🤖 AI Summary
This work addresses the need for reasoning in intuitionistic logic under temporal evolution and information updates by introducing modal and fixed-point operators to construct five intuitionistic dynamic logics. Building on bi-relational Kripke semantics, it develops corresponding Hilbert-style axiomatizations and circular sequent calculi, and employs model checking and compositional analysis to systematically investigate expressiveness, the finite model property, decidability, and computational complexity. The main contributions include the first analytic circular sequent calculus for intuitionistic epistemic and public announcement logics, a proof of the finite model property and decidability for bi-intuitionistic modal logic, and a resolution of the long-standing open problem of finite axiomatizability for intuitionistic linear temporal logic.
📝 Abstract
This thesis develops the mathematical theory of intuitionistic dynamic logics - extensions of intuitionistic propositional logic with modalities and fixed point operators. Such systems provide formal tools for reasoning about change, such as encountered in mathematical systems evolving over time or in the knowledge state of an agent after an information update.
We investigate five intuitionistic dynamic logics: intuitionistic master modality, intuitionistic common knowledge logic, intuitionistic linear temporal logic, bi-intuitionistic modal logic and bi-intuitionistic linear temporal logic. On the proof theoretic side we develop sound and complete Hilbert-style axiomatizations as well as non-wellfounded and cyclic sequent calculi. On the semantic side we study these logics over various classes of dynamic models, which are birelational Kripke models satisfying confluence and frame conditions. We establish expressivity results, the finite model property, decidability, as well as complexity bounds.
The main contributions are threefold. First, we develop analytic cyclic sequent calculi for intuitionistic master modality and common knowledge logic, where completeness is obtained by a robust proof search argument. Second, we obtain the finite model property and decidability for bi-intuitionistic modal logic via an intricate combinatorial analysis of dynamic models. Third, we develop a sound and complete axiomatization for intuitionistic linear temporal logic featuring the temporal operators next, eventually and henceforth, thereby providing a positive answer to the long-standing open question concerning the existence of a finite axiomatization.