Linear-time Logic Research Articles

Towards the hierarchical verification of reactive systems

The hierarchical design approach for action based systems that is known as action refinement has been studied in the literature extensively. In a paper of M. Huhn published in CONCUR 1996 a refinement operator on a linear time logic is presented that mimics precisely a semantic action refinement on synchronisation structures. We present here an alternative approach where our starting point is a process algebraic setting with a syntactic action refinement. We present a refinement operator on the Modal Mu-calculus that conforms with the process algebraic refinement in the following sense: Provided some reasonable conditions are met, the transition system induced by a process term P satisfies a Modal Mu-Calculus-specification ϕ if and only if the system which is induced by a refinement of P satisfies a particular refinement of ϕ . Alleviating these conditions, we show that each of the two implications in the equivalence assertion above can be separately proven valid for a particular fragment of the Modal Mu-calculus. We demonstrate that the obtained results can indeed be used as a hierarchical verification technique. As a further application of our results, we explain how they can be employed as an abstraction technique in order to enhance model checking techniques.

Representing and Reasoning about Temporal Granularities

In this paper, we propose a new logical approach to represent and to reason about different time granularities. We identify a time granularity as an infinite sequence of time points properly labelled with proposition symbols marking the starting and ending points of the corresponding granules, and we symbolically model sets of granularities by means of linear time logic formulas. Some real-world granularities are provided, from a clinical domain and from the Gregorian Calendar, to motivate and exemplify our approach. Different formulas are introduced, which represent relations between different granularities. The proposed framework permits one to algorithmically solve the consistency, the equivalence, and the classification problems in a uniform way, by reducing them to the validity problem for the considered linear time logic.

Towards the hierarchical verification of reactive systems

The hierarchical design approach for action based systems that is known as action refinement has been studied in the literature extensively. In a paper of M. Huhn published in CONCUR 1996 a refinement operator on a linear time logic is presented that mimics precisely a semantic action refinement on synchronisation structures. We present here an alternative approach where our starting point is a process algebraic setting with a syntactic action refinement. We present a refinement operator on the Modal Mu-calculus that conforms with the process algebraic refinement in the following sense: Provided some reasonable conditions are met, the transition system induced by a process term P satisfies a Modal Mu-Calculus-specification ϕ if and only if the system which is induced by a refinement of P satisfies a particular refinement of ϕ. Alleviating these conditions, we show that each of the two implications in the equivalence assertion above can be separately proven valid for a particular fragment of the Modal Mu-calculus. We demonstrate that the obtained results can indeed be used as a hierarchical verification technique. As a further application of our results, we explain how they can be employed as an abstraction technique in order to enhance model checking techniques.

An Expressively Complete Linear Time Temporal Logic for Mazurkiewicz Traces

A basic result concerning LTL , the propositional temporal logic of linear time, is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL -specifications. We show that LTrL , our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also constitutes a characterisation of the so-called trace consistent (robust) LTL -specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.

An Infinite Hierarchy of Temporal Logics over Branching Time

Many temporal logics have been suggested as branching time specification formalisms during the past 20 years. These logics were compared against each other for their expressive power, model checking complexity, and succinctness. Yet, unlike the case for linear time logics, no canonical temporal logic of branching time was agreed upon. We offer an explanation for the multiplicity of temporal logics over branching time and provide an objective quantified yardstick to measure these logics. We define an infinite hierarchy BTLk of temporal logics and prove its strictness. We examine the expressive power of commonly used branching time temporal logics. We show that CTL* has no finite base, and that almost all of its many sublogics suggested in the literature are inside the second level of our hierarchy. We introduce new Ehrenfeucht–Fraissé games on trees and use them as our main tool to prove inexpressibility.

