The Transpension Type: Technical Report

The purpose of these notes is to give a categorical semantics for the transpension type (Nuyts and Devriese, Transpension: The Right Adjoint to the Pi-type, pre-print, 2020), which is right adjoint to a potentially substructural dependent function type. In section 2 we discuss some prerequisites. In section 3, we define multipliers and discuss their properties. In section 4, we study how multipliers lift from base categories to presheaf categories. In section 5, we explain how typical presheaf modalities can be used in the presence of the transpension type. In section 6, we study commutation properties of prior modalities, substitution modalities and multiplier modalities

Transpension: The Right Adjoint to the Pi-type

Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models, either to make the resulting language more expressive, or in order to carry out further reasoning internally, allowing greater abstraction and sometimes automated verification. While the constructions of presheaf models largely follow a common pattern, approaches towards internalization do not. Throughout the literature, various internal presheaf operators ($\surd$, $\Phi/\mathsf{extent}$, $\Psi/\mathsf{Gel}$, $\mathsf{Glue}$, $\mathsf{Weld}$, $\mathsf{mill}$, the strictness axiom and locally fresh names) can be found and little is known about their relative expressivenes. Moreover, some of these require that variables whose type is a shape (representable presheaf, e.g. an interval) be used affinely. We propose a novel type former, the transpension type, which is right adjoint to universal quantification over a shape. Its structure resembles a dependent version of the suspension type in HoTT. We give general typing rules and a presheaf semantics in terms of base category functors dubbed multipliers. Structural rules for shape variables and certain aspects of the transpension type depend on characteristics of the multiplier. We demonstrate how the transpension type and the strictness axiom can be combined to implement all and improve some of the aforementioned internalization operators (without formal claim in the case of locally fresh names)

Abstract Congruence Criteria for Weak Bisimilarity

We introduce three general compositionality criteria over operational semantics and prove that, when all three are satisfied together, they guarantee weak bisimulation being a congruence. Our work is founded upon Turi and Plotkin's mathematical operational semantics and the coalgebraic approach to weak bisimulation by Brengos. We demonstrate each criterion with various examples of success and failure and establish a formal connection with the simply WB cool rule format of Bloom and van Glabbeek. In addition, we show that the three criteria induce lax models in the sense of Bonchi et al

Discovering Argumentative Patterns in Energy Polylogues: A Macroscope for Argument Mining

A macroscope is proposed and tested here for the discovery of the unique argumentative footprint that characterizes how a collective (e.g., group, online community) manages differences and pursues disagreement through argument in a polylogue. The macroscope addresses broader analytic problems posed by various conceptualizations of large-scale argument, such as fields, spheres, communities, and institutions. The design incorporates a two-tier methodology for detecting argument patterns of the arguments performed in arguing by an interactive collective that produces views, or topographies, of the ways that issues are generated in the making and defending of standpoints. The design premises for the macroscope build on insights about argument patterns from pragma-dialectical theory by incorporating research and theory on disagreement management and the Argumentum Model of Topics. The design reconceptualizes prototypical and stereotypical argument patterns for characterizing large-scale argumentation. A prototype of the macroscope is tested on data drawn from six threads about oil-drilling and fracking from the subreddit Changemyview. The implementation suggests the efficacy of the macroscope’s design and potential for identifying what communities make controversial and how the disagreement space in a polylogue is managed through stereotypical argument patterns in terms of claims/premises, inferential relations, and presentational devices

Robust Notions of Contextual Fibrancy

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Contributions to Multimode and Presheaf Type Theory

Dependent type theory is a powerful logic for both secure programming and computer assisted proving about programs. Dependently typed languages such as Agda, Coq and Idris can therefore be used both as programming languages and as proof assistants. The goal of this PhD project is to establish directed dependent type theories (DDTT) by formulating them, implementing them, proving their consistency and demonstrating their use. By a DDTT, we mean a dependent type theory which has not only a built-in notion of equality, but also of structure preserving transformation. By including the notion of structure preservation in the foundations of the system, rather than defining it on an ad-hoc basis as is done in classical mathematics, we expect to obtain many theorems (including all parametricity theorems) and even some operations for free. For example, when we define a functorial type former, such as List, we expect its functorial action (fmap) to be implemented automatically. This is especially useful when reasoning about or programming with more complicated concepts, such as monad transformations, which implement the effects of one monad in terms of another, or transformations between mathematical structures. DDTT will also give us a better understanding of subtyping in the context of dependent types. This is of particular interest for the programming language Scala, which has subtyping at its core and was recently extended with some basic support for dependent types.status: publishe

Dependable Atomicity in Type Theory

status: publishe