Syntactic complexity in the modal μ calculus
Lehtinen, Maria Karoliina
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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.