The Variable Hierarchy for the Lattice μ-Calculus

Authors:
Walid Belkhir;Luigi Santocanale
Affiliations:
Laboratoire d'Informatique Fondamentale de Marseille, Université de Provence,;Laboratoire d'Informatique Fondamentale de Marseille, Université de Provence,
Venue:
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
Year:
2008

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Abstract

The variable hierarchy problem asks whether every μ -term t is equivalent to a μ -term t *** where the number of fixed-point variables in t *** is bounded by a constant. In this paper we prove that the variable hierarchy of the lattice μ -calculus --- whose standard interpretation is over the class of all complete lattices --- is infinite, meaning that such a constant does not exist if the μ -terms are built up using the basic lattice operations as well as the least and the greatest fixed point operators. The proof relies on the description of the lattice μ -calculus by means of games and strategies.