Gentzen-type Sequent Calculus Research Articles

Modal extension of ideal paraconsistent four-valued logic and its subsystem

This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this subsystem are proved. This subsystem is also shown to be decidable and embeddable into S4.

Kripke-Completeness and Cut-elimination Theorems for Intuitionistic Paradefinite Logics With and Without Quasi-Explosion

Two intuitionistic paradefinite logics N4C and N4C+ are introduced as Gentzen-type sequent calculi. These logics are regarded as a combination of Nelson’s paraconsistent four-valued logic N4 and Wansing’s basic constructive connexive logic C. The proposed logics are also regarded as intuitionistic variants of Arieli, Avron, and Zamansky’s ideal paraconistent four-valued logic 4CC. The logic N4C has no quasi-explosion axiom that represents a relationship between conflation and paraconsistent negation, but the logic N4C+ has this axiom. The Kripke-completeness and cut-elimination theorems for N4C and N4C+ are proved.

A derivation-loop method for temporal logic

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been considered in the literature. Cutfree Gentzen-type sequent calculi are convenient tools for backward proof-search search of formulas and sequents. In this paper we present a cut-free Gentzen type sequent calculus for PLTL with the operator

A proof-search system for the logic of likelihood

Abstract The cut-free Gentzen-type sequent calculus LLK for the logic of likelihood (LL) is introduced in the paper. It is proved that the calculus is sound and complete for LL. Using the introduced calculus LLK, a decision procedure for LL is presented.

Bi-Classical Connexive Logic and its Modal Extension: Cut-elimination, completeness and duality

In this study, a new paraconsistent four-valued logic called bi-classical connexive logic (BCC) is introduced as a Gentzen-type sequent calculus. Cut-elimination and completeness theorems for BCC are proved, and it is shown to be decidable. Duality property for BCC is demonstrated as its characteristic property. This property does not hold for typical paraconsistent logics with an implication connective. The same results as those for BCC are also obtained for MBCC, a modal extension of BCC.

Gentzen-Type Sequent Calculi for Extended Belnap–Dunn Logics with Classical Negation: A General Framework

Gentzen-type sequent calculi GBD+, GBDe, GBD1, and GBD2 are respectively introduced for De and Omori’s axiomatic extensions BD+, BDe, BD1, and BD2 of Belnap–Dunn logic by adding classical negation. These calculi are constructed based on a small modification of the original characteristic axiom scheme for negated implication. Theorems for syntactically and semantically embedding these calculi into a Gentzen-type sequent calculus LK for classical logic are proved. The cut-elimination, decidability, and completeness theorems for these calculi are obtained using these embedding theorems. Similar results excluding cut-elimination results are also obtained for alternative Gentzen-type sequent calculi gBD+, gBDe, gBD1, and gBD2 for BD+, BDe, BD1, and BD2, respectively. These alternative calculi are constructed based on the original characteristic axiom scheme for negated implication.

Kripke Completeness of Bi-intuitionistic Multilattice Logic and its Connexive Variant

In this paper, bi-intuitionistic multilattice logic, which is a combination of multilattice logic and the bi-intuitionistic logic also known as Heyting–Brouwer logic, is introduced as a Gentzen-type sequent calculus. A Kripke semantics is developed for this logic, and the completeness theorem with respect to this semantics is proved via theorems for embedding this logic into bi-intuitionistic logic. The logic proposed is an extension of first-degree entailment logic and can be regarded as a bi-intuitionistic variant of the original classical multilattice logic determined by the algebraic structure of multilattices. Similar completeness and embedding results are also shown for another logic called bi-intuitionistic connexive multilattice logic, obtained by replacing the connectives of intuitionistic implication and co-implication with their connexive variants.

