The Influence of Temporal Logic on Finite Automata

Theoretical Computer Science, 1992

In this paper we solve some open problems raised in recent publications of the Computer Science Temporal Logic school represented by Manna-Pnueli [11], [12], Abadi-Manna [5], Abadi [1]-[4]. These problems concern the proof theoretic powers of the following inference systems: T 0 introduced in [11], [12] and reformulated in [1]-[4]; the resolution system R of [5]; and T 1 , T 2 of [1]-[4]. We use first-order temporal logic (FTL) with modalities , [F ], and U denoting "nexttime", "always-in-the-future", and "until" respectively. Given a first-order similarity type or language L, the usual predicate etc. symbols of L are considered to be rigid, i.e. their meanings do not change in time. Similarly, individual variables x i (i ∈ ω) are rigid. To this we add an infinity y i (i ∈ ω) of flexible constants. That is, the meaning of y i is allowed to change in time. Other authors, see e.g. Abadi [1]-[4], add flexible predicates too, but we will not need them here though we will mention them occasionally. Our theorems remain true even if we allow flexible predicateand function symbols, as it will be very easy to see. F m(F T L) denotes the set of all FTL-formulas (of some fixed similarity type L) defined above. For semantic purposes, we use classical two-sorted models M = < T, D, f 0 ,. .. , f i ,. .. > i∈ω where D is a classical first-order structure of similarity type L, T = < T, 0, suc, ≤, +, × > is a structure of the same similarity type as the standard model N = < ω, 0, suc, ≤, +, × > of arithmetic, and for i ∈ ω, f i , a function from T into D, serves to interpret the flexible constant y i. T is called the time-frame of M, and, except for its language, is arbitrary. M od denotes the class of all models M of the above kind. (The members of M od are basically the same as Kripke models known from the

Axioms

The article discusses minimal temporal logic systems built on the basis of classical logic as well as intuitionistic logic. The constructions of these systems are discussed as well as their basic properties. The K t system was discussed as the minimal temporal logic system built based on classical logic, while the IK t system and its modification were discussed as the minimal temporal logic system built based on intuitionistic logic.

Proceedings of the 4th International Conference on Automated Technology For Verification and Analysis, 2006

In previous work, the timed logic TCTL was extended with an "almost everywhere" Until modality which abstracts negligible sets of positions (i.e. with a null duration) along a run of a timed automaton. We propose here an extension of this logic with more powerful modalities, in order to specify properties abstracting transient states, which are events that last for less than k time units. Our main result is that modelchecking is still decidable and PSPACE-complete for this extension. On the other hand, a second semantics is defined, in which we consider the total duration where the property does not hold along a run. In this case, we prove that model-checking is undecidable. evolve at the rate of time (as in timed automata), are sometimes not expressive enough, hybrid variables (with multiple slopes) have been considered. The resulting model of hybrid automata has been largely studied in the subsequent years [16]. However, while some decidability results could be obtained [3, 18], using stopwatches (i.e. variables with slopes 0 and 1) already leads to undecidability for the reachability problem [2]. Further research has thus been devoted to weaker models where hybrid variables are only used as observers, i.e. are not tested in the automaton and thus play no role during a computation. These variables, sometimes called costs or prices in this context can be used in an optimization criterium [3, 7, 8, 11] or as constraints in temporal logic formulas. For instance, the logic WCTL [12, 10], interpreted over timed automata extended with costs, adds cost contraints on modalities: it is possible to express that a given state is reachable within a fixed cost bound. Abstracting transient states. When practical examples are considered, the need for abstracting transient states often happens. For example, modeling the instantaneous changes of a variable may introduce artificial (and thus non pertinent) transient states in the model. This motivated the work in [9], where configurations with zero duration could be abstracted by introducing into TCTL the almost everywhere U a modality. However, this is not sufficient in some cases. Contribution. In this paper, we propose an extension of TCTL called TCTL ∆ , which brings out a powerful generalization of the results in [9]. We introduce a new modality U k , where k ∈ N is a parameter, in order to abstract events that do not last continuously for at least k time units (t.u). For example, AF 2 ≤100 alarm expresses that for any execution, the atomic proposition alarm becomes true before 100 t.u and will hold for at least 2 time units. One also could express the fact that an event a precedes an event b along any run, an event being actually considered iff it lasts for at least k time units: the formula ArequestP 3 grant states that along any run where grant has occurred for a duration greater than 3, a request has been emitted continusously for a duration greater than 3. We prove that model-checking for TCTL ∆ is still PSPACE-complete. While the analogous result for TCTL or the extended version of [9] relies on the standard notion of equivalent runs, we have to define a stronger form for this equivalence, in order to obtain the consistency of TCTL ∆-formulae on the regions of the timed automaton. Finally, we also consider a global semantics, called TCTL ∆ Σ , for which the global duration during which a property does not hold, is bounded by a fixed constant k. Although this semantics is more natural and uses only observer hybrid variables in the model, we prove that model-checking TCTL ∆ Σ is undecidable. Outline. Section 2 recalls the main features of timed automata model and gives definitions for the syntax and semantics of our extended logics. Sections 3 and 4 are devoted to the model-checking of TCTL ∆ and, in the last section, we show that model-checking the extended logic TCTL ∆ Σ is undecidable.

Lecture Notes in Computer Science, 2005

This paper attempts to improve our understanding of timed languages and their relation to timed automata. We start by giving a constructive proof of the folk theorem stating that timed languages specified by the past fragment of MITL, can be accepted by deterministic timed automata. On the other hand we provide a proof that certain languages expressed in the future fragment of MITL are not deterministic, 4 and analyze the reason for this asymmetry.

Handbook of Modal Logic, 2007

Theoretical Computer Science, 1998

2001

A key issue in the design of a model-checking tool is the choice of the formal language with which properties are specified. It is now recognized that a good language should extend linear temporal logic with the ability to specify all ω-regular properties. Also, designers, who are familiar with finite-state machines, prefer extensions based on automata than these based on fixed points or propositional quantification. Early extensions of linear temporal logic with automata use nondeterministic Büchi automata.

Lecture Notes in Computer Science, 1999

Elec Power Syst Res, 1993