Norihiro Kamide

Linear and affine logics with temporal, spatial and epistemic operators

A temporal spatial epistemic intuitionistic linear logic (TSEILL) is introduced, and the completeness theorem for this logic is proved with respect to Kripke semantics. TSEILL has three temporal modal operators: [F] (any time in the future), [N] (next time) and [P] (past), some spatial modal operators [li] (locations), two epistemic modal operators; [K] (know) and (K), and a linear modal operator ! (exponential). A basic normal modal intuitionistic affine logic (BIAL) and its normal extensions are also defined, and the completeness theorems for these logics are proved with respect to Kripke semantics. In the proposed semantic framework of these normal extensions, a simple correspondence can be given between frame conditions and Lemmon-Scott axioms. A dynamic intuitionistic affine logic (DIAL) is proposed as an affine version of (test-free) dynamic logic, and the completeness theorem for this logic is shown with respect to Kripke semantics. Finally, some intuitive interpretations, such as resource and informational interpretations, are given for the proposed logics and semantics. By using these logics, semantics and interpretations, various kinds of fine-grained resource-sensitive reasoning can be expressed.

Quantized Linear Logic, Involutive Quantales and Strong Negation

A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletiers (commutative) involutive quantales. Some cut-free sequent calculi with a new property “quantization principle” and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansings extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.

Combining Soft Linear Logic and Spatio-temporal Operators

A new logic, spatio-temporal soft linear logic, which has soft linear exponential, linear exponential and spatio-temporal operators, is introduced as a sequent calculus. A Petri net interpretation and an informational interpretation for this logic are given by using simple complete semantics. By using this logic and these interpretations, various kinds of spatio-temporal and resource-sensitive reasoning can be expressed.

Natural deduction systems for Nelson's paraconsistent logic and its neighbors

Firstly, a natural deduction system in standard style is introduced for Nelsons para-consistent logic N4, and a normalization theorem is shown for this system. Secondly, a natural deduction system in sequent calculus style is introduced for N4, and a normalization theorem is shown for this system. Thirdly, a comparison between various natural deduction systems for N4 is given. Fourthly, a strong normalization theorem is shown for a natural deduction system for a sublogic of N4. Fifthly, a strong normalization theorem is proved for a typed λ-calculus for a neighbor of N4. Finally, it is remarked that the natural deduction frameworks presented can also be adapted for Wansings basic connexive logic C.

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLScw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLScw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.

Linear Logics with Communication-merge

Cut-elimination property, relevance principle, interpolation property and a new property named communication principle are proved for a number of modal intuitionistic linear logics with communication-merge rules. A concurrent-computational interpretation for these logics is obtained based on a process algebra with communication-merge.

A note on dual‐intuitionistic logic

Dual-intuitionistic logics are logics proposed by Czermak (1977), Goodman (1981) and Urbas (1996). It is shown in this paper that there is a correspondence between Goodmans dual-intuitionistic logic and Nelsons constructive logic N−.

A spatial modal logic with a location interpretation

A spatial modal logic (SML) is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space (or location) interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut-elimination theorem for a modified subsystem of SML are also presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On a logic of involutive quantales

The logic just corresponding to (non-commutative) involutive quantales, which was introduced by Wendy MacCaull, is reconsidered in order to obtain a cut-free sequent calculus formulation, and the completeness theorem (with respect to the involutive quantale model ) for this logic is proved using a new admissible rule. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)