Completeness of the Bounded Satisfiability Problem for Constraint LTL

Bersani, Marcello M.; Frigeri, Achille; Rossi, Matteo; Pietro, Pierluigi San

doi:10.1007/978-3-642-24288-5_7

Cited by 15 publications

(20 citation statements)

References 6 publications

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“…Constraint LTL (CLTL [18,11]) is an extension of LTL allowing atomic formulae over a constraint system. Let AP be a finite set of atomic propositions, let V be a finite set of variables and let D " pD, Rq be a constraint system, where D is a specific domain of interpretation for variables and constants and R is a finite family of relations on D (of various arities).…”

Section: Constraint Ltl Over Clocksmentioning

confidence: 99%

“…We show that any CLTLoc formula φ over a set of clocks V can be translated into a suitable Büchi automaton A R φ that accepts words including both sequences of symbolic valuations [18,10] and constraints representing the clock regions induced by φ. Automaton A R φ is built using a slight variation of the construction of [18,11], where instead variables are not restricted to behave as clocks.…”

Section: Decidability Of Cltl Over Clocksmentioning

confidence: 99%

“…In Section 6, we provide a way to solve in practice the satisfiability of CLTLoc through the method used in [11] to solve satisfiability of CLTL. The technique relies on encoding CLTLoc formulae into formulae of a decidable fragment of firstorder logic, which can then be solved by off-the-shelf SMT solvers.…”

Section: Complexitymentioning

confidence: 99%

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A Tool for Deciding the Satisfiability of Continuous-Time Metric Temporal Logic

Bersani

Rossi

Pietro

2013

2013 20th International Symposium on Temporal Representation and Reasoning

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Constraint LTL over clocks is a variant of CLTL, an extension of lineartime temporal logic allowing atomic assertions in a concrete constraint system. Satisfiability of CLTL over clocks is here shown to be decidable by means of a reduction to a decidable SMT (Satisfiability Modulo Theories) problem. The result is a complete Bounded Satisfiability Checking procedure, which has been implemented by using standard SMT solvers. The importance of this technique derives from the possibility of translating various continuous-time metric temporal logics, such as MITL and QTL, into CLTL over clocks itself. Although standard decision procedures of these logics do exist, they have never been realized in practice. Suitable translations into CLTL over clocks have instead allowed us the development of the first prototype tool for deciding MITL and QTL. The paper also reports preliminary, but encouraging, experiments on some significant examples of MITL and QTL formulae.

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Section: Constraint Ltl Over Clocksmentioning

confidence: 99%

Section: Decidability Of Cltl Over Clocksmentioning

confidence: 99%

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A Tool for Deciding the Satisfiability of Continuous-Time Metric Temporal Logic

Bersani

Rossi

Pietro

2013

2013 20th International Symposium on Temporal Representation and Reasoning

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“…CLTLoc allows explicit clocks that, similarly to clocks of Timed Automata, can be compared with integer constants and reset to 0. In particular, an MITL formula is encoded into an equisatisfiable CLTLoc formula, which can be solved through the same techniques presented in [17,18,19]. The approach in generalizes BSC to CLTL, generating an encoding suitable for verification with standard Satisfiability Modulo Theories (SMT) solvers such as Z3 [20].…”

