A study on shuffle, stopwatches and independently evolving clocks

We show that stopwatch automata are equivalent with timed shuffle expressions, an extension of timed regular expressions with the shuffle operation. Since the emptiness problem is undecidable for stopwatch automata, and hence also for timed shuffle expressions, we introduce a decidable subclass of stopwatch automata called partitioned stopwatch automata. We give for this class an equivalent subclass of timed shuffle expressions and investigate closure properties by showing that partitioned stopwatch automata are closed under union, concatenation, star, shuffle and renaming, but not under intersection. We also show that partitioned stopwatch automata are equivalent with distributed time-asynchronous automata, which are asynchronous compositions of timed automata in which time may evolve independently.     

Language equations for timed alternating finite automata

Traditionally, finite state automata are untimed or asynchronous models of computation in which only the ordering of events, not the time at which events occur, would affect the result of a computation. For real-time systems, it is important to augment these models of computation with a notion of time. For this purpose timed automata have become a powerful canonical model for describing timed behaviors and an effective tool for modeling real-time computations. In this paper, we extend the notion of timed alternating finite automata (TAFA), a class of alternating finite automata (AFA) extended with a finite set of real-valued clocks, and we present an algebraic interpretation of TAFA which parallels that of timed regular expressions and language equations. We further extend the equational representation of AFA to describe timed alternating finite automata, and explore solutions for such equations over time languages.     

Axiomatising timed automata

Timed automata has been developed as a basic semantic model for real time systems. Its algorithmic aspects for automated analysis have been well studied. But so far there is still no satisfactory algebraic theory to allow the derivation of semantical equivalence of automata by purely syntactical manipulation. The aim of this paper is to provide such a theory. We present an inference system of timed bisimulation equivalence for timed automata based on a CCS-style regular language for describing timed automata. It consists of the standard monoid laws for bisimulation and a set of inference rules. The judgments of the proof system are conditional equations of the form where is a clock constraint and t,u are terms denoting timed automata. The inference system is shown to be sound and complete for timed bisimulation. The proof of the completeness result relies on the notion of symbolic timed bisimulation, adapted from the work on value–passing processes. Received: 10 May 2001 / 22 October 2001     

Modeling and verification of real-time embedded systems with urgency

Real-time embedded systems are often designed with different types of urgencies such as delayable or eager, that are modeled by several urgency variants of the timed automata model. However, most model checkers do not support such urgency semantics, except for the IF toolset that model checks timed automata with urgency against observers. This work proposes an Urgent Timed Automata (UTA) model with zone-based urgency semantics that gives the same model checking results as absolute urgency semantics of other existing urgency variants of the timed automata model, including timed automata with deadlines and timed automata with urgent transitions. A necessary and sufficient condition, called complete urgency, is formulated and proved for avoiding zone partitioning so that the system state graphs are simpler and model checking is faster. A novel zone capping method is proposed that is time-reactive, preserves complete urgency, satisfies all deadlines, and does not need zone partitioning. The proposed verification methods were implemented in the SGM CTL model checker and applied to real-time and embedded systems. Several experiments, comparing the state space sizes produced by SGM with that by the IF toolset, show that SGM produces much smaller state-spaces.     

A Kleene theorem for splitable signals

In this paper, we consider timed automata for piecewise constant signals and prove that they recognize exactly the languages denoted by signal regular expressions with intersection and renaming. The main differences from the usual timed automata are: time elapses on transitions (passing through a state is instantaneous), signals may be split on a run on an automaton and constraints on transitions correspond to unions of open intervals but should be satisfied on closed intervals. This makes exact rendez-vous impossible. The paper stresses on the similarities and differences from the usual model.     

<Emphasis Type="Italic">SetExp</Emphasis>: a method of transformation of timed automata into finite state automata

Real-time discrete event systems are discrete event systems with timing constraints, and can be modeled by timed automata. The latter are convenient for modeling real-time discrete event systems. However, due to their infinite state space, timed automata are not suitable for studying real-time discrete event systems. On the other hand, finite state automata, as the name suggests, are convenient for modeling and studying non-real time discrete event systems. To take into account the advantages of finite state automata, an approach for studying real-time discrete event systems is to transform, by abstraction, the timed automata modeling them into finite state automata which describe the same behaviors. Then, studies are performed on the finite state automata model by adapting methods designed for non real-time discrete event systems. In this paper, we present a method for transforming timed automata into special finite state automata called Set-Exp automata. The method, called SetExp, models the passing of time as real events in two types: Set events which correspond to resets with programming of clocks, and Exp events which correspond to the expiration of clocks. These events allow to express the timing constraints as events order constraints. SetExp limits the state space explosion problem in comparison to other transformation methods of timed automata, notably when the magnitude of the constants used to express the timing constraints are high. Moreover, SetExp is suitable, for example, in supervisory control and conformance testing of real-time discrete event systems.     

