Visibly Pushdown Automata Research Articles

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

We study the expressiveness and succinctness of history-deterministic pushdown automata (HD-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. These are also known as good-for-games pushdown automata. We prove that HD-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that HD-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than HD-PDA. We also study HDness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show that deciding HDness is ExpTime-complete. HD-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than HD-VPA. Both of these lower bounds are tight. We then compare HD-PDA with PDA for which composition with games is well-behaved, i.e. good-for-games automata. We show that these two notions coincide, but only if we consider potentially infinitely branching games. Finally, we study the complexity of resolving nondeterminism in HD-PDA. Every HDPDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of HD-VPA, but not those of HD-PDA. HD-PDA with finite-state resolvers are determinisable.

Context-free timed formalisms: Robust automata and linear temporal logics

The paper focuses on automata and linear temporal logics for real-time pushdown reactive systems bridging tractable formalisms specialized for expressing separately dense-time real-time properties and context-free properties though preserving tractability. As for automata, we introduce Event-Clock Nested Automata (ECNA), a formalism that combines Event Clock Automata (ECA) and Visibly Pushdown Automata (VPA). ECNA enjoy the same closure and decidability properties of ECA and VPA expressively extending any previous attempt of combining ECA and VPA. As for temporal logics, we introduce two formalisms for specifying quantitative timing context-free requirements: Event-Clock Nested Temporal Logic (EC_NTL) and Nested Metric Temporal Logic (NMTL). EC_NTL is an extension of both the logic CaRet and Event-Clock Temporal Logic having Exptime-complete satisfiability of EC_NTL and visibly model-checking of Visibly Pushdown Timed Systems (VPTS) against EC_NTL. NMTL is a context-free extension of standard Metric Temporal Logic (MTL) which is in general undecidable having, though, a fragment expressively equivalent to EC_NTL with Exptime-complete satisfiability and visibly model-checking of VPTS problems.

Weighted operator precedence languages

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 by the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). A different area of research investigates quantitative evaluations of formal languages by adding weights to strings. 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 Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.

Visibly pushdown transducers

Visibly pushdown transducers (VPT ) extend visibly pushdown automata (VPA) with outputs. They read nested words, i.e. finite words on a structured alphabet partitioned into call, return and internal symbols, and produce output words, which are not necessarily nested. As for VPA, the behavior of the stack is synchronized with the types of input symbols: on reading a call symbol, exactly one stack symbol is pushed onto the stack; on reading a return symbol, exactly one stack symbol is popped from the stack; and the stack is untouched when reading internal symbols. Along their transitions VPT can append words on their output tape, whose concatenation define the resulting output word. Therefore VPT define transformations from nested words to words and can be seen as a subclass of pushdown transducers. In this paper, we study the algorithmic, closure and expressiveness properties of VPT.

Higher-Order Operator Precedence Languages

Floyd's Operator Precedence (OP) languages are a deterministic context-free family having many desirable properties. They are locally and parallely parsable, and languages having a compatible structure are closed under Boolean operations, concatenation and star; they properly include the family of Visibly Pushdown (or Input Driven) languages. OP languages are based on three relations between any two consecutive terminal symbols, which assign syntax structure to words. We extend such relations to k-tuples of consecutive terminal symbols, by using the model of strictly locally testable regular languages of order k at least 3. The new corresponding class of Higher-order Operator Precedence languages (HOP) properly includes the OP languages, and it is still included in the deterministic (also in reverse) context free family. We prove Boolean closure for each subfamily of structurally compatible HOP languages. In each subfamily, the top language is called max-language. We show that such languages are defined by a simple cancellation rule and we prove several properties, in particular that max-languages make an infinite hierarchy ordered by parameter k. HOP languages are a candidate for replacing OP languages in the various applications where they have have been successful though sometimes too restrictive.