Modal Multilattice Logic

A modal extension of multilattice logic, called modal multilattice logic, is introduced as a Gentzen-type sequent calculus $$\hbox {MML}_n$$ . Theorems for embedding $$\hbox {MML}_n$$ into a Gentzen-type sequent calculus S4C (an extended S4-modal logic) and vice versa are proved. The cut-elimination theorem for $$\hbox {MML}_n$$ is shown. A Kripke semantics for $$\hbox {MML}_n$$ is introduced, and the completeness theorem with respect to this semantics is proved. Moreover, the duality principle is proved as a characteristic property of $$\hbox {MML}_n$$ .

Paraconsistent Double Negations as Classical and Intuitionistic Negations

A classical paraconsistent logic (CP), which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic paraconsistent logic (IP), and several versions of Glivenko and Godel-Gentzen translation theorems are proved for CP and IP.

Paraconsistent double negation as a modal operator

A paraconsistent modal-like logic, , is defined as a Gentzen-type sequent calculus. The modal operator in the modal logic can be simulated by the paraconsistent double negation in . Some theorems for embedding into a Gentzen-type sequent calculus for and vice versa are proved. The cut-elimination and completeness theorems for are also proved.

Bunched sequential information

It is known that the logic BI of bunched implications is a logic of resources. Many studies have reported on the applications of BI to computer science. In this paper, an extension BIS of BI by adding a sequence modal operator is introduced and studied in order to formalize more fine-grained resource-sensitive reasoning. By the sequence modal operator of BIS, we can appropriately express “sequential information” in resource-sensitive reasoning. A Gentzen-type sequent calculus SBIS for BIS is introduced, and the cut-elimination and decidability theorems for SBIS are proved. An extension of the Grothendieck topological semantics for BI is introduced for BIS, and the completeness theorem with respect to this semantics is proved. The cut-elimination, decidability and completeness theorems for SBIS and BIS are proved using some theorems for embedding BIS into BI.

Single-succedent System Approach to Boolean BI

Abstract It is known that the logic BI of bunched implications is useful for describing shared mutable data structures and resource-aware reasoning. It has recently been clarified that some classical versions of BI are especially useful for describing shared mutable data structures. In this paper, a single-succedent Gentzen-type sequent calculus GcBI for a classical version (called Boolean BI) of an intuitionistic BI is introduced. Some theorems for embedding GcBI into its intuitionistic version GiBI, which are analogous to the Glivenko and Godel-Gentzen embedding theorems of classical logic into intuitionistic logic, are proved.

Inconsistency-Tolerant Multi-Agent Calculus

Verifying and specifying multi-agent systems in an appropriate inconsistency-tolerant logic are of growing importance in Computer Science since computer systems are generally used by or composed of inconsistency-tolerant multi-agents. In this paper, an inconsistency-tolerant logic for representing multi-agents is introduced as a Gentzen-type sequent calculus. This logic (or calculus) has multiple negation connectives that correspond to each agent, and these negation connectives have the property of paraconsistency that guarantees inconsistency-tolerance. The logic proposed is regarded as a modified generalization of trilattice logics, which are known to be useful for expressing fine-grained truth-values in computer networks. The completeness, cut-elimination and decidability theorems for the proposed logic (or sequent calculus) are proved as the main results of this paper.

AN EIGHT-VALUED PRACONSISTENT LOGIC

A b s t r a c t. It is known that many-valued paraconsistent logics are useful for expressing uncertain and inconsistency-tolerant reasoning in a wide range of Computer Science. Some four-valued and sixteen-valued logics have especially been well-studied. Some four-valued logics are not so ne-grained, and some sixteen-valued logics are enough ne-grained, but rather complex. In this paper, a natural eight-valued paraconsistent logic rather than four-valued and sixteen-valued logics is introduced as a Gentzen-type sequent calculus. This eight-valued logic is enough ne-grained and simpler than sixteen-valued logic. A triplet valuation semantics is introduced for this logic, and the completeness theorem for this semantics is proved. The cut-elimination theorem for this logic is proved, and this logic is shown to be decidable.

