Parametric Linear Dynamic Logic

TL;DR

Parametric Linear Dynamic Logic (PLDL) extends LDL with parameters to express timing constraints and all omega-regular properties, enabling efficient model checking and realizability algorithms comparable to LTL.

Contribution

Introduction of PLDL, a logic combining omega-regular expressiveness with parameterized timing constraints, along with automata-based translation and complexity bounds.

Findings

PSPACE model checking algorithm for PLDL

Doubly-exponential realizability algorithm for PLDL

Tight bounds on optimal parameter values

Abstract

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all $\omega$-regular specifications while still maintaining many of LTL's desirable properties like an intuitive syntax and a translation into non-deterministic B\"uchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to express all $\omega$-regular properties and that subsumes parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our main technical contribution is a translation of PLDL formulas into non-deterministic B\"uchi word automata of exponential size via alternating automata. This yields a PSPACE model checking algorithm and a realizability algorithm with doubly-exponential running time. Furthermore, we give tight upper and lower bounds on optimal parameter values for both problems. These results show that PLDL model checking and realizability are not harder than LTL model checking and realizability.