Duration-constrained regular expressions

This paper investigates the logic-automata-connection for Duration Calculus. It has been frequently observed that Duration Calculus with linear duration terms comes close to being a logic of linear hybrid automata. We attempt to make this relation precise by constructing Kleene-connection between duration-constrained regular expressions and a subclass of linear hybrid automata called loop-reset automata in which any variable tested in a loop is reset in the same loop. The formalism of duration-constrained regular expressions is an extension of regular expressions with duration constraints, which are essentially formulas of Duration Calculus without negation, yet extended by a Kleene-star operator. In this paper, we show that this formalism is equivalent in expressive power to loop-reset automata by providing a translation procedure from expressions to automata and vice verse.Received June 1999Accepted in revised form September 2003 by M. R. Hansen and C. B. Jones     

有关时间自动机重置的若干问题的计算复杂性

自动机的重置序列也称为同步序列,具有以下特性：有限自动机通过运行重置序列w,可从任意一个未知的或无法观测到的状态q₀到达某个特定状态q_w.这仅依赖于w,而与开始运行w时的状态q₀无关.这一特性可用于部分可观察的复杂系统的自动恢复,而无需重启,甚至有时不能重启.基于此,重置问题自出现以来便得到关注和持续研究.最近几年,它被扩展到可以描述诸如分布式、嵌入式实时系统等复杂系统的无限状态模型上,比如时间自动机和寄存器自动机等.以时间自动机的重置问题的计算复杂性为研究对象,发现重置问题与可达性问题有着紧密的联系.主要贡献是：（1）利用时间自动机可达性问题的最新成果,完善完全的确定的时间自动机重置问题的计算复杂性结论;（2）对部分规约的确定的时间自动机,研究得出,即使在输入字母表大小减至2的情况下,其复杂性仍是PSPACE-完全的;特别地,在单时钟情况下是NLOGSPACE-完全的;（3）对完全的非确定的时间自动机,研究得出其D_i-可重置问题（i=1,2,3）是不可判定的,其重置问题与非确定的寄存器自动机重置问题在指数时间可以相互归约,通过证明指数时间归约相对高复杂性类具有封闭性,利用非确定的寄存器自动机的结论得出单时钟的时间自动机的重置问题是Ackermann-完全的、限界的重置问题是NEXPTIME-完全的.这些复杂性结论,说明关于时间自动机的重置问题大都是难解的,一方面,为时间系统的可重置性的检测和求解奠定坚实的理论基础,另一方面,为以后寻找具有高效算法的特殊结构的时间系统（即具有高效算法的问题子类）给予理论指导.     

Controller synthesis for dynamic hierarchical real-time plants using timed automata

We use timed I/O automata based timed games to synthesize task-level reconfiguration services for cost-effective fault tolerance in a case study. The case study shows that state-space explosion is a severe problem for timed games. By applying suitable abstractions, we dramatically improve the scalability. However, timed I/O automata do not facilitate algorithmic abstraction generation techniques. The case study motivates the development of timed process automata to improve modeling and analysis for controller synthesis of time-critical plants which can be hierarchical and dynamic. The model offers two essential features for industrial systems: (i) compositional modeling with reusable designs for different contexts, and (ii) state-space reduction technique. Timed process automata model dynamic networks of continuous-time communicating plant processes which can activate other plant processes. We show how to establish safety and reachability properties of timed process automata by reduction to solving timed games. To mitigate the state-space explosion problem, an algorithmic state-space reduction technique using compositional reasoning and aggressive abstractions is also proposed. In this article, we demonstrate the theoretical framework of timed process automata and the effectiveness of the proposed state-space reduction technique by extending the case study.     

Exact Acceleration of Real-Time Model Checking

Different time scales do often occur in real-time systems, e.g., a polling real-time system samples the environment many times per second, whereas the environment may only change a few times per second. When these systems are modeled as (networks of) timed automata, the validation using symbolic model checking techniques can significantly be slowed down by unnecessary fragmentation of the symbolic state space. This paper introduces a syntactical adjustment to a subset of timed automata that addresses this fragmentation problem and that can speed-up forward symbolic reachability analysis in a significant way. We prove that this syntactical adjustment does not alter reachability properties and that it indeed is effective. We illustrate our exact acceleration technique with run-time data obtained with the model checkers Uppaal and Kronos. Moreover, we demonstrate that automated application of our exact acceleration technique can significantly speed-up the verification of the run-time behavior of LEGO Mindstorms programs.     

Bounded Model Checking for Timed Automata

Given a timed automaton M, a linear temporal logic formula φ, and a bound k, bounded model checking for timed automata determines if there is a falsifying path of length k to the hypothesis that M satisfies the specification φ. This problem can be reduced to the satisfiability problem for Boolean constraint formulas over linear arithmetic constraints. We show that bounded model checking for timed automata is complete, and we give lower and upper bounds for the length k of counterexamples. Moreover, we define bounded model checking for networks of timed automata in a compositional way.     

