The Alternation Hierarchy for the Theory of mu-lattices

Abstract

The alternation hierarchy problem asks whether every mu-term,
that is a term built up using also 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 point of a different type
is bounded by a fixed number.
In this paper we give a proof that the alternation hierarchy
for the theory of mu-lattices is strict, meaning that such number
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.