Automata-based Verification of Linear Temporal Logic Models with Bounded Variability

Abstract

Abstract-A model has variability bounded by v/k when the state changes at most v times over any linear interval containing k time instants. When interpreted over models with bounded variability, specification formulae that contain redundant metric information-through the usage of next operators-can be simplified without affecting their validity. This paper shows how to harness this simplification in practice: we present a translation of LTL into Büchi automata that removes redundant metric information, hence makes for more efficient verification over models with bounded variability. To show the feasibility of the approach, we also implement a proofof-concept translation in ProMeLa and verify it using the Spin off-the-shelf model checker.

Bounded variability of metric temporal logic

“…Even if the asymptotic worst-case complexity of MTL over discrete time does not change with bounded variability, our previous work [21,22] indicates that the complexity decreases in practice for MTL formulas that can be expressed with certain syntactic restrictions, where the exponential blow-up due to succinct encoding of constants is essentially avoided. Therefore, we can still assert that bounded variability may help simplify MTL decidability in some specific cases.…”

“…One approach to restricting MTL expressiveness consists in introducing semantic constraints. We pursued this approach in previous work [18,21,22], where we showed that validity of MTL over dense time becomes decidable under the restriction of bounded variability. A nonnegative integer v and a time duration V identify all models with variability bounded by v/V : models where there are no more than v events (such as change of state) over any interval of length V .…”

“…Wilke first studied bounded variability for timed automata and monadic logic [62]; his results imply the decidability of MTL over dense time with bounded variability. Previous work of ours studied the complexity of MTL with bounded variability [18,20], and extended some of the techniques to the case of discrete time [21,22], where we showed how the LTL validity problem can be simplified under the assumption that only v < V change events happen every V discrete time steps. Section 4 recalls in detail these works of ours, and connects them to the present paper's results.…”

Bounded Variability of Metric Temporal Logic

“…Bounded variability is a natural semantic restriction over dense time, which has been applied to various formalisms including timed automata [27], duration calculus [10], and, in our previous work, MTL [12]. Recently, we also applied it to LTL over discrete time; while it is obvious that every discrete-time model has bounded variability (given by the fixed duration associated with one discrete time unit), in [14,15] we showed how the LTL validity problem can be simplified under the assumption that only v < V change events happen every V discrete time steps. Therefore, bounded variability can be a simplifying assumption also for discrete time.…”

“…The validity of ψ ⇒ B v,V can thus be decided in singly exponential time (Lemma 14), as opposed to BV N (v, V ) or the validity of φ which are EXPSPACE-complete: solving them for a generic φ takes time doubly exponential in |φ|. We can thus decide whether ψ ⇒ B v,V is valid in singly exponential time; if it is, φ has bounded variability a fortiori, and hence we can study its validity with the simplified algorithms [14,15].…”