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.

Online verification in cyber‐physical systems: Practical bounds for meaningful temporal costs

“…Satisfiability checking is a verification approach that is alternative to model‐checking where a logical formula defines the system behavior instead of an operational model (like, for instance, automata or transition systems). This approach is helpful when verification is required for system that are specified by means of descriptive specifications as claimed in Pradella et al Although the standard technique to prove the satisfiability of CLTL and CLTLoc formulae is based on the construction of a Büchi automata, the evidence has turned out that it is rather expensive in practice, even in the case of LTL (the size of the automaton is exponential with respect to the size of the formula) and that new techniques should be investigated. Bounded satisfiability checking tackles the complexity of checking the satisfiability for CLTL formulae by avoiding the unfeasible construction of the whole automaton.…”

“…Constraint LTL over clocks (CLTLoc) is a semantic restriction of constraint LTL (CLTL) allowing atomic formulae over where the arithmetical variables behave like clocks of timed automata (TA) . A clock x measures the time elapsed since the last time when x =0, ie, the last “reset” of x .…”

“…Looking for a bounded representation of the models of a formula is equivalent to verifying the existence of looping paths in the automaton recognizing the language of the formula. Bersani et al show that BSC for CLTLoc is complete and that is reducible to a decidable satisfiability modulo theory (SMT) problem. The results prove that any CLTLoc formula can be translated into the decidable theory of quantifier‐free formulae with equality and uninterpreted functions combined with the theory of Reals .…”

“…They are evaluated at runtime when the online engine is inquired to verify properties on the running systems. Upon a service request, OLIVE: 1Extracts the set of predicates from the request and the current configuration of the system; 2Generates an updated model, based on a set of CLTLoc formulae where the initial value of all the variables is set with the current configuration of the server; 3Verifies the satisfiability of the model by means of a e 2 z o t tool …”

“…Cyber-physical systems require that such a time is determined a priori. In this paper, this is achieved through a priori simulations of the system adaptation process by solving a satisfiability problem of a logical model, defined in Constraint LTL over clocks (CLTLoc) 6 that captures the semantics of the server behavior.…”