Visibly Linear Temporal Logic

We introduce Visibly Linear Temporal Logic (VLTL), a linear-time temporal logic that captures the full class of Visibly Pushdown Languages over infinite words. The novel logic avoids fix points and instead provides natural temporal operators with simple and intuitive semantics. We prove that the complexities of the satisfiability and visibly pushdown model checking problems are the same as for other well known logics, like CaRet and the nested word temporal logic NWTL, which in contrast are strictly more limited in expressive power than VLTL. Moreover, formulas of CaRet and NWTL can be translated inductively and in linear-time into VLTL.

Ordered multi-stack visibly pushdown automata

Visibly Pushdown Automata (VPA) are a special case of pushdown machines where the stack operations are driven by the input. In this paper, we consider VPA with multiple stacks, namely n-VPA, with n>1. These automata introduce a useful model to effectively describe concurrent pushdown systems using a simple communication mechanism between stacks. We show that n-VPA are strictly more expressive than VPA. Indeed, n-VPA accept some context-sensitive languages that are not context-free and some context-free languages that are not accepted by any VPA. Nevertheless, we show that the class of languages accepted by n-VPA is closed under union and intersection. On the contrary, this class turns out to be neither closed under complementation nor determinizable. Moreover, we show that the emptiness problem for n-VPA is undecidable. By adding an ordering constraint on stacks (n-OVPA), decidability of emptiness can be recovered, as well as the closure under complementation. Using these properties along with the automata-theoretic approach, one can prove that the model checking problem over n-OVPA models against n-OVPA specifications is decidable.

Trimming visibly pushdown automata

We study the problem of trimming visibly pushdown automata (VPA). We first describe a polynomial time procedure which, given a visibly pushdown automaton that accepts only well-nested words, returns an equivalent visibly pushdown automaton that is trimmed. We then show how this procedure can be lifted to the setting of arbitrary VPA. Furthermore, we present a way of building, given a VPA, an equivalent VPA which is both deterministic and trimmed. Last, our trimming procedures can be applied to weighted VPA.

Visibly rational expressions

Regular expressions (RE) are an algebraic formalism for expressing regular languages, widely used in string search and as a specification language in verification. In this paper, we introduce and investigate visibly rational expressions (VRE), an extension of RE for the class of visibly pushdown languages (VPL). We show that VRE capture precisely the class of VPL. Moreover, we identify an equally expressive fragment of VRE which admits a quadratic time compositional translation into the automata acceptors of VPL. We also prove that, for this fragment, universality, inclusion and language equivalence are EXPTIME-complete. Finally, we provide an extension of VRE for VPL over infinite words.

High-performance complex event processing over hierarchical data

While Complex Event Processing (CEP) constitutes a considerable portion of the so-called Big Data analytics, current CEP systems can only process data having a simple structure, and are otherwise limited in their ability to efficiently support complex continuous queries on structured or semistructured information. However, XML-like streams represent a very popular form of data exchange, comprising large portions of social network and RSS feeds, financial feeds, configuration files, and similar applications requiring advanced CEP queries. In this article, we present the XSeq language and system that support CEP on XML streams, via an extension of XPath that is both powerful and amenable to an efficient implementation. Specifically, the XSeq language extends XPath with natural operators to express sequential and Kleene-* patterns over XML streams, while remaining highly amenable to efficient execution. In fact, XSeq is designed to take full advantage of the recently proposed Visibly Pushdown Automata (VPA), where higher expressive power can be achieved without compromising the computationally attractive properties of finite state automata. Besides the efficiency and expressivity benefits, the choice of VPA as the underlying model also enables XSeq to go beyond XML streams and be easily applicable to any data with both sequential and hierarchical structures, including JSON messages, RNA sequences, and software traces. Therefore, we illustrate the XSeq's power for CEP applications through examples from different domains and provide formal results on its expressiveness and complexity. Finally, we present several optimization techniques for XSeq queries. Our extensive experiments indicate that XSeq brings outstanding performance to CEP applications: two orders of magnitude improvement is obtained over the same queries executed in general-purpose XML engines.

