232 research outputs found
Categorical structures for deduction
We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context.
We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models.
Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality
Interpolation in Linear Logic and Related Systems
We prove that there are continuum-many axiomatic extensions of the full
Lambek calculus with exchange that have the deductive interpolation property.
Further, we extend this result to both classical and intuitionistic linear
logic as well as their multiplicative-additive fragments. None of the logics we
exhibit have the Craig interpolation property, but we show that they all enjoy
a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic
extensions of each of these logics without the deductive interpolation
property
Mathematical Foundations for a Compositional Account of the Bayesian Brain
This dissertation reports some first steps towards a compositional account of
active inference and the Bayesian brain. Specifically, we use the tools of
contemporary applied category theory to supply functorial semantics for
approximate inference. To do so, we define on the `syntactic' side the new
notion of Bayesian lens and show that Bayesian updating composes according to
the compositional lens pattern. Using Bayesian lenses, and inspired by
compositional game theory, we define fibrations of statistical games and
classify various problems of statistical inference as corresponding sections:
the chain rule of the relative entropy is formalized as a strict section, while
maximum likelihood estimation and the free energy give lax sections. In the
process, we introduce a new notion of `copy-composition'.
On the `semantic' side, we present a new formalization of general open
dynamical systems (particularly: deterministic, stochastic, and random; and
discrete- and continuous-time) as certain coalgebras of polynomial functors,
which we show collect into monoidal opindexed categories (or, alternatively,
into algebras for multicategories of generalized polynomial functors). We use
these opindexed categories to define monoidal bicategories of cilia: dynamical
systems which control lenses, and which supply the target for our functorial
semantics. Accordingly, we construct functors which explain the bidirectional
compositional structure of predictive coding neural circuits under the free
energy principle, thereby giving a formal mathematical underpinning to the
bidirectionality observed in the cortex. Along the way, we explain how to
compose rate-coded neural circuits using an algebra for a multicategory of
linear circuit diagrams, showing subsequently that this is subsumed by lenses
and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment
of statistical games. A number of errors also fixed, and some presentation
improved. Comments most welcom
A Complete V-Equational System for Graded lambda-Calculus
Modern programming frequently requires generalised notions of program
equivalence based on a metric or a similar structure. Previous work addressed
this challenge by introducing the notion of a V-equation, i.e. an equation
labelled by an element of a quantale V, which covers inter alia (ultra-)metric,
classical, and fuzzy (in)equations. It also introduced a V-equational system
for the linear variant of lambda-calculus where any given resource must be used
exactly once.
In this paper we drop the (often too strict) linearity constraint by adding
graded modal types which allow multiple uses of a resource in a controlled
manner. We show that such a control, whilst providing more expressivity to the
programmer, also interacts more richly with V-equations than the linear or
Cartesian cases. Our main result is the introduction of a sound and complete
V-equational system for a lambda-calculus with graded modal types interpreted
by what we call a Lipschitz exponential comonad. We also show how to build such
comonads canonically via a universal construction, and use our results to
derive graded metric equational systems (and corresponding models) for programs
with timed and probabilistic behaviour
Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences
Mathematical fuzzy logic (MFL) specifically targets many-valued logic and has significantly contributed to the logical foundations of fuzzy set theory (FST). It explores the computational and philosophical rationale behind the uncertainty due to imprecision in the backdrop of traditional mathematical logic. Since uncertainty is present in almost every real-world application, it is essential to develop novel approaches and tools for efficient processing. This book is the collection of the publications in the Special Issue “Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences”, which aims to cover theoretical and practical aspects of MFL and FST. Specifically, this book addresses several problems, such as:- Industrial optimization problems- Multi-criteria decision-making- Financial forecasting problems- Image processing- Educational data mining- Explainable artificial intelligence, etc
Twisted Cubes and their Applications in Type Theory
This thesis captures the ongoing development of twisted cubes, which is a
modification of cubes (in a topological sense) where its homotopy type theory
does not require paths or higher paths to be invertible. My original motivation
to develop the twisted cubes was to resolve the incompatibility between cubical
type theory and directed type theory. The development of twisted cubes is still
in the early stages and the intermediate goal, for now, is to define a twisted
cube category and its twisted cubical sets that can be used to construct a
potential definition of (infinity, n)-categories. The intermediate goal above
leads me to discover a novel framework that uses graph theory to transform
convex polytopes, such as simplices and (standard) cubes, into base categories.
