Categories for Dynamic Epistemic Logic

The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspective -- which emphasizes a family of objects and structural relationships among them, as opposed to a single object and structure on it. Also, the categorical semantics of DEL can be merged straightforwardly with a standard categorical semantics for first-order logic, providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Stochastic Relational Presheaves and Dynamic Logic for Contextuality

Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.Comment: In Proceedings QPL 2014, arXiv:1412.810

Topos Semantics for Higher-Order Modal Logic

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos $\mathcal{E}$. In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier $\Omega_{\mathcal{E}}$, but rather by a suitable complete Heyting algebra $H$. The canonical map relating $H$ and $\Omega_{\mathcal{E}}$ both serves to interpret equality and provides a modal operator on $H$ in the form of a comonad. Examples of such structures arise from surjective geometric morphisms $f : \mathcal{F} \to \mathcal{E}$, where $H = f_\ast \Omega_{\mathcal{F}}$. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion

Generalized Topological Semantics for First-Order Modal Logic

This dissertation provides a new semantics for first-order modal logic. It is philosophicallymotivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities.(i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke's relational notion of accessibility. It requires instead a more general, topological notion of accessibility.(ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke's semantics for quantified modal logic is inadequate; its logic is free logic as opposed to classical logic.(iii) More importantly, Kripke's semantics comes with a restriction that is too strong to let us semantically express, for instance, that the identity of Hesperus and Phosphorus, even if metaphysically necessary, can still be a matter of epistemic discovery.To provide a semantics that accommodates the three desiderata, I show, on the one hand, howthe desideratum (i) can be achieved with topological semantics, and more generally neighborhood semantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out that David Lewis's counterpart theory is helpful at least technically. Even though Lewis's ownformulation is too liberal---in contrast to Kripke's being too restrictive---to achieve our goals, this dissertation provides a unification of the two frameworks, Kripke's and Lewis's. Through a series of both formal and conceptual comparisons of their ontologies and semantic ideas, it is shown that structures called sheaves are needed to unify the ideas and achieve the desiderata (ii) and (iii). In the end, I define a category of sheaves over a neighborhood frame with certain properties, and show that it provides a semantics that naturally unifies neighborhood semantics for propositional modal logic, on the one hand, and semantics for first-order logic on the other. Completeness theorems are proved

A Biset-Enriched Categorical Model for Proto-Quipper with Dynamic Lifting

Quipper and Proto-Quipper are a family of quantum programming languages that, by their nature as circuit description languages, involve two runtimes: one at which the program generates a circuit and one at which the circuit is executed, normally with probabilistic results due to measurements. Accordingly, the language distinguishes two kinds of data: parameters, which are known at circuit generation time, and states, which are known at circuit execution time. Sometimes, it is desirable for the results of measurements to control the generation of the next part of the circuit. Therefore, the language needs to turn states, such as measurement outcomes, into parameters, an operation we call dynamic lifting. The goal of this paper is to model this interaction between the runtimes by providing a general categorical structure enriched in what we call "bisets". We demonstrate that the biset-enriched structure achieves a proper semantics of the two runtimes and their interaction, by showing that it models a variant of Proto-Quipper with dynamic lifting. The present paper deals with the concrete categorical semantics of this language, whereas a companion paper deals with the syntax, type system, operational semantics, and abstract categorical semantics.Comment: In Proceedings QPL 2022, arXiv:2311.0837

A tutorial introduction to quantum circuit programming in dependently typed Proto-Quipper

We introduce dependently typed Proto-Quipper, or Proto-Quipper-D for short, an experimental quantum circuit programming language with linear dependent types. We give several examples to illustrate how linear dependent types can help in the construction of correct quantum circuits. Specifically, we show how dependent types enable programming families of circuits, and how dependent types solve the problem of type-safe uncomputation of garbage qubits. We also discuss other language features along the way.Comment: Added a section on related work and a paragraph explaining qubit initialization and terminatio

Contextuality, Cohomology and Paradox

Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions

外国人散在地域住民の外国人受け入れをめぐる意識の考察－石川県白山市によるアンケート調査の報告－

Although Hakusan City in Ishikawa Prefecture is a region with little immigration experience, the acceptance of immigrant workers has rapidly increased, and multicultural symbiosis of community development is promoted. This study analyzes the results of a questionnaire survey of the chairmans of neighborhood associations regarding the acceptance of residents toward immigrants. The results of the questionnaire indicated that the respondents in areas without immigrants have expectations and concerns about the possible increase in the number of immigrants; contrastingly, the respondents in the areas with immigrants did not feel that the increase in the number of immigrants influenced their lives. In this background, it was hypothesized that management were provided to the technical interns by their employers. This contributes to reducing opportunities for contact between the residents and immigrant interns. Simultaneously, it ensures the smooth functioning of society and fostered its frosty acceptance of immigrants. However, analysis of the “free description” column in the questionnaire detailed the level of people’s awareness. Some of the residents expressed a desire for immigrants to follow Japanese customs; other residents expressed concern about lumping diverse immigrants and commented on the need for change and understanding in Japanese residents. This indicates that the awareness of residents leads to multicultural coexistence and exchange