An Improved Decision Procedure for Linear Time Mu-Calculus

TL;DR

This paper introduces an improved Present Future form for linear time $ u$-calculus, enabling a more efficient decision procedure for satisfiability and model checking, with practical implementation and visual counterexamples.

Contribution

It presents a new PF form for $ u$-calculus, an algorithm for constructing PFGs, and an efficient satisfiability checking method with implementation.

Findings

Achieves the best time complexity among existing methods.

Provides a practical C++ implementation.

Enables visual counterexamples in model checking.

Abstract

An improved Present Future form (PF form) for linear time $\mu$-calculus ($\nu$TL) is presented in this paper. In particular, the future part of the new version turns into the conjunction of elements in the closure of a formula. We show that every closed $\nu$TL formula can be transformed into the new PF form. Additionally, based on the PF form, an algorithm for constructing Present Future form Graph (PFG), which can be utilized to describe models of a formula, is given. Further, an intuitive and efficient decision procedure for checking satisfiability of the guarded fragment of $\nu$TL formulas based on PFG is proposed and implemented in C++. The new decision procedure has the best time complexity over the existing ones despite the cost of exponential space. Finally, a PFG-based model checking approach for $\nu$TL is discussed where a counterexample can be obtained visually when a model violates a property.