arXiv subjects

Katsuhiko Sano

Publications and source records attributed to Katsuhiko Sano.

14 recordsLinked to original sources

arXiv2026

Phase Semantic Cut-elimination for Intuitionistic Linear Logic with Least and Greatest Fixed Points

This paper establishes the cut-elimination theorem for intuitionistic propositional multiplicative-additive linear logic with the least and greatest fixpoints ($μ$IMALL) by means of its phase semantics. A classical first-order multiplicative-additive linear logic system with the least and greatest fixpoints was introduced by Baelde and Miller (2007). Its intuitionistic fragment was discussed in Baelde (2012), but the cut-elimination theorem for this fragment has not yet been proved. We introduce a propositional fragment of this system, $μ$IMALL, and establish the cut-elimination theorem. To prove the theorem, we define phase semantics for $μ$IMALL and show the following two statements: (1) Soundness: if a formula is provable in $μ$IMALL, then it is true in all phase models, and (2) Cut-free Completeness: if a formula is true in all phase models, then it is provable in $μ$IMALL without Cut. Okada (1999, 2002) employed a phase semantic method to prove the cut-elimination theorems for classical and intuitionistic linear logic systems. De et al. (2022) applied this method to a propositional fragment of classical propositional multiplicative-additive linear logic with the least and greatest fixpoints. We refine and apply their arguments to prove the cut-elimination theorem for $μ$IMALL.

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arXiv2026

Analytic Cut in Epistemic Logics with Distributed Knowledge

Distributed knowledge is a notion of group knowledge studied in multi-agent epistemic logic. Semantically, the distributed knowledge of a group is interpreted via an accessibility relation given by the intersection of the epistemic accessibility relations of the agents in that group. This paper investigates sequent calculi for epistemic logics of distributed knowledge based on K45, KD45, and S5. While cut elimination holds in existing sequent calculi for modal logics K45 and KD45, it fails in all the systems mentioned above. Instead, we establish the analytic cut property for all three systems by adapting Takano' s (2018) strategy, which restricts the cut formulas to the set of subformulas of the conclusion of the cut rule. As a corollary, the Craig interpolation theorem holds for all logics considered. We also show that all proof-theoretic results remain valid when the empty group is allowed for the distributed-knowledge operator, in which case the distributed knowledge for the empty group is interpreted as the global modality.

cs.MA↗

arXiv2026

Uniform Interpolation of Basic Tense Logic

This paper establishes the uniform interpolation theorem for basic tense logic, which is also known as two-way modal logic or modal logic with converse. First introduced by Arthur Prior, basic tense logic is a syntactic expansion of basic modal logic with a converse modality. Its corresponding accessibility relation is defined as the converse of the standard accessibility relation in a given Kripke model. Although basic tense logic has been widely studied since its introduction, its uniform interpolation property has yet to be fully established. For basic modal logic K, Albert Visser (1996) provided a semantic argument formulated in terms of layered (or bounded) bisimulation, explicitly attributing the uniform interpolation property of K to Silvio Ghilardi. This paper extends Visser's semantic argument to demonstrate that basic tense logic also enjoys the uniform interpolation property.

cs.LO↗

arXiv2025

Undecidability of Linear Logics without Weakening

The goal of this paper is to establish that it remains undecidable whether a sequent is provable in two systems in which a weakening rule for an exponential modality is completely omitted from classical propositional linear logic $\mathbf{CLL}$ introduced by Girard (1987), which is shown to be undecidable by Lincoln et al. (1992). We introduce two logical systems, $\mathbf{CLLR}$ and $\mathbf{CLLRR}$. The first system, $\mathbf{CLLR}$, is obtained by omitting the weakening rule for the exponential modality of $\mathbf{CLL}$. The system $\mathbf{CLLR}$ has been studied by several authors, including Meliès-Tabareau (2010), but its undecidability was unknown. This paper shows the undecidability of $\mathbf{CLLR}$ by reducing it to the undecidability of $\mathbf{CLL}$, where the units $\mathbf{1}$ and $\bot$ play a crucial role in simulating the weakening rule. We also omit these units from the syntax and inference rules of $\mathbf{CLLR}$ in order to define the second system, $\mathbf{CLLRR}$. The undecidability of $\mathbf{CLLRR}$ is established by showing that the system can simulate any two-counter machine proposed by Minsky (1961).

cs.LO↗

arXiv2025

Bounded Inquisitive Logics: Sequent Calculi and Schematic Validity

Propositional inquisitive logic is the limit of its $n$-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti, who also found complete axiomatizations of $n$-bounded inquisitive logics $\mathsf{InqBQ}_{n}$, for every fixed $n$. We introduce cut-free labelled sequent calculi for these logics. We illustrate the intricacies of \textit{schematic validity} in such systems by showing that the well-known Casari formula is \textit{atomically} valid in (a weak sublogic of) predicate inquisitive logic $\mathsf{InqBQ}$, fails to be schematically valid in it, and yet is schematically valid under the finite boundedness assumption. The derivations in our calculi, however, are guaranteed to be schematically valid whenever a single specific rule is not used.