Section: Introductionmentioning

confidence: 99%

An SMT-based approach to satisfiability checking of MITL

Bersani¹,

Rossi²,

Pietro³

2015

Information and Computation

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We present a satisfiability-preserving reduction from MITL interpreted over finitely-variable continuous behaviors to Constraint LTL over clocks, a variant of CLTL that is decidable, and for which an SMT-based bounded satisfiability checker is available. The result is a new complete and effective decision procedure for MITL. Although decision procedures for MITL already exist, the automata-based techniques they employ appear to be very difficult to realize in practice, and, to the best of our knowledge, no implementation currently exists for them. A prototype tool for MITL based on the encoding presented here has, instead, been implemented and is publicly available. IntroductionComputer systems are inherently discrete-time objects, but their application to real-time control and monitoring often requires to deal with external asynchronous events that may not always happen at integer-valued times. Hence, a discrete-time assumption requires to approximate continuous time by choosing some fixed minimal interval, thus limiting the accuracy of modeling, verification and validation of such systems. To overcome this restriction, many continuoustime models have been developed, most notably Timed Automata [4], a densetime operational model based on finite-state machines, but also descriptive models such as the continuous-time temporal logics MTL (Metric Temporal Logic) [5,6] and MITL (Metric Interval Temporal Logic) [6]. In general, the role of temporal logics in verification and validation is two-fold. First, temporal logic allows abstract, concise and convenient expression of required properties of $ This research was supported by the Programme IDEAS-ERC, Project 227977-SMScom, and by PRIN Project 2010LYA9RH. $$ Parts of this work were previously published in [1], [2] and [3], of which this article is an extension.Email addresses: marcellomaria.bersani@polimi.it (Marcello M. Bersani), matteo.rossi@polimi.it (Matteo Rossi), pierluigi.sanpietro@polimi.it (Pierluigi San Pietro) Preprint submitted to ElsevierJanuary 9, 2015 a system. Linear Temporal Logic (LTL) is often used with this goal in the verification of finite-state models, e.g., in model checking [7]. Second, temporal logic allows a descriptive approach to specification and modeling (see, e.g., [8,9]). A descriptive model is based on axioms, written in some (temporal) logic, defining a system by means of its general properties, rather than by an operational model based on some kind of machine (e.g., a Timed Automaton) behaving in the desired way. In this case, verification typically consists of satisfiability checking of the conjunction of the model and of the (negation of) its desired properties. An example of the latter approach is Bounded Satisfiability Checking (BSC) [10], where MTL specifications and properties on discrete time are translated into Boolean logic, in an approach similar to Bounded Model Checking of LTL properties of finite-state machines.In general, verification of continuous-time temporal logics is not as well supported as for discre...

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“…Unlike MTL, clocks are explicit resources in CLTLoc and, as in TA, they can be compared with constants over N (or Q). In [7], we prove that satisfiability of CLTLoc is PSPACE-complete, by combining results on the decidability of CLTL [11], [5] over R with Region Graphs [1] capturing time behavior of variables. Moreover, the satisfiability of CLTLoc can be reduced to an instance of a SMT (satisfiability Modulo Theory) problem.…”

Section: Introductionmentioning

confidence: 99%

A Logical Characterization of Timed (non-)Regular Languages

Bersani

Rossi

Pietro

2014

Lecture Notes in Computer Science

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Abstract. CLTLoc (Constraint LTL over clocks) is a quantifier-free extension of LTL allowing variables behaving like clocks over real numbers. CLTLoc is in PSPACE [9] and its satisfiability can polynomially be reduced to a SMT problem, allowing a feasible implementation of a decision procedure. We used CLTLoc to capture the semantics of metric temporal logics over continuous time, such as Metric Interval Temporal Logic (MITL), resulting in the first successful implementation of a tool for checking MITL satisfiability [7]. In this paper, we assess the expressive power of CLTLoc, by comparing it with various temporal formalisms over dense time. When interpreted over timed words, CLTLoc is equivalent to Timed Automata. We also define a monadic theory of orders, extending the one introduced by Kamp, which is expressively equivalent to CLTLoc. We investigate a decidable extension with an arithmetical next operator, which allows the expression of timed non-ω-regular languages.

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Completeness of the Bounded Satisfiability Problem for Constraint LTL

Cited by 15 publications

References 6 publications

A Tool for Deciding the Satisfiability of Continuous-Time Metric Temporal Logic

A Tool for Deciding the Satisfiability of Continuous-Time Metric Temporal Logic

An SMT-based approach to satisfiability checking of MITL

A Logical Characterization of Timed (non-)Regular Languages

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