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.
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Bustan, D., Kupferman, O., Vardi, M.Y. (2004). A Measured Collapse of the Modal μ-Calculus Alternation Hierarchy. In: Diekert, V., Habib, M. (eds) STACS 2004. STACS 2004. Lecture Notes in Computer Science, vol 2996. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24749-4_46
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