Operator precedence and the visibly pushdown property

Floydʼs operator precedence grammars and languages (FG, FL) are a classical subclass of deterministic context-free (DCF) grammars and languages. We prove that several recently introduced language families motivated by the needs of model checking and of specifying XML-like languages are proper subsets of FL. The main cases considered include visibly pushdown languages (VPL) and balanced languages (BALAN), which are characterized by restricted precedence relations. FL have all the closure properties available for regular languages and generally viewed as necessary for application to model checking: reversal, prefixing and suffixing, concatenation, Kleene star, and boolean operations. All but the last results are new, and some require complex proofs, due to the necessary changes of syntax structure. Thus FL are the largest known subfamily of DCF having the same closure properties as VPL. FG, unlike VPL grammars, which are intended for abstract syntax modelling, are structurally adequate to specify real programming languages.

Rewriting of visibly pushdown languages for XML data integration

In this work, we focus on XML data integration by studying rewritings of XML target schemas in terms of source schemas. Rewriting is very important in data integration systems where the system is asked to find and assemble XML documents from the data sources and produce documents that satisfy a target schema. As schema representation, we consider Visibly Pushdown Automata (VPAs), which accept Visibly Pushdown Languages (VPLs). The latter have been shown to coincide with the family of (word-encoded) regular tree languages, which are the basis of formalisms for specifying XML schemas. Furthermore, practical semi-formal XML schema specifications (defined by simple pattern conditions on XML) compile into VPAs that are exponentially more concise than other representations based on tree automata. Notably, VPLs enjoy a “well-behavedness” that facilitates us in addressing rewriting problems for XML data integration. Based on VPAs, we positively solve these problems, and present detailed complexity analyses.

Evolving schemas for streaming XML

In this paper we model schema evolution for XML by defining formal language operators on Visibly Pushdown Languages (VPLs). Our goal is to provide a framework for efficient validation of streaming XML in the realistic setting where the schemas of the exchanging parties evolve and thus diverge from one another. We show that Visibly Pushdown Languages are closed under the defined language operators and this enables us to expand the schemas (for XML) in order to account for flexible or constrained evolution.

Checking On-the-Fly Universality and Inclusion Problems of Visibly Pushdown Automata

Visibly pushdown automata (VPA), introduced by Alur and Madhusuan in 2004, is a subclass of pushdown automata whose stack behavior is completely determined by the input symbol according to a fixed partition of the input alphabet. Since it was introduced, VPA have been shown to be useful in various contexts, e.g., as specification formalism for verification and as an automaton model for processing XML streams. However, implementation of formal verification based on VPA framework is a challenge. In this paper, we propose on-the-flyalgorithms to test universality and inclusion problems of this automata class. In particular, we first present a slight improvement on the upper bound for determinization of VPA. Next, in order to check universality of a nondeterministic VPA, we simultaneously determinize this VPA and apply the P-automata technique to compute a set of reachable configurations of the target determinized VPA. When a rejecting configuration is found, the checking process stops and reports that the original VPA is not universal. Otherwise, if all configurations are accepting, the original VPA is universal. Furthermore, to strengthen the algorithm, we define a partial ordering over transitions of P-automaton, and only minimal transitions are used to incrementally generate the P-automaton. The purpose of this process is to keep the determinization step implicitly for generating reachable configurations as minimum as possible. This improvement helps to reduce not only the size of the P-automaton but also the complexity of the determinization phase. We implement the proposed algorithms in a prototype tool, named VPAchecker. Finally, we conduct experiments on randomly generated VPA. The experimental results show that the proposed method outperforms the standard one by several orders of magnitude.

A Tighter Bound for the Determinization of Visibly Pushdown Automata

Visibly pushdown automata (VPA), introduced by Alur and Madhusuan in 2004, is a subclass of pushdown automata whose stack behavior is completely determined by the input symbol according to a fixed partition of the input alphabet. Since its introduce, VPAs have been shown to be useful in various context, e.g., as specification formalism for verification and as automaton model for processing XML streams. Due to high complexity, however, implementation of formal verification based on VPA framework is a challenge. In this paper we consider the problem of implementing VPA-based model checking algorithms. For doing so, we first present an improvement on upper bound for determinization of VPA. Next, we propose simple on-the-fly algorithms to check universality and inclusion problems of this automata class. Then, we implement the proposed algorithms in a prototype tool. Finally, we conduct experiments on randomly generated VPAs. The experimental results show that the proposed algorithms are considerably faster than the standard ones.

