On decidability of recursive weighted logics

Abstract

In this paper, we develop and study two recursive weighted logics (RWLs) $\mathcal {L}^w$ and $\mathcal {L}^t$, which are multi-modal logics that express qualitative and quantitative properties of labelled weighted transition systems (LWSs). LWSs are transition systems describing systems with quantitative aspects. They have labels with both actions and real-valued quantities representing the costs of transitions with respect to various resources. RWLs use first-order variables to measure local costs. The main syntactic operators are similar to the ones of timed logics for real-time systems. $\mathcal {L}^w$ has operators that constrain the value of resource-variables at the current state. $\mathcal {L}^t$ extends $\mathcal {L}^w$ by having quantitative constraints on the transition modalities as well. This extension makes sure that $\mathcal {L}^t$ is adequate, i.e. the semantic equivalence induced by $\mathcal {L}^t$ coincides with the weighted bisimilarity of LWSs. In addition, our logic is endowed, with simultaneous recursive equations, which allow encoding of properties of infinite behaviours. We prove that unlike in the case of the timed logics, the satisfiability problems for RWLs are decidable. The proofs use a variant of the region construction technique used in the literature with timed automata, which we adapt to the specific settings of RWLs. For $\mathcal {L}^t$, we also propose an attractive alternative proof which makes use of the algorithm for $\mathcal {L}^w$.