Related papers: Types, equations, dimensions and the Pi theorem
Matrix syntax is a formal model of syntactic relations in language. The purpose of this paper is to explain its mathematical foundations, for an audience with some formal background. We make an axiomatic presentation, motivating each axiom…
A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound \Delta (a parameter of the theory) is unrestricted, the resulting dimension is precisely the…
In previous work we gave a mathematical foundation, referred to as DisCoCat, for how words interact in a sentence in order to produce the meaning of that sentence. To do so, we exploited the perfect structural match of grammar and…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
In spite of its fundamental importance, inference has not been an inherent function of multidimensional models and analytical applications. These models are mainly aimed at numeric (quantitative) analysis where the notions of inference and…
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the…
In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own…
The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in probability distributions and relational databases, using the…
In this short note we present several infinite dimensional theorems which generalize corresponding facts from the finite dimensional differential inclusions theory.
We present a novel approach to construction of a formal semantics for a programming language. Our approach, using a parametric denotational semantics, allows the semantics to be easily extended to support new language features, and…
Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical…
This document is an introduction to and review of two-dimensional mathematical physics. The reader is introduced to the subject matter primarily through problems, which are presented along with detailed worked solutions. For each chapter,…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…
There is no known way of giving a domain-theoretic semantics to higher-order probabilistic languages, in such a way that the involved domains are continuous or quasi-continuous - the latter is required to do any serious mathematics. We…
Session types capture precise protocol structure in concurrent programming, but do not specify properties of the exchanged values beyond their basic type. Refinement types are a form of dependent types that can address this limitation,…
The goal of the presented paper is to provide an introduction to the basic computational models used in quantum information theory. We review various models of quantum Turing machine, quantum circuits and quantum random access machine…
The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these…
Formalizations of quantum information theory in category theory and type theory, for the design of verifiable quantum programming languages, need to express its two fundamental characteristics: (1) parameterized linearity and (2) metricity.…
Gradual typing is an approach to integrating static and dynamic typing within the same language, and puts the programmer in control of which regions of code are type checked at compile-time and which are type checked at run-time. In this…
We discuss several ways of illustrating fundamental concepts in statistical and thermal physics by considering various models and algorithms. We emphasize the importance of replacing students' incomplete mental images by models that are…