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

Abstract

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.

A Logical Characterization of Timed (non-)Regular Languages

“…Constraint LTL over clocks [7] (CLTLoc) is a semantic fragment of CLTL [11] where formulae are defined with respect to a finite set AP of atomic propositions, a finite set V of clocks and a pair D = (R, {<, =}).…”

“…More precisely, a clock c q ∈ X Q is associated with each control state q ∈ Q; the value of c q is 0 whenever A is in q, and it is left to grow (i.e., c q > 0) otherwise. Since in CLTLoc, unlike in TA, a clock cannot be read and reset at the same time, following the approach of [7] for each x ∈ X we introduce two CLTLoc clocks, x 1 and x 2 , which are alternately reset. In addition, we introduce a third clock, x 12 , which is used to keep track of whether x 1 < x 2 (x 12 is 0 if, and only if, x 1 < x 2 ).…”

“…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.…”

“…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.…”

How bit-vector logic can help improve the verification of LTL specifications over infinite domains

“…CLTL-over-clocks (CLTLoc) [9,12] is the variant of CLTL where variables behave as real-valued clocks. It is decidable, expressively equivalent to Timed Automata [10], and a decision procedure for solving it is implemented in the Zot tool.…”