cs.LO↗

arXiv2023

Semantic Incompleteness of Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic

The updated version of this paper has already been published in The Australasian Journal of Logic. You can access to the paper from the following link: https://ojs.victoria.ac.nz/ajl/article/view/7696. This paper shows Hilbert system $(\mathbf{C+J})^{-}$, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system $(\mathbf{C+J})^{-}$ contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system $(\mathbf{C+J})^{-}$ is semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979).

cs.LO↗

arXiv2022

Combining First-Order Classical and Intuitionistic Logic

This paper studies a first-order expansion of a combination C+J of intuitionistic and classical propositional logic, which was studied by Humberstone (1979) and del Cerro and Herzig (1996), from a proof-theoretic viewpoint. While C+J has both classical and intuitionistic implications, our first-order expansion adds classical and intuitionistic universal quantifiers and one existential quantifier to C+J. This paper provides a multi-succedent sequent calculus G(FOC+J) for our combination of the first-order intuitionistic and classical logic. Our sequent calculus G(FOC+J) restricts contexts of the right rules for intuitionistic implication and intuitionistic universal quantifier to particular forms of formulas. The cut-elimination theorem is established to ensure the subformula property. As a corollary, G(FOC+J) is conservative over both first-order intuitionistic and classical logic. Strong completeness of G(FOC+J) is proved via a canonical model argument.

cs.LO↗

arXiv2019

Axiomatizing Epistemic Logic of Friendship via Tree Sequent Calculus

This paper positively solves an open problem if it is possible to provide a Hilbert system to Epistemic Logic of Friendship (EFL) by Seligman, Girard and Liu. To find a Hilbert system, we first introduce a sound, complete and cut-free tree (or nested) sequent calculus for EFL, which is an integrated combination of Seligman's sequent calculus for basic hybrid logic and a tree sequent calculus for modal logic. Then we translate a tree sequent into an ordinary formula to specify a Hilbert system of EFL and finally show that our Hilbert system is sound and complete for the intended two-dimensional semantics.

cs.LO↗

arXiv2018

Model Theory and Proof Theory of Coalgebraic Predicate Logic

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result.

cs.LO↗

arXiv2018

Characterizing Relative Frame Definability in Team Semantics via the Universal Modality

Let ML(U^+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterize the relative definability of ML(U^+) relative to finite transitive frames in the spirit of the well-known Goldblatt-Thomason theorem. We show that a class F of Kripke frames is definable in ML(U^+) relative to finite transitive frames if and only if F is closed under taking generated subframes and bounded morphic images. In addition, we study modal definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. With respect to global model definability we obtain a trichotomy and with respect to frame definability a dichotomy. As a corollary we obtain relative Goldblatt--Thomason -style theorems for each of the logics listed above.

math.LO↗

arXiv2018

Characterising Modal Definability of Team-Based Logics via the Universal Modality

We study model and frame definability of various modal logics. Let ML(A+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML(A+) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML(A+) in the spirit of the well-known Goldblatt--Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML(A+) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition we study frame definability relative to finite transitive frames and give an analogous characterisation of ML(A+)-definability relative to finite transitive frames. Finally, we initiate the study of model and frame definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. We establish strict linear hierarchies with respect to model definability and frame definability, respectively. We show that, with respect to model and frame definability, the before mentioned team-based logics, except modal dependence logic, either coincide with ML(A+) or plain modal logic ML. Thus as a corollary we obtain model theoretic characterisation of model and frame definability for the team-based logics.

math.LO↗

arXiv2017

Strong Completeness and the Finite Model Property for Bi-Intuitionistic Stable Tense Logics

Bi-Intuitionistic Stable Tense Logics (BIST Logics) are tense logics with a Kripke semantics where worlds in a frame are equipped with a pre-order as well as with an accessibility relation which is 'stable' with respect to this pre-order. BIST logics are extensions of a logic, BiSKt, which arose in the semantic context of hypergraphs, since a special case of the pre-order can represent the incidence structure of a hypergraph. In this paper we provide, for the first time, a Hilbert-style axiomatisation of BISKt and prove the strong completeness of BiSKt. We go on to prove strong completeness of a class of BIST logics obtained by extending BiSKt by formulas of a certain form. Moreover we show that the finite model property and the decidability hold for a class of BIST logics.

cs.LO↗

arXiv2014

Axiomatizing Propositional Dependence Logics

We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi.

cs.LO↗

arXiv2014

The Expressive Power of Modal Dependence Logic

We study the expressive power of various modal logics with team semantics. We show that exactly the properties of teams that are downward closed and closed under team k-bisimulation, for some finite k, are definable in modal logic extended with intuitionistic disjunction. Furthermore, we show that the expressive power of modal logic with intuitionistic disjunction and extended modal dependence logic coincide. Finally we establish that any translation from extended modal dependence logic into modal logic with intuitionistic disjunction increases the size of some formulas exponentially.

cs.LO↗