The alternation hierarchy for the theory of $\mu$- lattices

Abstract

The alternation hierarchy problem asks whether every $\mu$-term $\phi$, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a $\mu $-term where the number of nested fixed points of a different type is bounded by a constant independent of $\phi$. In this paper we give a proof that the alternation hierarchy for the theory of $\mu$-lattices is strict, meaning that such a constant does not exist if $\mu$-terms are built up from the basic lattice operations and are interpreted as expected. The proof relies on the explicit characterization of free $\mu$-lattices by means of games and strategies.