Grail: A C++ Library for Automata and Expressions

Grail is a package for symbolic manipulation of finite-state automata and regular expressions. It provides most standard operations on automata and expressions, including minimization, subset construction, conversion between automata and regular expressions, and language enumeration and testing. Grail 's objects are parameterizable; users can provide their own classes to define the input alphabet of automata and expressions. Grail's operations are accessible either as individual programs or directly through a C++ class library.     

Interval Duration Logic: Expressiveness and Decidability

We investigate a variant of dense-time Duration Calculus which permits model checking using timed/hybrid automata. We define a variant of the Duration Calculus, called Interval Duration Logic, (IDL), whose models are timed state sequences [1].A subset LIDL of IDL consisting only of located time constraints is presented. As our main result, we show that the models of an LIDL formula can be captured as timed state sequences accepted by an event-recording integrator automaton. A tool called IDLVALID for reducing LIDL formulae to integrator automata is briefly described. Finally, it is shown that LIDL has precisely the expressive power of event-recording integrator automata, and that a further subset LIDL- corresponds exactly to event-recording timed automata [2]. This gives us an automata-theoretic decision procedure for the satisfiability of LIDL– formulae.     

Timed substitutions for regular signal-event languages

In the classical framework of formal languages, a refinement operation is modeled by a substitution and an abstraction by an inverse substitution. These mechanisms have been widely studied, because they describe a change in the specification level, from an abstract view to a more concrete one, or conversely. For timed systems, there is up to now no uniform notion of substitution. In this paper, we study timed substitutions in the general framework of signal-event languages, where both signals and events are taken into account. We prove that regular signal-event languages are closed under substitution and inverse substitution. To obtain these results, we use in a crucial way a “well known” result: regular signal-event languages are closed under intersection. In fact, while this result is indeed easy for languages defined by Alur and Dill’s timed automata, it turns out that the construction is much more tricky when considering the most involved model of signal-event automata. We give here a construction working on finite and infinite signal-event words and taking into account signal stuttering, unobservability of zero-duration τ-signals and Zeno runs. Note that if several constructions have been proposed in particular cases, it is the first time that a general construction is provided.     

An algebraic approach to data languages and timed languages

Algebra offers an elegant and powerful approach to understand regular languages and finite automata. Such framework has been notoriously lacking for timed languages and timed automata. We introduce the notion of monoid recognizability for data languages, which includes timed languages as special case, in a way that respects the spirit of the classical situation. We study closure properties and hierarchies in this model and prove that emptiness is decidable under natural hypotheses. Our class of recognizable languages properly includes many families of deterministic timed languages that have been proposed until now, and the same holds for non-deterministic versions.     

A Kleene–Schützenberger theorem for weighted timed automata

During the last years, weighted timed automata have received much interest in the real-time community. Weighted timed automata form an extension of timed automata and allow us to assign weights (costs) to both locations and edges. This model, introduced by Alur et al. (2001) and Behrmann et al. (2001), permits the treatment of continuous consumption of resources and has led to much research on scheduling problems, optimal reachability and model checking. Also, several authors have derived Kleene-type characterizations of (unweighted) timed automata and their accepted timed languages. The goal of this paper is to provide a characterization of the behaviours of weighted timed automata by rational power series. We define weighted timed automata with weights taken in an arbitrary semiring, resulting in a model that subsumes several weighted timed automata concepts of the literature. For our main result, we combine the methods of Schützenberger, a recent approach for a Kleene-type theorem for unweighted timed automata by Bouyer and Petit as well as new techniques. Our main result also implies Kleene-type theorems for several subclasses of weighted timed automata investigated before, e.g., for timed automata and timed automata with stopwatch observers.     

Reactive automata

A reactive automaton has extra links whose role is to change the behaviour of the automaton. We show that these links do not increase the expressiveness of finite automata but that they can be used to reduce dramatically their state number both in the deterministic case and the non-deterministic case.Typical examples of regular expressions associated with deterministic automata of exponential size according to the length of the expression show that reactive links provide an alternative representation of total linear size for the language.     

Algorithms for learning regular expressions from positive data

We describe algorithms that directly infer very simple forms of 1-unambiguous regular expressions from positive data. Thus, we characterize the regular language classes that can be learned this way, both in terms of regular expressions and in terms of (not necessarily minimal) deterministic finite automata.     

Series Parallel Digraphs with Loops

In the conversion of finite automata to regular expressions, an exponential blowup in size can generally not be avoided. This is due to graph-structural properties of automata which cannot be directly encoded by regular expressions and cause the blowup combinatorially. In order to identify these structures, we generalize the class of arc-series-parallel digraphs beyond the acyclic case. The resulting digraphs are shown to be reversibly encoded by linear-sized regular expressions. Also, a characterization of this new class by a set of seven forbidden substructures is given. Automata that require expressions of superlinear size must contain some of these substructures.