Nested Sibling Tree Automata

In the XML standard, data are represented as unranked labeled ordered trees. Regular unranked tree automata provide a useful formalism for the validation of schemas enforcing regular structural constraints on XML documents. However some concrete application contexts need the expression of more general constraints than the regular ones. In this paper we propose a new framework in which context-free style structural constraints can be expressed and validated. This framework is characterized by: (i) the introduction of a new notion of trees, the so-called typed unranked labeled trees (tulab trees for short) in which each node receives one of three possible types (up, down or fix), and (ii) the definition of a new notion of tree automata, the so-called nested sibling tulab tree automata, able to enforce context-free style structural constraints on tulab tree languages. During their structural control process, such automata are using visibly pushdown languages of words [R. Alur and P. Madhusudan, Visibly pushdown languages, 36th ACM symposium on Theory of Computing, Chicago, USA (2004) 202–211] on their alphabet of states. We show that the resulting class NSTL of tulab tree languages recognized by nested sibling tulab tree automata is robust, i.e. closed under Boolean operations and with decision procedures for the classical membership, emptiness and inclusion problems. We then give three characterizations of NSTL : a logical characterization by defining an adequate logic in which NSTL happens to coincide with the models of monadic second order sentences; the two other characterizations are using adequate encodings and map together languages of NSTL with some regular sets of 3-ary trees or with particular sets of binary trees.

Beyond Language Equivalence on Visibly Pushdown Automata

We study (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods for proving complexity upper and lower bounds for a number of studied preorders and equivalences like simulation, completed simulation, ready simulation, 2-nested simulation preorders/equivalences and bisimulation equivalence. Our main results are that all the mentioned equivalences and preorders are EXPTIME-complete on visibly pushdown automata, PSPACE-complete on visibly one-counter automata and P-complete on visibly BPA. Our PSPACE lower bound for visibly one-counter automata improves also the previously known DP-hardness results for ordinary one-counter automata and one-counter nets. Finally, we study regularity checking problems for visibly pushdown automata and show that they can be decided in polynomial time.

Cloud Warehousing

The exact validation of streaming XML documents can be realised by using visibly push-down automata (VPA) that are defined by Extended Document Type Def- initions (EDTD). It is straightforward to represent such an automaton as an Abstract State Machine (ASM). In doing so we enable computations on abstract states that are defined by a certain class of automata, in this case VPAs. In this paper we elaborate on this approach by taking also updates of XML documents into account. In this way the ASM-approach combines vertical refinements, which first make states explicit and then instantiate by a specific EDTD, with horizontal refinements, which replace streaming XML documents by stored ones and then add updates. Furthermore, as the EDTD appears as part of the abstract state, updating it is another natural extension by hor- izontal refinement. In this way we obtain consistently integrated updates and schema updates for XML documents, which can even be extended to become fault-tolerant by taking at most k errors in the document into consideration. It further provides an example of ASM-based computation with automata-defined states.

Propositional dynamic logic with recursive programs

We extend the propositional dynamic logic PDL of Fischer and Ladner with a restricted kind of recursive programs using the formalism of visibly pushdown automata [R. Alur, P. Madhusudan, Visibly pushdown languages, in: Procceings of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), 2004, ACM, pp. 202–211]. We show that the satisfiability problem for this extension remains decidable, generalising known decidability results for extensions of PDL by non-regular programs. Our decision procedure establishes a 2-E xpT ime upper complexity bound, and we prove a matching lower bound that applies already to rather weak extensions of PDL with non-regular programs. Thus, we also show that such extensions tend to be more complex than standard PDL.