Completeness of the aconjunctive matching μ-logic fragment

Establish whether the aconjunctive fragment of matching μ-logic has a complete proof system, while determining how aconjunctivity should be defined in the presence of quantified element variables and universal quantification.

Background

The paper proves that validity is not recursively enumerable for a small fragment of definedness-free matching μ-logic, using a conjunctive fixpoint whose bound variable occurs in multiple conjuncts. This obstruction does not apply directly to the aconjunctive fragment.

The modal μ-calculus has elementary completeness results for its aconjunctive fragment, but transferring that notion to matching μ-logic is nontrivial because universal quantification is represented using negated existential quantification, which can introduce conjunctions. The authors therefore identify completeness of this fragment as a focused unresolved target.