A Hierarchy of Weak Double Negations

In this paper, a way of constructing many-valued paraconsistent logics with weak double negation axioms is proposed. A hierarchy of weak double negation axioms is addressed in this way. The many-valued paraconsistent logics constructed are defined as Gentzen-type sequent calculi. The completeness and cut-elimination theorems for these logics are proved in a uniform way. The logics constructed are also shown to be decidable.

Representing any-time and program-iteration by infinitary conjunction

Two new infinitary modal logics are simply obtained from a Gentzen-type sequent calculus for infinitary logic by adding a next-time operator, and a program operator, respectively. It is shown that an any-time operator and a program-iteration operator can respectively be expressed using infinitary conjunction in these logics. The cut-elimination and completeness theorems for these logics are proved using some theorems for embedding these logics into (classical) infinitary logic.

Temporal BI: Proof system, semantics and translations

The logic BI of bunched implications is a combination of intuitionistic logic and multiplicative intuitionistic linear logic. In this paper, a temporal extension tBI of BI is introduced and studied. A Gentzen-type sequent calculus GtBI for tBI is introduced, and the cut-elimination and decidability theorems for GtBI are proved using a theorem for syntactically embedding GtBI into a sequent calculus GBI for BI. A semantics for GtBI is introduced extending the Grothendieck topological semantics for BI, and the completeness theorem with respect to this semantics is proved using a theorem for semantically embedding tBI into BI. A semantics for GtBI without additive falsity constant is introduced extending the Kripke resource semantics for BI without additive falsity constant, and the completeness theorem with respect to this semantics is proved in a similar way. Moreover, an intuitionistic temporal linear logic, ITLL, is introduced as a Gentzen-type sequent calculus, and a theorem for embedding GtBI into ITLL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic.

Inconsistency-Tolerant Bunched Implications

It is known that logical systems with the property of paraconsistency can deal with inconsistency-tolerant and uncertainty reasoning more appropriately than systems which are non-paraconsistent. It is also known that the logic BI of bunched implications is useful for formalizing resource-sensitive reasoning. In this paper, a paraconsistent extension PBI of BI is studied. The logic PBI is thus intended to formalize an appropriate combination of inconsistency-tolerant reasoning and resource-sensitive reasoning. A Gentzen-type sequent calculus SPBI for PBI is introduced, and the cut-elimination and decidability theorems for SPBI are proved. An extension of the Grothendieck topological semantics for BI is introduced for PBI, and the completeness theorem with respect to this semantics is proved.

Reasoning about Context Information in Cloud Computing Environments

The notion of context provides flexibility and adaptation to cloud computing services. Location, time identity and activity of users are examples of primary context types. The motivation of this paper is to formalize reasoning about context information in cloud computing environments. To formalize such context-aware reasoning, the logic LCM of context-mixture is introduced based on a Gentzen-type sequent calculus for an extended resource-sensitive logic. LCM has a specific inference rule called the context-mixture rule, which can naturally represent a mechanism for merging formulas with context information. Moreover, LCM has a specific modal operator called the sequence modal operator, which can suitably represent context information. The cut-elimination and embedding theorems for LCM are proved, and a fragment of LCM is shown to be decidable. These theoretical results are intended to provide a logical justification of context-aware cloud computing service models such as a flowable service model.

A logic of sequences

The notion of “sequences” is fundamental to practical reasoning in computer science, because it can appropriately represent “data (information) sequences”, “program (execution) sequences”, “action sequences”, “time sequences”, “trees”, “orders” etc. The aim of this paper is thus to provide a basic logic for reasoning with sequences. A propositional modal logic LS of sequences is introduced as a Gentzen-type sequent calculus by extending Gentzen’s LK for classical propositional logic. The completeness theorem with respect to a sequence-indexed semantics for LS is proved, and the cut-elimination theorem for LS is shown. Moreover, a first-order modal logic FLS of sequences, which is a first-order extension of LS, is introduced. The completeness theorem with respect to a first-order sequence-indexed semantics for FLS is proved, and the cut-elimination theorem for FLS is shown. LS and the monadic fragment of FLS are shown to be decidable.