Reasoning about Infinite Computations (original) (raw)
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An automata-theoretic approach to linear temporal logic
Logics for concurrency, 1996
The automata-theoretic approach to linear temporal logic uses the theory of automata as a unifying paradigm for program specification, verification, and synthesis. Both programs and specifications are in essence descriptions of computations. These computations can be viewed as words over some alphabet. Thus, programs and specifications can be viewed as descriptions of languages over some alphabet. The automata-theoretic perspective considers the relationships between programs and their specifications as relationships between languages. By translating programs and specifications to automata, questions about programs and their specifications can be reduced to questions about automata. More specifically, questions such as satisfiability of specifications and correctness of programs with respect to their specifications can be reduced to questions such as nonemptiness and containment of automata. Unlike classical automata theory, which focused on automata on finite words, the applications to program specification, verification, and synthesis, use automata on infinite words, since the computations in which we are interested are typically infinite. This paper provides an introduction to the theory of automata on infinite words and demonstrates its applications to program specification, verification, and synthesis.
Temporalized logics and automata for time granularity
Theory and Practice of Logic Programming, 2004
The ability of providing and relating temporal representations at different 'grain levels' of the same reality is an important research theme in computer science and a major requirement for many applications, including formal specification and verification, temporal databases, data mining, problem solving, and natural language understanding. In particular, the addition of a granularity dimension to a temporal logic makes it possible to specify in a concise way reactive systems whose behaviour can be naturally modeled with respect to a (possibly infinite) set of differently-grained temporal domains.
Reasoning about infinite computation paths
24th Annual Symposium on Foundations of Computer Science (sfcs 1983), 1983
We investigate extensions of tenlporal logic by finite automata on infinite words. There are three different types of acceptance conditions (finite, looping and repeating) that one can give for these finite automata. This gives rise to three different logics. It turns out, ho\vever. that these logics have the same expressive po\ver but differ in the complexity of their decision problem. V/e also investigate the addition of alternation and sho\v that it does not increase the complexity of the decision problem.
A Decidable Temporal Logic of Repeating Values
Lecture Notes in Computer Science, 2007
Various logical formalisms with the freeze quantifier have been recently considered to model computer systems even though this is a powerful mechanism that often leads to undecidability. In this paper, we study a linear-time temporal logic with past-time operators such that the freeze operator is only used to express that some value from an infinite set is repeated in the future or in the past. Such a restriction has been inspired by a recent work on spatio-temporal logics. We show decidability of finitary and infinitary satisfiability by reduction into the verification of temporal properties in Petri nets. This is a surprising result since the logic is closed under negation, contains future-time and past-time temporal operators and can express the nonce property and its negation. These ingredients are known to lead to undecidability with a more liberal use of the freeze quantifier.
Towards an automata-theoretic counterpart of combined temporal logics
2001
In this paper, we define a new class of combined automata, called temporalized automata, which can be viewed as the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Furthermore, we successfully apply temporalized automata to provide the full secondorder theory of k-refinable downward unbounded layered structures with a temporal logic counterpart. Finally, we show how temporalized automata can be used to deal with relevant classes of reactive systems, such as granular reactive systems and mobile reactive systems.
Theoretical Aspects of Computing – ICTAC 2018, 2018
We show that Branching-time temporal logics CTL and CTL * , as well as Alternating-time temporal logics ATL and ATL * , are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for CTL, as well as for ATL, with a single variable is EXPTIME-complete, while satisfiability for CTL * , as well as for ATL * , with a single variable is 2EXPTIME-complete,-i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiability for arbitrary formulas.
The Influence of Temporal Logic on Finite Automata
Al-Rafidain Engineering Journal (AREJ)
The theory of automata combines ideas from engineering, linguistics, mathematics, philosophy, etc. The Entscheidungsproblem asks if it is possible to design a series of steps that replaces a mathematician. An automaton is an abstract machine that processes data. C. Shannon's theory is today's most popular despite having no relationship with the other. The Kt system is called "minimal" because it makes no assumptions about the structure of time. In LKt, we have four monary temporal operators, F, P, G and H, which are mutually interdefinable. Interdefinability means that we will pass logic in the future is the same as saying I will never fail logic, interpreting not passing logic as failing logic. The minimal system syntax of temporal logic introduces operators that have the property of being defined in terms of others. Modal logic studies the reasoning that involves the use of expressions "necessarily" and "possibly". In this article, we will represent through a finite automaton the temporal logic formula Fp. It allows us to see an acceptance pattern for Fp by considering two variables: p and q. Kt's axiomatic system of time expresses the idea that both the present and the past are fixed, if it has always been in the past that it will be some time in the future that p is now. No philosophical argument supports deterministic time flow; the logic of time must be open.Temporal logic has revived many old problems, from the Megaric-Stoics to the minimal system of temporal logic. Our work suggests that the future operators of system Kt follow an evaluation pattern, but we must be cautious because this pattern can only apply to models whose time flow is based on instants and precedence relations.
On the strength of temporal proofs
Theoretical Computer Science, 1991
In this paper we investigate the reasoning powers or proof theoretic powers of various established temporal logics used in Computer Science. In Sections 1-6 we concentrate on provability of various program properties while in Sections 7-9 we investigate provability of temporal formulas in general. In Sections 1-6 we consider both deterministic and nondeterministic programs. Our investigations are ~wofoid: (1) compare the reasoning powers of the various logics, and (2) characterize their reasoning powers. The investigations in (2) are often called completeness issues, because a good characterization amounts to finding a nice and mathematically transparent semantics for which our logic is complete (cf. e.g. [4] and [19]). In doing (2), we follow the methodology called Correspondence Theory in philosophical logic (see [19, Chap. II.4]) which was first elaborated for temporal logics of programs in the 1978 version of Sain's [43] (cf. also [10] and [21]; both [43] and [10] were based on the Computer Science temporal logics in [6]), in the framework called time oriented Nonstandard Logics of Programs (NLP). The same is used in [1-4]. In particular, the semantics denoted as "~-o P(...)'" by Abadi was first introduced as "(lnd+ Tord)~'" in the above quoted NLP literature (cf. e.g. the historical notes in [38, 42, 21, 49]), and will also play a central role here. Among others, we will obtain new strong (hereditarily in a sense) incompleteness results w.r.t, this semantics for proof systems of [5] and [33]. No number of new axioms, but a single new modality can eliminate this incompleteness (see [40]). In Section 8 we solve some of the problems raised in recent publications of the famous temporal logic school represented by Manna and Pnueli [33, 34], Abadi and Manna [5], and Abadi [ 1-4]. These problems concern the strongest among the inference systems designed so far for computer science oriented first-order temp',ral logics. Here we consider only inference systems which are (at least theoretically) relevant i'or machine implementation.
Timed Context-Free Temporal Logics
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.