Quantitative temporal reasoning

Real-time Systems, 1990

This paper is motivated by the need for a formal specification method for real-time systems. In these systems quantitative temporal properties play a dominant role. We first characterize real-time systems by giving a classification of such quantitative temporal properties. Next, we extend the usual models for temporal logic by including a distance function to measure time and analyze what restrictions should be imposed on such a function. Then we introduce appropriate temporal operators to reason about such models by turning qualitative temporal operators into (quantitative) metric temporal operators and show how the usual quantitative temporal properties of real-time systems can be expressed in this metric temporal logic. After we illustrate the application of metric temporal logic to real-time systems by several examples, we end this paper with some conclusions.

Electronic Proceedings in Theoretical Computer Science, 2018

The paper is focused on temporal logics for the description of the behaviour of real-time pushdown reactive systems. The paper is motivated to bridge tractable logics specialized for expressing separately dense-time real-time properties and context-free properties by ensuring decidability and tractability in the combined setting. To this end we introduce two real-time linear temporal logics for specifying quantitative timing context-free requirements in a pointwise semantics setting: Event-Clock Nested Temporal Logic (EC NTL) and Nested Metric Temporal Logic (NMTL). The logic EC NTL is an extension of both the logic CaRet (a context-free extension of standard LTL) and Event-Clock Temporal Logic (a tractable real-time logical framework related to the class of Event-Clock automata). We prove that satisfiability of EC NTL and visibly model-checking of Visibly Pushdown Timed Automata (VPTA) against EC NTL are decidable and EXPTIME-complete. The other proposed logic NMTL is a context-free extension of standard Metric Temporal Logic (MTL). It is well known that satisfiability of future MTL is undecidable when interpreted over infinite timed words but decidable over finite timed words. On the other hand, we show that by augmenting future MTL with future context-free temporal operators, the satisfiability problem turns out to be undecidable also for finite timed words. On the positive side, we devise a meaningful and decidable fragment of the logic NMTL which is expressively equivalent to EC NTL and for which satisfiability and visibly model-checking of VPTA are EXPTIME-complete. * The work by Adriano Peron and Aniello Murano has been partially supported by the GNCS project Formal methods for verification and synthesis of discrete and hybrid systems and by Dept. project MODAL MOdel-Driven Analysis of Critical Industrial Systems.

1989

Research on the testing and debugging of distributed realtime programs now focuses on more formal approaches to specification and testing. Temporal logic is a natural candidate for this since it can specify properties of event and state sequences. However, the absence of any concept of realtime limits the application of temporal logic to non real-time behavior. This paper presents an extension of the interval logic of Schwartz et al. [SMSVP83], by increasing the expressive power of the logic (with respect to real time) while retaining its intuitive appeal and understandability. The extensions are added in a "layer" that can be stripped away if formal verification is the goal, or retained if timing behavior must be tested. The extensions include: the ability to deal with real time (as in [JM86b, JM86a, OW87, NA88]); more powerful interval specification mechanisms; a limited form of quantification; and the direct expression of event predicates (as in [LeD86]). Since our work is intended to yield practical tools for software testers, we emphasize the ease of expressing the complex timing properties of real-time software (e.g. periodic behavior, performance constraints), and we demonstrate the use of the interval logic on some realtime examples that represent a test of the expressiveness and understandability of the notation.

Electronic Proceedings in Theoretical Computer Science, 2017

Theoretical Computer Science, 1999

This paper presents a framework for the specification and verification of timing properties of reactive systems using Temporal Logic with Clocks (TLC). Reactive systems usually contain a number of parallel processes, therefore, it is essential to study and analyse each process based on its own local time. TLC is a temporal logic extended with multiple clocks, and it is in particular suitable for the specification of reactive systems. In our framework, the behavior of a reactive system is described through a formal specification; its timing properties, including safety and liveness properties, are expressed by TLC formulas. We also propose several demonstration techniques, such as an application of local reasoning and deriving fixed-time rules from the proof system of TLC, for proving that a reactive system meets its temporal specification. Under the proposed framework, the timing properties of a reactive system can therefore be directly reasoned about from the formal specification of the system.

2001

This paper redefines RTL within classical many-sorted logic with natural number and real arithmetic. In doing so, RTL is generalised in a number of ways. In particular, functionality is handled through the use of timed variables. Various models of time for RTL are discussed, and it is argued that, providing events satisfy a countable occurrence property, time in RTL can be continuous. A number of useful RTL theorems are stated, and it is shown that RTL can naturally express all the usual temporal requirements that are placed on real-time systems. RTL is compared with other timed logics.

