Interpolation for the two-way modal μ-calculus

The two-way modal μ-calculus is the extension of the (standard) one-way μ-calculus with converse (backward-looking) modalities. For this logic we introduce two new sequent-style proof calculi: a non-wellfounded system admitting infinite branches and a finitary, cyclic version of this that employs annotations. As is common in sequent systems for two-way modal logics, our calculi feature an analytic cut rule. What distinguishes our approach is the use of so-called trace atoms, which serve to apply Vardi’s two-way automata in a proof-theoretic setting. We prove soundness and completeness for both systems and subsequently use the cyclic calculus to show that the two-way μ-calculus has the (local) Craig interpolation property, with respect to both propositions and modalities. Our proof uses a version of Maehara’s method adapted to cyclic proof systems. As a corollary we prove that the two-way μ-calculus also enjoys Beth’s definability property.