Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL (full version)

TL;DR

This paper extends parameterized linear temporal logics to include cost bounds, demonstrating that model checking and game solving remain computationally comparable to standard LTL, thus maintaining desirable algorithmic properties.

Contribution

It generalizes PLTL to systems with costs, showing that complexity remains similar to LTL for model checking and game solving, and discusses extensions to multiple costs.

Findings

Model checking for PLTL with costs is not harder than for LTL.

Solving games for PLTL with costs retains similar complexity to LTL.

Extensions to PLDL with costs and multiple cost functions are discussed.

Abstract

We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all omega-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.