Annual Review in Automatic Programming, 1991

This paper introduces a knowledge-based approach to ensure the integrity of information in a real-time database (rtdb). A form of explicit clock temporal logic (called TLrt) useful in specifying timing constraints on controller actions, rtdb items, and constraints in a real-time constraint base (rtcb), is presented. Timing as well as other forms of constraints are stored in the rtcb.

2001

[lo], TILCO [5], Temporal logics are typically used for the specification Logic-based languages for modeling temporal constraints can be based on time points, or on time in-Of systems since they are capable Of describing tempora1 constraints among events and actions: tervals. For a review, please see [Ill. Interval logic formulE have typically a higher level of abstraction, properties of invariance, precedence, periodicity, re-These logics usually have specific operators to express peated occurrences, liveness and safety conditions, etc. the between intervals (meet, before, This paper describes an Of the Tempora1 Int e r) , operators to combine intervals (e.g,, the chop operator), or operator? that specify the interval that con-terval Logic called TILCO. TILCO is based on time inwith time In this paper, TILCO-X extension logics are typically concise than point-based tern-tervals and can express tempora1 constraints stitutes the context of temporal formulE [123. Interval Of 's presented' proposes two new poral logics for the specification of real-time systems. operators that strongly increase conciseness and readability of specifications to allow describing (i) the ternporal ordering of events without distinction between According to [5i 1 where logic (Ternwith ComPositional Operators) Poral Interva1 past and future, and (ii) predicates depending on the has been presented by the authors, it has been shown number of Occurrences of events in intervals, TILCO-that none of the temporal logics presented in the past is capable to describe specifications by using a lower number of quantifications and of nesting levels betweed temporal This paper also presents the semantics of TILCO-X and related examples showing the few years is completely satisfactory for real-time system specification. In fact, most of them have no metric for time, thus allowing only specification of qualitative temporal requirementse.g., [2], [13], [4]. In the litpower of the model proposed in terms of conciseness, erature, only a few examples of quantitative temporal Index formal specification language, first or-logics exist. In these cases, an operator expressing the der logic, temporal interval logic, verification and validation, real-time systems, temporal operators. distance between time points is usually defined. Those first-order temporal logics that provide a metric for time usually allow quantification over the temporal domaine.g., RTL, MTL, T R I O [ll]-whereas a prohibition of this kind of quantification has been shown F~~ the specification of system behavior many to be a necessary condition for the existence of feasifactors have to be considered, ~~~i~~l l~, their specifi-ble automated verification mechanisms [14]. All these cation includes the definition of a set of relationships temporal logics are based on time points and provide a expressing the temporal constraints among events and sharp distinction between past and future. In that conactions [I] : properties of invariance, precedence among ditions, when a simple constraints imposes its validity events, periodicity, liveness and safety conditions, etc. always it has to be specified by using three different T~ this end, many temporal logics for temporal rea-pieces: one for the past, one for the current evaluation soning have been proposed; e.g., [a], [SI. [4], [5], [GI, time instant and one for the future. TILCO does not allow the quantification over time and present unique Several different temporal logics with different de-Operators to Specify the events and action from past to grees of expressiveness have been proposed. Some of future. them are based on propositional logice.g., PTL [a], The authors defined TILCO temporal logic in order T P T L [7], RTTL, ITLand adopt the 0 and op-to cover the above-mentioned problems, with a special erators. Other logics are based on first or higher order emphasis on temporal logic expressiveness and conlogice.g., TRIO [8], MTL [9], interval temporal logic ciseness for the specification of real-time systems 151.

Journal of Automated Reasoning, 2005

First-order temporal logic, the extension of first-order logic with operators dealing with time, is a powerful and expressive formalism with many potential applications. This expressive logic can be viewed as a framework in which to investigate problems specified in other logics. The monodic fragment of first-order temporal logic is a useful fragment that possesses good computational properties such as completeness and sometimes even decidability. Temporal logics of knowledge are useful for dealing with situations where the knowledge of agents in a system is involved. In this paper we present a translation from temporal logics of knowledge into the monodic fragment of first-order temporal logic. We can then use a theorem prover for monodic first-order temporal logic to prove properties of the translated formulas. This allows problems specified in temporal logics of knowledge to be verified automatically without needing a specialized theorem prover for temporal logics of knowledge. We present the translation, its correctness, and examples of its use.

Lecture Notes in Computer Science, 1997

This paper proposes an expressive extension to Propositional Linear Temporal Logic dealing with real time correctness properties and gives an automata-theoretic model checking algorithm for the extension. The algorithm has been implemented and applied to examples.