Visibly Linear Dynamic Logic

TL;DR

Visibly Linear Dynamic Logic (VLDL) extends LTL with visibly pushdown language guards, enabling the expression of recursive program properties and providing exponential-time decision procedures.

Contribution

Introduction of VLDL, a logic that captures recursive properties with a translation to visibly pushdown automata, and analysis of its computational complexity.

Findings

VLDL exactly characterizes ω-visibly pushdown languages.

Decidability results with exponential-time algorithms for satisfiability, validity, and model checking.

VLDL games are solvable in triply-exponential time.

Abstract

We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the $\omega$-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of this work is a translation of VLDL into $\omega$-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time. We prove all these problems to be complete for their respective complexity classes.