A Cut-free Sequent Calculus for Basic Intuitionistic Dynamic Topological Logic

Abstract: As part of a broader family of logics, [1, 3] introduced two key logical systems: $\mathsf{iK_{d}}$, which encapsulates the basic logical structure of dynamic topological systems, and $\mathsf{iK_{d*}}$, which provides a well-behaved yet sufficiently general framework for an abstract notion of implication. These logics have been thoroughly examined through their algebraic, Kripke-style, and topological semantics. To complement these investigations with their missing proof-theoretic analysis, this paper introduces a cut-free G3-style sequent calculus for $\mathsf{iK_{d}}$ and $\mathsf{iK_{d*}}$. Using these systems, we demonstrate that they satisfy the disjunction property and, more broadly, admit a generalization of Visser's rules. Additionally, we establish that $\mathsf{iK_{d}}$ enjoys the Craig interpolation property and that its sequent system possesses the deductive interpolation property.

• The paper presents a cut-free G3-style sequent calculus for logics iK_d and iK_{d*} that formalizes dynamic topological systems.
• It establishes key properties such as disjunction, Craig and deductive interpolation, along with generalizations of Visser's rules.
• The study confirms the conservativity of iK_d over iK_{d*} and demonstrates height-preserving admissibility of contraction and cut elimination.

Cut-Free Sequent Calculus for Intuitionistic Dynamic Topological Logic

This paper (2502.09456) introduces a cut-free G3-style sequent calculus for the logics $iK_d$ and $iK_{d*}$, which are basic logical systems that encapsulate the structure of dynamic topological systems and provide a framework for abstract implication. The paper uses these systems to demonstrate that $iK_d$ and $iK_{d*}$ satisfy the disjunction property and admit a generalization of Visser's rules. It also establishes that $iK_d$ enjoys the Craig interpolation property and that its sequent system possesses the deductive interpolation property.

Background and Motivation

The paper addresses the proof-theoretic analysis of the logics $iK_d$ and $iK_{d*}$, systems introduced in prior work to formalize dynamic topological systems and abstract implications. Dynamic topological systems, which model the evolution of states in a topological space, have been approached using modal logics. The logics $iK_d$ and $iK_{d*}$ were previously studied through algebraic, Kripke-style, and topological semantics, but lacked a corresponding proof theory due to the absence of a cut-free sequent calculus.

Abstract Implications and Dynamic Topological Systems

The paper frames its investigation within the broader context of abstract implications, which aim to internalize the meta-linguistic provability order into the formal language itself. The authors adopt the abstract notion of implication proposed in previous work, interpreting implications as a logical mechanism to represent the provability order $A \vdash B$ between propositions as a proposition $A \to B$ within the formal language. The paper references $\nabla$-algebras, which provide a unifying framework that generalizes both bounded lattices and Heyting algebras, and are also connected to temporal interpretations. The study of normal Heyting $\nabla$-algebras corresponds to the algebraic study of dynamic topological systems.

Sequent Calculi $iK_d$ and $iK_{d*}$

The authors introduce two cut-free sequent calculi, $iK_d$ and $iK_{d*}$, for the logics $iK_d$ and $iK_{d*}$, respectively. The calculus $iK_d$ is defined over a language including the Heyting implication, while $iK_{d*}$ is defined over a fragment excluding this implication. These calculi are designed to address the limitations of existing sequent calculi for these logics, which are not analytic and do not admit cut elimination. The paper presents the axioms and rules of $iK_d$ and $iK_{d*}$, highlighting key differences from previous systems.

Properties of $iK_d$ and $iK_{d*}$

The paper establishes fundamental properties of the systems $iK_d$ and $iK_{d*}$, including the provability of certain sequents, $\nabla$-analyticity, and conservativity. It is shown that $iK_d$ is conservative over $iK_{d*}$, meaning that if a sequent over the language of $iK_{d*}$ is provable in $iK_d$, it is also provable in $iK_{d*}$. An inversion lemma is presented for certain rules of $iK_d$ and $iK_{d*}$, and the height-preserving admissibility of contraction is proven. The admissibility of the cut rule is demonstrated, which is a key result in establishing the equivalence of these calculi with existing systems.

Equivalence and Deduction Theorem

The paper proves that $iK_d$ and $iK_{d*}$ are equivalent to existing sequent calculi for these logics, meaning they prove precisely the same set of sequents. This establishes that the logics $iK_d$ and $iK_{d*}$ are the logics of the calculi $iK_d$ and $iK_{d*}$, respectively. Furthermore, a modified version of the deduction theorem is presented for $iK_d$ and $iK_{d*}$, which relates provability with and without assumptions in the calculi.

Visser's Rules and Interpolation Theorems

The paper introduces generalizations of Visser's rules and proves their admissibility in the logics $iK_d$ and $iK_{d*}$. Since the languages are more expressive than usual propositional language, the authors consider three types of Visser's rules: disjunctive, implicative, and Heyting implicative. It is also shown that $iK_d$ satisfies the Craig interpolation property, meaning that if $A \to B$ is provable, there exists an interpolant formula $C$ such that $A \to C$ and $C \to B$ are provable, and the variables in $C$ are contained in both $A$ and $B$. Finally, the deductive interpolation property for $iK_d$ is established.

Conclusion

This paper (2502.09456) makes a significant contribution by providing a cut-free sequent calculus for the logics $iK_d$ and $iK_{d*}$. By establishing key proof-theoretic properties such as cut admissibility, conservativity, and interpolation, the authors advance the understanding and applicability of these logics in formalizing dynamic topological systems and abstract implications. The results of the paper open avenues for further research in intuitionistic modal logic and its applications in computer science and AI.