On Guarded Transformation In The Modal Mu-Calculus

Guarded normal form requires occurrences of fixpoint variables in a {\mu}-calculus-formula to occur under the scope of a modal operator. The literature contains guarded transformations that effectively bring a {\mu}-calculus-formula into guarded normal form. We show that the known guarded transformations can cause an exponential blowup in formula size, contrary to existing claims of polynomial behaviour. We also show that any polynomial guarded transformation for {\mu}-calculus-formulas in the more relaxed vectorial form gives rise to a polynomial solution algorithm for parity games, the existence of which is an open problem. We also investigate transformations between the {\mu}-calculus, vectorial form and hierarchical equation systems, which are an alternative syntax for alternating parity tree automata.