Tomorrow and All our Yesterdays: MTL Satisfiability over the Integers

Abstract

Abstract. We investigate the satisfiability problem for metric temporal logic (MTL) with both past and future operators over linear discrete bi-infinite time models isomorphic to the integer numbers, where time is unbounded both in the future and in the past. We provide a technique to reduce satisfiability over the integers to satisfiability over the wellknown mono-infinite time model of natural numbers, and we show how to implement the technique through an automata-theoretic approach. We also prove that MTL satisfiability over the integers is EXPSPACEcomplete, hence the given algorithm is optimal in the worst case.

Satisfiability Checking for Mission-Time LTL

“…We use π i (|π| > i ≥ 0) to represent the suffix of π starting from position i (including i). Let a, b ∈ I, a ≤ b; we define that π models (satisfies) an MLTL formula ϕ, denoted as π |= ϕ, as follows: Compared to the traditional MTL-over-naturals 1 [16], the Until formula in MLTL is interpreted in a slightly different way. In MTL-over-naturals, the satisfaction of ϕ U I ψ requires ϕ to hold from position 0 to the position where ψ holds (in I), while in MLTL ϕ is only required to hold within the interval I, before the time ψ holds.…”

Checking EMTLK Properties of Timed Interpreted Systems Via Bounded Model Checking

“…In this paper we consider MTLK which is an epistemic extension of Metric Temporal Logic (MTL) [10] that cannot be translated into LTL (because of the considered semantics), and which allows for the representation of the quantitative temporal evolution of epistemic states of the agents. We interpret MTLK over discrete timed models generated by TISs.…”

Checking MTL Properties of Discrete Timed Automata via Bounded Model Checking

“…A number of formalisms, which use a discrete time domain, have been proposed in the literature to model the behaviour of these systems: discrete timed automata [3], discrete timed Petri nets [14], and others. To express the requirements of the systems mostly standard temporal logics are used: computation tree logic (CTL) [10], the soft real-time CTL (RTCTL) [13], linear temporal logic (LTL) [10], and metric temporal logic (MTL) [17,15,25]. RTCTL is a propositional branching-time temporal logic with bounded operators, which was introduced to permit specification and reasoning about time-critical correctness properties.…”

“…RTCTL is interpreted over discrete time models and it is simply CTL, but with an exponentially succinct encoding [13]. MTL (or metric LTL) is a propositional linear-time temporal logic with bounded operators, and interpreted over discrete time models is simply LTL, but with an exponentially succinct encoding [15]. MTL can express multiple timing constraints on computations, which is really interesting for writing specifications.…”

“…Namely, [25,24] describe a SAT-based BMC method for MTL, which is based on a translation to LTL, and which has been implemented in the ZOT tool 1 . Next, [15] provides an automata model checking technique for MTL that is also based on a translation to LTL, and it is implemented on top of the Spin model-checker [16]. Further, [19] shows a model checking technique of the two classes of MTL properties: bounded response and minimum separation.…”