## Syntactic complexity in the modal μ calculus

##### Abstract

This thesis studies how to eliminate syntactic complexity in Lμ, the modal μ calculus.
Lμ is a verification logic in which a least fixpoint operator μ, and its dual v, add recursion
to a simple modal logic. The number of alternations between μ and v is a measure
of complexity called the formula’s index: the lower the index, the easier a formula is
to model-check. The central question of this thesis is a long standing one, the Lμ index
problem: given a formula, what is the least index of any equivalent formula, that is to
say, its semantic index?
I take a syntactic approach, focused on simplifying formulas. The core decidability
results are (i) alternative, syntax-focused decidability proofs for ML and Pμ
1 , the
low complexity classes of μ; and (ii) a proof that Ʃμ
2 , the fragment of Lμ with one
alternation, is decidable for formulas in the dual class Pμ
2 .
Beyond its algorithmic contributions, this thesis aims to deepen our understanding
of the index problem and the tools at our disposal. I study disjunctive form and
related syntactic restrictions, and how they affect the index problem. The main technical
results are that the transformation into disjunctive form preserves Pμ
2 -indices but
not μ
2 -indices, and that some properties of binary trees are expressible with a lower
index using disjunctive formulas than non-deterministic automata. The latter is part
of a thorough account of how the Lμ index problem and the Rabin–Mostowski index
problem for parity automata are related.
In the final part of the thesis, I revisit the relationship between the index problem
and parity games. The syntactic index of a formula is an upper bound on the descriptive
complexity of its model-checking parity games. I show that the semantic index of
a formula Ψ is bounded above by the descriptive complexity of the model-checking
games for Ψ. I then study whether this bound is strict: if a formula Ψ is equivalent to
a formula in an alternation class C, does a formula of C suffice to describe the winning
regions of the model-checking games of Ψ? I prove that this is the case for ML, Pμ
1 ,
Ʃμ
2 , and the disjunctive fragment of any alternation class. I discuss the practical implications
of these results and propose a uniform approach to the index problem, which
subsumes the previously described decision procedures for low alternation classes.
In brief, this thesis can be read as a guide on how to approach a seemingly complex
Lμ formula. Along the way it studies what makes this such a difficult problem
and proposes novel approaches to both simplifying individual formulas and deciding
further fragments of the alternation hierarchy.