arXiv subjects

Kenji Tokuo

Publications and source records attributed to Kenji Tokuo.

7 recordsLinked to original sources

arXiv2026

Spectral Sign Implication for Quantum Logic

We study the spectral sign implication for quantum logic, defined by the nonnegative spectral projection of the operator obtained by subtracting the antecedent projection from the consequent projection. The construction agrees with classical material implication on commuting projections and compares arbitrary pairs of projections through the spectral structure of their difference. It satisfies Hardegree's four minimal implicative conditions, his law of contraposition, and a falsity condition. It differs from the standard polynomial implications in that its value need not belong to the ortholattice generated by its arguments. Within a uniform class of Borel constructions for pairs of projections, the operations satisfying entailment, contraposition, and the falsity condition correspond exactly to measurable choices of spectral branch. In the continuous subclass, these three conditions determine the spectral sign implication uniquely. In finite dimensions, the operation is also the largest among the acceptance projections of optimal projective tests for Helstrom discrimination between subspace states with priors proportional to rank. These results connect quantum implication with the operator theory of two projections and binary quantum discrimination.

quant-ph↗

arXiv2026

Multimodal Logic Programming with Full Formulas

This paper presents a first-order multimodal logic programming system called MMLP. The system accepts arbitrary formulas as both programs and queries, without restricting either side to Horn clauses or a separate goal grammar. Its declarative semantics is given independently by a Hilbert system for selected D, T, I, B, 4, and 5 modal principles. Execution uses a nested proof calculus with finite grammar certificates for modal propagation. Certificate reachability is equivalent to the associated Horn closure, certificate existence is decidable, and the calculus is cut-free complete. For quantified answer computation, we give a unification algorithm based on permission sets that controls eigenparameter scope. The algorithm always terminates and fails exactly when no admissible solution exists. A successful run returns a unifier that is itself admissible and through which all admissible solutions factor. Computed answers are declaratively correct, and each declaratively correct answer is an ordinary output instance of a computed answer. In proof search, MMLP admits syntactic focalization and a fair and complete enumeration of focused answers. It represents exactly the same answer substitutions as Nguyen's pure logical KDI4s5-MPROLOG on their common fragment, while admitting a strictly larger program and query language.

math.LO↗

arXiv2026

QBism Logic

QBism interprets quantum theory as a normative discipline for an agent's probability assignments and their revision across possible experience. This paper develops a logical formalization of that picture. A well-formed core datum consists of an admissible prior space, a finite family of actual measurements, Born kernels, and update kernels. For each such datum, we introduce a guarded dynamic language for histories and posterior states and prove a global reduction theorem. We next consider effectively semialgebraic data over an effectively presented real closed field. For data in this class, we translate the fragment without dynamic operators into first-order formulas in the corresponding language of ordered rings, thereby reducing validity to first-order reasoning over real closed fields. Together, the reduction and first-order translation yield a sound and complete recursive calculus and a decision procedure for validity. Finally, assuming a symmetric informationally complete (SIC) reference measurement, we show that quantum theory in finite dimensions realizes the framework through SIC coordinates, POVMs, and quantum instruments. We also prove that the corresponding SIC image satisfies the standard qplex geometry conditions, namely the consistency bounds and the lower polar condition, and that under explicit coefficient field hypotheses the resulting quantum datum is effectively semialgebraic.

quant-ph↗

arXiv2026

Component Modalities of Quantum Logic

This paper determines the structural and proof-theoretic consequences of the forcing condition in relational quantum modal logic, under which every modal transition available at a world is also available at every world compatible with it. We prove that modal successor sets are constant on compatibility components, so boxed truth sets belong to the Boolean algebra of unions of these components. The relation holding exactly between worlds in the same component assigns to each stable proposition its greatest lower and least upper approximations by unions of components, and in hard superselection models these are exactly the approximations by central propositions. We adopt local validity for sequents with multiple conclusions to give a semantics for modal excluded middle on frames with several components. A connectedization obtained by adding one point then shows that component frames, equivalence frames satisfying the forcing condition, and connected compatibility frames with universal modal accessibility have the same logic for sequents with one conclusion. Finally, maximal consistent pairs yield a canonical model, and the calculus obtained by adding T, 4, and B is proved sound and complete for the three frame classes. These results characterize the logical scope of the forcing condition and establish a complete proof theory for component modalities.

cs.LO↗

arXiv2026

Decidability of Quantum Modal Logic

The decidability of a logical system refers to the existence of an algorithm that can determine whether any given formula in that system is a theorem. In this paper, Harrop's lemma is used to prove the decidability of quantum modal logic.

cs.LO↗

arXiv2025

Quantum modal logic

A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

cs.LO↗

arXiv2007

Logic of Simultaneity

A logical model of spatiotemporal structures is pictured as a succession of processes in time. One usual way to formalize time structure is to assume the global existence of time points and then collect some of them to form time intervals of processes. Under this set-theoretic approach, the logic that governs the processes acquires a Boolean structure. However, in a real distributed system or a relativistic universe where the message-passing time between different locations is not negligible, the logic has no choice but to accept time interval instead of time point as a primitive concept. From this modeling process of spatiotemporal structures, orthologic, the most simplified version of quantum logic, emerges naturally.

math.LO↗