ACTUALIZATION OF THE CATEGORICAL SEMANTICS OF SYNCHRONOUS SIMULTANEITY

Категориальная семантика синхронной одновременности актуализируется в немецких высказываниях с таксисными предлогами темпоральной и обстоятельственной семантики während, in, bei, mit, infolge, trotz, vor, wegen, aus, für, zu и др. Актуализация примарно-таксисных и секундарно-таксисных категориальных ситуаций синхронной одновременности наблюдается при соотнесении между собой и полном (синхронном) совпадении в рамках целостного временного периода процессов (состояний, действий, событий) девербативов и глаголов. The categorical semantics of synchronous simultaneity is actualized in German statements with taxis prepositions of temporal and adverbial semantics während, in, bei, mit, infolge, trotz, vor, wegen, aus, für, zu, etc. Actualization of primary-taxis and secondary-taxis categorical situations of synchronous simultaneity is observed when correlating with each other and complete (synchronous) coincidence within an integral time period of processes (states, actions, events) of deverbatives and verbs.

Dagger linear logic for categorical quantum mechanics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

The Non-Prototypical Taxis and Interlanguage Variability

The purpose of this article is to consider the issue of actualizing of the categorical semantics of the non-prototypical taxis of simultaneity in German, Russian, Dutch, English and Polish. The study was conducted on the material of different structural languages within the framework of comparative typological and functional grammatical approaches to the study of taxis meanings, taking into account the phenomenon of interlanguage variability. The non-prototypic taxis of simultaneity is the secondary taxis of simultaneity, which is actualized in statements with taxis prepositions of adverbial semantics (conditional, causal, concessive, consecutive, final, modal, instrumental, medial) in German, Russian, Dutch, English and Polish. Taxis prepositions are distinguished by their interlanguage variability and polyfunctionality. They mark the semantics of the secondary taxis of simultaneity, in particular the causal, conditional, modal, concessive, consecutive, final and conditional taxis in German, Russian, English, Polish, Dutch, as well as the categorical semantics of the instrumental/medial taxis in Germanic languages: German, English and Dutch. Within the framework of this study, we distinguish the following variants of the non-prototypical taxis of simultaneity: modal taxis (secondary taxis of manner), conditional taxis, causal taxis, consequential taxis, concessive taxis, final taxis in German, English, Dutch, Russian and Polish, as well as instrumental and medial taxis in German, English and Dutch. The examined statements represent such secondary taxis categorical situations of simultaneity as: secondary taxis categorical situations of the manner, conditional-taxis, causal-taxis, concessive-taxis, consecutive-taxis, final-taxis (in German, Russian, English, Polish, Dutch), as well as instrumental-taxis and medial-taxis categorical situations of simultaneity (in German, English and Dutch).

Higher-order probabilistic adversarial computations: categorical semantics and program logics

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

Categorical Semantics of Cyber-Physical Systems Theory

Cyber-physical systems require the construction and management of various models to assure their correct, safe, and secure operation. These various models are necessary because of the coupled physical and computational dynamics present in cyber-physical systems. However, to date the different model views of cyber-physical systems are largely related informally, which raises issues with the degree of formal consistency between those various models of requirements, system behavior, and system architecture. We present a category-theoretic framework to make different types of composition explicit in the modeling and analysis of cyber-physical systems, which could assist in verifying the system as a whole. This compositional framework for cyber-physical systems gives rise to unified system models, where system behavior is hierarchically decomposed and related to a system architecture using the systems-as-algebras paradigm. As part of this paradigm, we show that an algebra of (safety) contracts generalizes over the state of the art, providing more uniform mathematical tools for constraining the behavior over a richer set of composite cyber-physical system models, which has the potential of minimizing or eliminating hazardous behavior.

Curry–Howard–Lambek Correspondence for Intuitionistic Belief

This paper introduces a natural deduction calculus for intuitionistic logic of belief$$\mathsf {IEL}^{-}$$ IEL - which is easily turned into a modal$$\lambda $$ λ -calculus giving a computational semantics for deductions in $$\mathsf {IEL}^{-}$$ IEL - . By using that interpretation, it is also proved that $$\mathsf {IEL}^{-}$$ IEL - has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in $$\mathsf {IEL}^{-}$$ IEL - showing the general structure of such a modality for belief in an intuitionistic framework.