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

Doron Bustan*, Orna Kupferman, Moshe Y. Vardi

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

12 Scopus citations


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 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.

Original languageEnglish
Title of host publication STACS 2004
Subtitle of host publication21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings
EditorsVolker Diekert, Michel Habib
PublisherSpringer Verlag
Number of pages12
ISBN (Print)3-540-21236-1
StatePublished - 2004
Event21st Annual Symposium on Theoretical Aspects of Computer Science, STACS 2004 - Montpellier, France
Duration: 25 Mar 200427 Mar 2004
Conference number: 21

Publication series

Name Lecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference21st Annual Symposium on Theoretical Aspects of Computer Science, STACS 2004
Internet address


Dive into the research topics of 'A Measured Collapse of the Modal µ-Calculus Alternation Hierarchy.'. Together they form a unique fingerprint.

Cite this