Intuitively, an n-dimensional polytope is transformed into a directed graph
consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of
the polytope as its edges. Then, we define the base category as the full
subcategory of the graph category induced by the family of these graphs from
all n-dimensional cases. With this framework, the modification from cubes to
twisted cubes can formally be done by reversing some edges of cube graphs.
Equivalently, the twisted n-cube graph is the result of a certain endofunctor
being applied n times to the singleton graph; this endofunctor (called twisted
prism functor) duplicates the input, reverses all edges in the first copy, and
then pairwisely links nodes from the first copy to the second copy. The core
feature of a twisted graph is its unique Hamiltonian path, which is useful to
prove many properties of twisted cubes. In particular, the reflexive transitive
closure of a twisted graph is isomorphic to the simplex graph counterpart,
which remarkably suggests that twisted cubes not only relate to (standard)
cubes but also simplices.Comment: PhD thesis (accepted at the University of Nottingham), 162 page
Bifibrations of polycategories and classical multiplicative linear logic
In this thesis, we develop the theory of bifibrations of polycategories.
We start by studying how to express certain categorical structures as
universal properties by generalising the shape of morphism. We call this
phenomenon representability and look at different variations, namely the
correspondence between representable multicategories and monoidal categories,
birepresentable polycategories and -autonomous categories, and
representable virtual double categories and double categories.
We then move to introduce (bi)fibrations for these structures. We show that
it generalises representability in the sense that these structures are
(bi)representable when they are (bi)fibred over the terminal one. We show how
to use this theory to lift models of logic to more refined ones. In particular,
we illustrate it by lifting the compact closed structure of the category of
finite dimensional vector spaces and linear maps to the (non-compact)
-autonomous structure of the category of finite dimensional Banach spaces
and contractive maps by passing to their respective polycategories. We also
give an operational reading of this example, where polylinear maps correspond
to operations between systems that can act on their inputs and whose outputs
can be measured/probed and where norms correspond to properties of the systems
that are preserved by the operations.
Finally, we recall the B\'enabou-Grothendieck correspondence linking
fibrations to indexed categories. We show how the B-G construction can be
defined as a pullback of virtual double categories and we make use of
fibrational properties of vdcs to get properties of this pullback. Then we
provide a polycategorical version of the B-G correspondence.Comment: 250 pages, 15 figures, PhD thesis in the Theory Group at the Computer
Science School of the University of Birmingham under the supervision of Noam
Zeilberger and Paul Lev
Cartesian institutions with evidence: Data and system modelling with diagrammatic constraints and generalized sketches
Data constraints are fundamental for practical data modelling, and a
verifiable conformance of a data instance to a safety-critical constraint
(satisfaction relation) is a corner-stone of safety assurance. Diagrammatic
constraints are important as both a theoretical concepts and a practically
convenient device. The paper shows that basic formal constraint management can
well be developed within a finitely complete category (hence the reference to
Cartesianity in the title). In the data modelling context, objects of such a
category can be thought of as graphs, while their morphisms play two roles: of
data instances and (when being additionally labelled) of constraints.
Specifically, a generalized sketch consists of a graph and a set of
constraints declared over , and appears as a pattern for typical
data schemas (in databases, XML, and UML class diagrams). Interoperability of
data modelling frameworks (and tools based on them) very much depends on the
laws regulating the transformation of satisfaction relations between data
instances and schemas when the schema graph changes: then constraints are
translated co- whereas instances contra-variantly. Investigation of this
transformation pattern is the main mathematical subject of the paperComment: 35 pages. The paper will be presented at the conference on Applied
Category Theory, ACT'2
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