Automata operating on strings of nested brackets, known as input-driven pushdown automata, and as visibly pushdown automata, have been studied since the 1980s. They were extended to the case of infinite strings by Alur and Madhusudan ("Visibly pushdown languages", STOC 2004). This paper investigates the properties of these automata under the assumption that a given infinite string is always well-nested. This restriction enables a complete characterization of the corresponding $ω$-languages in terms of classical $ω$-regular languages and input-driven automata on finite strings. This characterization leads to a determinization result for these automata, as well as to the first results on their Wadge degrees.
Recently there has been a significant effort to handle quantitative properties in formal verification and synthesis. While weighted automata over finite and infinite words provide a natural and flexible framework to express quantitative properties, perhaps surprisingly, some basic system properties such as average response time cannot be expressed using weighted automata, nor in any other know decidable formalism. In this work, we introduce nested weighted automata as a natural extension of weighted automata which makes it possible to express important quantitative properties such as average r…
In this paper we introduce and study Event-Clock Nested Automata (ECNA), a formalism that combines Event Clock Automata (ECA) and Visibly Pushdown Automata (VPA). ECNA allow to express real-time properties over non-regular patterns of recursive programs. We prove that ECNA retain the same closure and decidability properties of ECA and VPA being closed under Boolean operations and having a decidable language-inclusion problem. In particular, we prove that emptiness, universality, and language-inclusion for ECNA are EXPTIME-complete problems. As for the expressiveness, we have that ECNA properly…
Automata over infinite alphabets have recently come to be studied extensively as potentially useful tools for solving problems in verification and database theory. One popular model of automata studied is the Class Memory Automata (CMA), for which the emptiness problem is equivalent to Petri Net Reachability. We identify a restriction - which we call weakness - of CMA, and show that their emptiness problem is equivalent to Petri Net Coverability. Further, we show that in the deterministic case they are closed under all Boolean operations. We clarify the connections between weak CMA and existin…
While weighted automata provide a natural framework to express quantitative properties, many basic properties like average response time cannot be expressed with weighted automata. Nested weighted automata extend weighted automata and consist of a master automaton and a set of slave automata that are invoked by the master automaton. Nested weighted automata are strictly more expressive than weighted automata (e.g., average response time can be expressed with nested weighted automata), but the basic decision questions have higher complexity (e.g., for deterministic automata, the emptiness quest…
Nested weighted automata (NWA) present a robust and convenient automata-theoretic formalism for quantitative specifications. Previous works have considered NWA that processed input words only in the forward direction. It is natural to allow the automata to process input words backwards as well, for example, to measure the maximal or average time between a response and the preceding request. We therefore introduce and study bidirectional NWA that can process input words in both directions. First, we show that bidirectional NWA can express interesting quantitative properties that are not express…
Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Büchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider …
Hankel matrices (aka connection matrices) of word functions and graph parameters have wide applications in automata theory, graph theory, and machine learning. We give a characterization of real-valued functions on nested words recognized by weighted visibly pushdown automata in terms of Hankel matrices on nested words. This complements C. Mathissen's characterization in terms of weighted monadic second order logic.
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Slovak University of Technology in Bratislava（布拉迪斯拉发斯洛伐克工业大学）；CISPA Helmholtz Center for Information Security（CISPA赫尔霍夫信息安全中心）；Sabancı University（萨班奇大学）
Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from ne…
In 2007 Kambites presented an algebraic interpretation of Chomsky-Schutzenberger theorem for context-free languages. We give an interpretation of the corresponding theorem for the class of displacement context-free languages which are equivalent to well-nested multiple context-free languages. We also obtain a characterization of k-displacement context-free languages in terms of monoid automata and show how such automata can be simulated on two stacks. We introduce the simultaneous two-stack automata and compare different variants of its definition. All the definitions considered are shown to …
Input-driven pushdown automata (also known as visibly pushdown automata and as nested word automata) are a subclass of deterministic pushdown automata and a superclass of the parenthesis languages. Nguyen and Ogawa ("Event-clock visibly pushdown automata", SOFSEM 2009) defined a timed extension of these automata under the event-clock model, and showed that this model can be determinized using the method of region construction. This paper defines a further extension of this model with the event clock on the call-return operations, and proposes a new, direct determinization procedure for these a…
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. As trees and more generally hedges can be linearized into (well) nested words, VPTs are a natural formalism to express tree transformations evaluated in streaming. This paper aims at characterizing precisely the expressive power of VPTs with respect to other tree transducer models.
The Emptiness Problem for Quantum Finite Automata with Classical States
National Taipei University of Technology（国立台北科技大学）；School of Computer Science and Informatics, University of Liverpool（利物浦大学计算机科学与信息学院）；National Institute of Informatics and SOKENDAI（日本国立情报学研究所与综合研究大学院大学）
In this paper we adopt a category-theoretic approach to the conception of automata classes enjoying minimization by design. The main instantiation of our construction is a new class of automata that are hybrid between deterministic automata and automata weighted over a field.
Regular nested word languages (a.k.a. visibly pushdown languages) strictly extend regular word languages, while preserving their main closure and decidability properties. Previous works have shown that considering languages of 2-nested words, i.e. words enriched with two matchings (a.k.a. 2-visibly pushdown languages), is not as successful: the corresponding model of automata is not closed under determinization. In this work, inspired by homomorphic representations of indexed languages, we identify a subclass of 2-nested words, which we call 2-wave words. This class strictly extends the class …
We follow language theoretic approach to synchronizing automata and Černý's conjecture initiated in a series of recent papers. We find a precise lower bound for the reset complexity of a principal ideal languages. Also we show a strict connection between principal left ideals and synchronizing automata. We characterize regular languages whose minimal deterministic finite automaton is synchronizing and possesses a reset word belonging to the recognized language.
