arXiv subjects

Lukas Zenger

Publications and source records attributed to Lukas Zenger.

4 recordsLinked to original sources

arXiv2026

Intuitionistic Dynamic Logic

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.

cs.LO↗

arXiv2026

Non-Wellfounded and Cyclic Proofs for LTL: A Syntactic Correspondence with Linear Nested Sequents

We introduce and investigate non-wellfounded and cyclic linear nested sequent calculi, and, as a case study, develop such systems for linear temporal logic (LTL). The paper addresses two central problems, which we call 'cycle recognition' and 'unraveling.' Cycle recognition concerns identifying cycles in non-wellfounded proofs in order to extract corresponding cyclic proofs, while unraveling studies the converse transformation, from cyclic proofs to non-wellfounded ones. Although these processes are well understood for Gentzen sequents, they have received little attention for more expressive sequent formalisms and become more challenging in the linear nested sequent setting. To address cycle recognition, we show the completeness of non-wellfounded proofs relative to a particular normal form exhibiting a property we call 'saturation recurrence,' which enables the systematic extraction of cyclic proofs. To address unraveling, we introduce a specialized procedure that shifts rule applications forward along linear nested sequents, allowing non-wellfounded proofs to be reconstructed from cyclic ones. Overall, our work provides new proof-theoretic techniques for cycle recognition and unraveling in expressive multisequent formalisms.

cs.LO↗

arXiv2026

Intuitionistic Common Knowledge

We study an intuitionistic version of common knowledge logic (CK), called ICK, which was introduced by J\"ager and Marti. ICK extends intuitionistic propositional logic (IPL) by multiple box modalities interpreted as knowledge operators for various agents and a common knowledge operator. Formulae are interpreted over birelational Kripke models satisfying a simple interaction principle between the intuitionistic order and the modal accessibility relations. Furthermore, we consider the restriction to reflexive, S4 and S5 models. We present axiomatizations as well as analytic cyclic sequent calculi for all considered logics and prove them to be sound and complete. Furthermore, we establish the finite model property and decidability, show that proof-search in the cyclic calculi can be automated, provide a translation of CK over S5 into ICK over S5 and establish that the proof-search and validity problems of all considered logics can be solved in exponential time.

math.LO↗

arXiv2025

Coalgebraic proof translations for non-wellfounded proofs

Non-wellfounded proof theory results from allowing proofs of infinite height in proof theory. To guarantee that there is no vicious infinite reasoning, it is usual to add a constraint to the possible infinite paths appearing in a proof. Among these conditions, one of the simplest is enforcing that any infinite path goes through the premise of a rule infinitely often. Systems of this kind appear for modal logics with conversely well-founded frame conditions like GL or Grz. In this paper, we provide a uniform method to define proof translations for such systems, guaranteeing that the condition on infinite paths is preserved. In addition, as particular instance of our method, we establish cut-elimination for a non-wellfounded system of the logic Grz. Our proof relies only on the categorical definition of corecursion via coalgebras, while an earlier proof by Savateev and Shamkanov uses ultrametric spaces and a corresponding fixed point theorem.

math.LO↗