CYDER's key objective consists in the definition of proof systems for modal fixpoint logics with frame conditions.
Modal fixpoint logics are modal logics including "recursive modalities", which are operators allowing to capture recursively defined notions, such as iterative behaviours of programs or common knowledge. These logics represent a formal tool to reason about recursion, and find several prominent applications in computer science, most notably in formal verification and knowledge representation. Much as in basic modal logic, the semantics of modal fixpoint logics can be enriched with additional properties, called "frame conditions", to capture specific attitudes.
Proof theory, the discipline studying proof systems, plays a fundamental role in the analysis of logical systems. A proof system is defined by a set of axioms and inference rules, used to check the validity of a formula by constructing a relevant derivation, that is, a proof tree composed from the axioms and inference rules of the proof system. Despite their wide range of application, the proof theory of modal fixpoint logics is currently underdeveloped. More specifically, the proof theory of modal fixpoint logics with frame conditions is much less advanced than the proof theory of modal or intuitionistic logics, due to non-trivial combinatorial challenges.
CYDER will meet the ambitious aim of defining proof systems for modal fixpoint logics extended with frame conditions by bringing together methods from two research areas: the study of proof systems for modal logics and of cyclic proof systems.
Analytic proof systems for modal logics have been defined by enriching the Gentzen-style sequent calculus. While sequents for classical propositional logic are multisets of formulas, sequents for modal logics are characterised by more complex data structures, reflecting the rich semantics of their logics. Specifically, two formalisms have been defined in the literature: labelled sequents and structured calculi. Labelled sequents encode graphs of formulas by adding semantic information within the language of the calculi. To the structured approach there belong several kinds of calculi, encoding various structures: for instance, nested sequents are trees of formulas, while hypersequents are multisets of sequents. Both labelled and structured sequents have been defined for large families of modal logics extended with frame conditions. CYDER plans to exploit the modular proof-theoretical methods for modal logics to introduce labelled and structured sequents into the richer framework of modal fixpoint logics.
Cyclic proofs are an elegant and analytic proof system to capture logics with fixpoint modalities. Cyclic proofs are a special kind of infinitary sequent proofs, treating recursion by means of rules which “unfold” the recursive operators infinitely often. A cyclic proof is regular, meaning that it contains only finitely many distinct subtrees, and each of its cyclic branches satisfies a progress condition, guaranteeing soundness. Cyclic proofs admit finite representation, in the form of finite (cyclic) graphs, as depicted in the figure. In the literature, cyclic proofs have been mostly defined for Gentzen-style sequent calculus. CYDER plans to extend the scope of cyclic proofs by defining such objects within the labelled and nested frameworks.
CYDER will introduce cyclic versions of labelled and structured proof systems for modal fixpoint logics extended with frame conditions. Thus, CYDER will develop original and general proof-theoretical frameworks for modal fixpoint logics with frame conditions, significantly advancing the state of the art of both cyclic proofs and proof systems for modal logics.
CYDER will initially focus on epistemic and temporal modal fixpoint logics, and then extend the investigation to more complex systems: more specifically, it will introduce cyclic proofs for Intuitionistic modal logics (IML) and Dynamic Epistemic Logics (DEL), both extended with recursive modalities. These logics have significant applications in formal verification and knowledge representation, but have been studied only from a model-theoretical viewpoint.
