-complete; this uses the fact that pushdown transducers are sufficient to compute the stack heights of all real-time height-deterministic pushdown automata, and yields a candidate arithmetization of LogDCFL that is no harder than LogDCFL (our main result).

Topology and Ambiguity in Omega Context Free Languages

Abstract: We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by B\"uchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by B\"uchi 2-tape automata and we get first results in that direction.

Deciding probabilistic bisimilarity over infinite-state probabilistic systems

Abstract: We prove that probabilistic bisimilarity is decidable over probabilistic extensions of BPA and BPP processes. For normed subclasses of probabilistic BPA and BPP processes we obtain polynomial-time algorithms. Further, we show that probabilistic bisimilarity between probabilistic pushdown automata and finite-state systems is decidable in exponential time. If the number of control states in PDA is bounded by a fixed constant, then the algorithm needs only polynomial time.

Two-way coupled finite automata and its usage in translators

Abstract: This article defines two-way coupled finite automata. Two-way coupled finite automaton enable to make a translation from input language to output language and from output language to input language too. There is discussed deterministic parsing by using coupled finite automaton in this article. For example, this deterministic model can be used for translation between assembly language and a binary code.