Parametric Linear Dynamic Logic (full version)

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by adding temporal operators equipped with parameters that bound their scope. LDL itself was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all ω-regular specifications while still maintaining many of LTL’s desirable properties like intuitive syntax and semantics and a translation into non-deterministic Büchi automata of exponential size. However, LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic which is able to express all ω-regular properties and which subsumes parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our main technical contribution is a translation of PLDL formulas into non-deterministic Büchi automata of exponential size via alternating automata. This yields polynomial space algorithms for model checking and assume-guarantee model checking and a realizability algorithm with doubly-exponential running time. All three problems are also shown to be complete for these complexity classes. Moreover, we give tight upper and lower bounds on optimal parameter values for model checking and realizability. Using these bounds, we present a polynomial space procedure for model checking optimization and an algorithm with triply-exponential running time for realizability optimization. Our results show that PLDL model checking, assume-guarantee model checking, and realizability are no harder than their respective (parametric) LTL counterparts.