STACS 2004: STACS 2004 pp 522-533

# A Measured Collapse of the Modal μ-Calculus Alternation Hierarchy

• Doron Bustan
• Orna Kupferman
• Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)

## Abstract

The μ-calculus model-checking problem has been of great interest in the context of concurrent programs. Beyond the need to use symbolic methods in order to cope with the state-explosion problem, which is acute in concurrent settings, several concurrency related problems are naturally solved by evaluation of μ-calculus formulas. The complexity of a naive algorithm for model checking a μ-calculus formula ψ is exponential in the alternation depth d of ψ. Recent studies of the μ-calculus and the related area of parity games have led to algorithms exponential only in $$\frac{d}{2}$$. No symbolic version, however, is known for the improved algorithms, sacrificing the main practical attraction of the μ-calculus.

The μ-calculus can be viewed as a fragment of first-order fixpoint logic. One of the most fundamental theorems in the theory of fixpoint logic is the Collapse Theorem, which asserts that, unlike the case for the μ-calculus, the fixpoint alternation hierarchy over finite structures collapses at its first level. In this paper we show that the Collapse Theorem of fixpoint logic holds for a measured variant of the μ-calculus, which we call μ #-calculus. While μ-calculus formulas represent characteristic functions, i.e., functions from the state space to {0,1}, formulas of the μ #-calculus represent measure functions, which are functions from the state space to some measure domain. We prove a Measured-Collapse Theorem: every formula in the μ-calculus is equivalent to a least-fixpoint formula in the μ #-calculus. We show that the Measured-Collapse Theorem provides a logical recasting of the improved algorithm for μ-calculus model-checking, and describe how it can be implemented symbolically using Algebraic Decision Diagrams. Thus, we describe, for the first time, a symbolic μ-calculus model-checking algorithm whose complexity matches the one of the best known enumerative algorithm.

## Keywords

Model Check Binary Decision Diagram Kripke Structure Progress Measure Alternation Level
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Doron Bustan
• 1
• Orna Kupferman
• 2
• Moshe Y. Vardi
• 1
1. 1.Department of Computer ScienceRice UniversityHoustonU.S.A.
2. 2.School of Engineering and Computer ScienceHebrew UniversityJerusalemIsrael