A BELIEF-GOAL-ROLE THEORY FOR MULTIAGENT SYSTEMS

Agent-oriented techniques represent a promising approach for engineering complex systems where interaction is probably the most important single characteristic. Accordingly, the recent years have witnessed the emergence of different approaches for the study of intelligent agent-based systems. One such architecture views the system as a rational agent having certain mental attitudes of Belief, Desire and Intention (BDI). This paper explores a particular type of rational agent, a Belief-Goal-Role agent. Unlike most previous work, our approach has been to characterize the mental state of the agents that leads them to take part in cooperative action. Hence, beliefs, goals and roles are relevant to our study of cooperation which have lead to the identification of communication concepts (beliefs and goals) and organization concepts (roles). The model is formalized by expressing it as a theory in a first-order, multi-modal, and linear-time logic. We use labeled transition systems to deal with the truth conditions of the formulae of our theory as well as with the behavioral semantics of our agents. We illustrate our work with the well-known prey/predator game.

Reasoning about actions in dynamic linear time temporal logic

In this paper we present a theory for reasoning about actions which is based on Dynamic Linear Time Temporal Logic (DLTL). DLTL is a simple extension of propositional temporal logic of linear time in which regular programs of propositional dynamic logic can be used for indexing temporal modalities. The action theory we define allows to reason with incomplete initial states, to do postdiction and to deal with ramifications and with nondeterministic actions, which are captured by possibly alternative extensions (temporal models). The expressiveness of temporal logic is exploited to enhance the action language by allowing the definition of general temporal constraints as well as complex actions in the specification of the domain description. We show that the temporal projection problem and the planning problem can be modelled as satisfiability problems in DLTL.

TALplanner: A temporal logic based forward chaining planner

We present TALplanner, a forward-chaining planner based on the use of domain-dependent search control knowledge represented as formulas in the Temporal Action Logic (TAL). TAL is a narrative based linear metric time logic used for reasoning about action and change in incompletely specified dynamic environments. TAL is used as the formal semantic basis for TALplanner, where a TAL goal narrative with control formulas is input to TALplanner which then generates a TAL narrative that entails the goal and control formulas. The sequential version of TALplanner is presented. The expressivity of plan operators is then extended to deal with an interesting class of resource types. An algorithm for generating concurrent plans, where operators have varying durations and internal state, is also presented. All versions of TALplanner have been implemented. The potential of these techniques is demonstrated by applying TALplanner to a number of standard planning benchmarks in the literature.

Dynamic linear time temporal logic

A simple extension of the propositional temporal logic of linear time is proposed. The extension consists of strengthening the until operator by indexing it with the regular programs of propositional dynamic logic. It is shown that DLTL, the resulting logic, is expressively equivalent to the monadic second-order theory of ω-sequences. In fact, a sublogic of DLTL which corresponds to propositional dynamic logic with a linear time semantics is already expressively complete. We show that DLTL has an exponential time decision procedure and admits a finitary axiomatization. We also point to a natural extension of the approach presented here to a distributed setting.

Decidability of model checking for infinite-state concurrent systems

We study the decidability of the model checking problem for linear and branching time logics, and two models of concurrent computation, namely Petri nets and Basic Parallel Processes.

A Product Version of Dynamic Linear Time Temporal Logic

We present here a linear time temporal logic which simultaneously extends LTL, the propositional temporal logic of linear time, along two dimensions. Firstly, the until operator is strengthened by indexing it with the regular programs of propositional dynamic logic (PDL). Secondly, the core formulas of the logic are decorated with names of sequential agents drawn from fixed finite set. The resulting logic has a natural semantics in terms of the runs of a distributed program consisting of a finite set of sequential programs that communicate by performing common actions together. We show that our logic, denoted DLTL, admits an exponential time decision procedure. We also show that DLTL is expressively equivalent to the so called regular product languages.

An Expressively Complete Linear Time Temporal Logic for Mazurkiewicz Traces

<p>A basic result concerning LTL, the propositional temporal logic of linear time, is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL-specifications.</p><p>We show that LTrL, our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also provides a syntactic characterisation of the so-called trace consistent (robust) LTL-specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.</p>

A Temporal Logic for Proving Properties of Topologically General Executions

We present a generalization of the temporal propositional logic of linear time which is useful for stating and proving properties of the generic execution sequence of a parallel program or a non-deterministic program. The formal system we present is exactly that same as the third of three logics presented by Lehmann and Shelah ( Information and Control 53 , 165–198 (1982)), but we give it a different semantics. The models are tree models of arbitrary size similar to those used in branching time temporal logic. The formulation we use allows us to state properties of the “co-meagre” family of paths, where the term “co-meagre” refers to a set whose complement is of the first category in Baire's classification looking at the set of paths in the model as a metric space. Our system is decidable, sound, and, complete for models of arbitrary size, but it has the finite model property; namely, every sentence having a model has a finite model.