We show that the shuffle $L \unicode{x29E2} F$ of a piecewise-testable language $L$ and a finite language $F$ is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of $L \unicode{x29E2} F$.
In this paper we regard languages and their acceptors -- such as deterministic or weighted automata, transducers, or monoids -- as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be applied to several…
Patterned self-assembly is a process whereby coloured tiles self-assemble to build a rectangular coloured pattern. We propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. The adva…
We use a semigroup-theoretic construction by Peter Higgins in order to produce, for each even $n$, an $n$-state and 3-letter synchronizing automaton with the following two features: 1) all its input letters act as idempotent selfmaps of rank $\dfrac{n}2$; 2) its reset threshold is asymptotically equal to $\dfrac{n^2}2$. In the revised version a few inaccuracies (spotted by the anonymous referees of the previous version) have been removed and several relevant references have been added.
Extensions to finite-state automata on strings, such as multi-head automata or multi-counter automata, have been successfully used to encode many infinite-state non-regular verification problems. In this paper, we consider a generalization of automata-theoretic infinite-state verification from strings to labeled series-parallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on series-parallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of series-parallel graphs de…
Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions([14]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in [4]. In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linearbounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extend…
A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its syntactic monoid has a zero element, by means of Eilenberg's variety theoretic approach. Our proof gives an effective automata characterisation of the zero-one law for regular languages, and it leads to a linear time algorithm for testing whether a given regular language is zero-one. In addition, we discuss the logical aspects of the zero-one law for regular langua…
Visibly pushdown transducers form a subclass of pushdown transducers that (strictly) extends finite state transducers with a stack. Like visibly pushdown automata, the input symbols determine the stack operations. In this paper, we prove that functionality is decidable in PSpace for visibly pushdown transducers. The proof is done via a pumping argument: if a word with two outputs has a sufficiently large nesting depth, there exists a nested word with two outputs whose nesting depth is strictly smaller. The proof uses technics of word combinatorics. As a consequence of decidability of functiona…
We introduce an automata-theoretic method for the verification of distributed algorithms running on ring networks. In a distributed algorithm, an arbitrary number of processes cooperate to achieve a common goal (e.g., elect a leader). Processes have unique identifiers (pids) from an infinite, totally ordered domain. An algorithm proceeds in synchronous rounds, each round allowing a process to perform a bounded sequence of actions such as send or receive a pid, store it in some register, and compare register contents wrt. the associated total order. An algorithm is supposed to be correct indepe…
We define a new kind of automata recognizing properties of data words or data trees and prove that the automata capture all queries definable in Regular XPath. We show that the automata-theoretic approach may be applied to answer decidability and expressibility questions for XPath.
The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with ${\le_{tt}^p}(\mathsf{NP})$, that is, the class of decision problems polynomial-time truth-table reducible to problems in $\mathsf{NP}$. An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select…
University of Leicester（莱斯特大学）；Gran Sasso Science Institute（格兰萨索科学研究所）
We investigate a learning algorithm in the context of nominal automata, an extension of classical automata to alphabets featuring names. This class of automata captures nominal regular languages; analogously to the classical language theory, nominal automata have been shown to characterise nominal regular expressions with binders. These formalisms are amenable to abstract modelling resource-aware computations. We propose a learning algorithm on nominal regular languages with binders. Our algorithm generalises Angluin's L* algorithm with respect to nominal regular languages with binders. We sho…
We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.
We consider infinite-state Attacker-Defender games with reachability objectives. The results of the paper are twofold. Firstly we prove a new language-theoretic result for weighted automata on infinite words and show its encoding into the framework of Attacker-Defender games. Secondly we use this novel concept to prove undecidability for checking existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games.
Unordered, variable-sized inputs arise in many settings across multiple fields. The ability for set- and multiset-oriented neural networks to handle this type of input has been the focus of much work in recent years. We propose to represent multisets using complex-weighted multiset automata and show how the multiset representations of certain existing neural architectures can be viewed as special cases of ours. Namely, (1) we provide a new theoretical and intuitive justification for the Transformer model's representation of positions using sinusoidal functions, and (2) we extend the DeepSets m…
Given the versatility of timed automata a huge body of work has evolved that considers extensions of timed automata. One extension that has received a lot of interest is timed automata with a, possibly unbounded, stack, also called the pushdown timed automata (PDTA) model. While different algorithms have been given for reachability in different variants of this model, most of these results are purely theoretical and do not give rise to efficient implementations. One main reason for this is that none of these algorithms (and the implementations that exist) use the so-called zone-based abstracti…
Timed automata are a common formalism for the verification of concurrent systems subject to timing constraints. They extend finite-state automata with clocks, that constrain the system behavior in locations, and to take transitions. While timed automata were originally designed for safety (in the wide sense of correctness w.r.t. a formal property), they were progressively used in a number of works to guarantee security properties. In this work, we review works studying security properties for timed automata in the last two decades. We notably review theoretical works, with a particular focus o…
The locality of a graph problem is the smallest distance $T$ such that each node can choose its own part of the solution based on its radius-$T$ neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem $Π$, we would like to determine if $Π$ is solvable and what is the asymptotic locality of $Π$ as a function of the size of the graph. Put otherwise, we seek to automatically synthesize ef…
