# On the universal and existential fragments of the mu-calculus

Henzinger TA, Kupferman O, Majumdar R. 2006. On the universal and existential fragments of the mu-calculus. Theoretical Computer Science. 354(2), 173–186.

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Henzinger, Thomas A

^{ISTA}^{}; Kupferman, Orna; Majumdar, Ritankar SAbstract

One source of complexity in the μ-calculus is its ability to specify an unbounded number of switches between universal (AX) and existential (EX) branching modes. We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the μ-calculus, in which only one branching mode is allowed. The universal fragment is rich enough to express most specifications of interest, and therefore improved algorithms are of practical importance. We show that while the satisfiability and validity problems become indeed simpler for the existential and universal fragments, this is, unfortunately, not the case for model checking and implication. We also show the corresponding results for the alternation-free fragment of the μ-calculus, where no alternations between least and greatest fixed points are allowed. Our results imply that efforts to find a polynomial-time model-checking algorithm for the μ-calculus can be replaced by efforts to find such an algorithm for the universal or existential fragment.

Publishing Year

Date Published

2006-03-28

Journal Title

Theoretical Computer Science

Volume

354

Issue

2

Page

173 - 186

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### Cite this

Henzinger TA, Kupferman O, Majumdar R. On the universal and existential fragments of the mu-calculus.

*Theoretical Computer Science*. 2006;354(2):173-186. doi:10.1016/j.tcs.2005.11.015Henzinger, T. A., Kupferman, O., & Majumdar, R. (2006). On the universal and existential fragments of the mu-calculus.

*Theoretical Computer Science*. Elsevier. https://doi.org/10.1016/j.tcs.2005.11.015Henzinger, Thomas A, Orna Kupferman, and Ritankar Majumdar. “On the Universal and Existential Fragments of the Mu-Calculus.”

*Theoretical Computer Science*. Elsevier, 2006. https://doi.org/10.1016/j.tcs.2005.11.015.T. A. Henzinger, O. Kupferman, and R. Majumdar, “On the universal and existential fragments of the mu-calculus,”

*Theoretical Computer Science*, vol. 354, no. 2. Elsevier, pp. 173–186, 2006.Henzinger, Thomas A., et al. “On the Universal and Existential Fragments of the Mu-Calculus.”

*Theoretical Computer Science*, vol. 354, no. 2, Elsevier, 2006, pp. 173–86, doi:10.1016/j.tcs.2005.11.015.