Foundations of linear-time logic programming

This paper lays down foundations of logic programming based on a linear-time temporal logic with unbounded past and future. The resulting logic programming language is called Chronolog(ℒ). The language is suitable for applications involving the notion of dynamic change such as modeling periodical changes, historical databases and non-terminating computations. The declarative semantics of Chronolog(ℒ) programs is given in terms of temporal Herbrand models and the operational semantics in terms of a resolution-type proof procedure called TiSLD-resolution. It is shown that TiSLD-resolution is a sound and complete proof procedure. The equivalence of the declarative and the operational semantics of Chronolog(ℒ) programs is established. Extensions of Chronolog (ℒ) with extra temporal and modal operators such as “eventually” and “henceforth” are considered.

Nondeterminism within $P^ * $

Classes of machines using very limited amounts of nondeterminism are studied. The P=? NP question is related to questions about classes lying within P. Complete sets for these classes are given.

Compositionality in the temporal logic of concurrent systems (extended abstract)

In this paper we develop a new method for describing in temporal logic in a compositional manner sequential composition, its iterated version (loops) and its interaction with (nested) parallelism. The logic we will use will be a linear time logic with as sole temporal operator the until. We will illustrate this method in the construction of a compositional proof system for a CSP-like language. We will prove this system to be sound and (relative) complete.

A complete axiomatic characterization of first-order temporal logic of linear time

As shown in (Szalas, 1986, 1986, 1987) there is no finitistic and complete axiomatization of First-Order Temporal Logic of linear and discrete time. In this paper we give an infinitary proof system for the logic. We prove that the proof system is sound and complete. We also show that any syntactically consistent temporal theory has a model. As a corollary we obtain that the Downward Theorem of Skolem, Lowenheim and Tarski holds in the case of considered logic.

Modalities for model checking: branching time logic strikes back

We consider automatic verification of finite state concurrent programs. The global state graph of such a program can be viewed as a finite (Kripke) structure, and a model checking algorithm can be given for determining if a given structure is a model of a specification expressed in a propositional temporal logic. In this paper, we present a unified approach for efficient model checking under a broad class of generalized fairness constraints in a branching time framework extending that of Clarke et al. (1983). Our method applies to any type of fairness expressed in a certain canonical form. Almost all ‘practical’ types of fairness from the literature, including the fundamental notions of impartiality, weak fairness, and strong fairness, can be succintly written in our canonical form. Moreover, our branching time approach can easily be adapted to handle types of fairness (such as fair reachability of a predicate) which cannot even be expressed in a linear temporal logic. We go on to argue that branching time logic is always better than linear time logic for model checking. We show that given any model checking algorithm for any system of linear time logic (in particular, for the usual system of linear time logic) there is a model checking algorithm of the same order of complexity (in both the structure and formula size) for the corresponding full branching time logic which trivially subsumes the linear time logic in expressive power (in particular, for the system of full branching time logic CTL *). We also consider an application of our work to the theory of finite automata on infinite strings.

Reasoning with time and chance

The temporal propositional logic of linear time is generalized to an uncertain world, in which random events may occur. The formulas do not mention probabilities explicitly, i.e., the only probability appearing explicitly in formulas is probability one. This logic is claimed to be useful for stating and proving properties of probabilistic programs. It is convenient for proving those properties that do not depend on the specific distribution of probabilities used in the program's random draws. The formulas describe properties of execution sequences. The models are stochastic systems, with state transition probabilities. Three different axiomatic systems are proposed and shown complete for general models, finite models, and models with bounded transition probabilities, respectively. All three systems are decidable, by the results of Rabin ( Trans. Amer. Math. Soc. 141 (1969), 1–35).