Periodic Reporting for period 1 - CYDER (CYclic DErivations for Recursive operators)
Berichtszeitraum: 2022-09-01 bis 2024-08-31
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.
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.
In its initial phase, CYDER mostly focused on the definition of cyclic proofs in the labelled formalism for the basic system of modal logic K with a simple recursive modality (transitive closure) and no frame condition. This task turned out to be significantly more complex than anticipated, and was completed after an in-depth study of methods to detect repeating behaviour in labelled proofs. This study was carried out in collaboration with Roman Kuznets, Sonia Marin, Marianela Morales and Lutz Straßurger, and resulted in proofs of decidability of intuitionistic modal logics IK and IS4 (without recursive modalities) through an innovative method to bound the size of proofs constructed in labelled calculi for the logics. The loop detection mechanism developed in the context of labelled calculi for IK and IS4 gave precious insights in the definition of cyclic labelled proofs for K extended with a transitive closure operator, a result achieved in collaboration with Jan Rooduijn. The definition of cyclic labelled proof systems for modal logics with recursive modalities and frame conditions is object of current study, and consists in a relatively extensions of the methods developed for the basic system.
CYDER achieved significant results in the definition of cyclic proofs in the structured formalism. Specifically, in collaboration with Anupam Das we defined a cyclic proof system for Transitive Closure Logic based on hypersequents. Transitive Closure Logic is a first-order logic together with a predicate expressing transitive closure of binary relations. This results represents a first step in defining structured systems for modal fixpoint logics, which is object of ongoing work.
Overall, CYDER developed innovative and solid methods to define cyclic proofs within the labelled and structured formalism. These methods are suitable to be generalised to modal fixpoint logics with frame conditions, even though this has not been achieved in the time span of the project.
CYDER achieved significant results in the definition of cyclic proofs in the structured formalism. Specifically, in collaboration with Anupam Das we defined a cyclic proof system for Transitive Closure Logic based on hypersequents. Transitive Closure Logic is a first-order logic together with a predicate expressing transitive closure of binary relations. This results represents a first step in defining structured systems for modal fixpoint logics, which is object of ongoing work.
Overall, CYDER developed innovative and solid methods to define cyclic proofs within the labelled and structured formalism. These methods are suitable to be generalised to modal fixpoint logics with frame conditions, even though this has not been achieved in the time span of the project.
CYDER developed several results and methodologies that go beyond the state of the art.
The decidability proof of intuitionistic modal logic IS4 is the first result of this kind for the logic. Moreover, the method to establish decidability by detecting repeating behaviours in labelled calculi is innovative, and suitable to be fruitfully applied to many other logics and proof systems.
The cyclic labelled proofs for modal logic K with a transitive closure operator required non-trivial modifications of the notion of progress and regularity as defined for cyclic proofs in the literature. This proof system advances the state of the art of both cyclic proofs and labelled calculi, and it is suitable to be generalised to capture modal fixpoint logics with frame conditions. Similarly, the cyclic hypersequent system for Transitive Closure Logic employs a novel progress condition, which again is suitable to be generalised to modal fixpoint logics with frame conditions.
The decidability proof of intuitionistic modal logic IS4 is the first result of this kind for the logic. Moreover, the method to establish decidability by detecting repeating behaviours in labelled calculi is innovative, and suitable to be fruitfully applied to many other logics and proof systems.
The cyclic labelled proofs for modal logic K with a transitive closure operator required non-trivial modifications of the notion of progress and regularity as defined for cyclic proofs in the literature. This proof system advances the state of the art of both cyclic proofs and labelled calculi, and it is suitable to be generalised to capture modal fixpoint logics with frame conditions. Similarly, the cyclic hypersequent system for Transitive Closure Logic employs a novel progress condition, which again is suitable to be generalised to modal fixpoint logics with frame conditions.