549 research outputs found
Physical computation and compositionality
Developments in quantum computing and, more in general, non-standard
computing systems, represent a clear indication that the very notion of what a
physical computing device is and does should be recast in a rigorous and sound
framework. Physical computing has opened a whole stream of new research aimed
to understand and control how information is processed by several types of
physical devices. Therefore, classical definitions and entire frameworks need
to be adapted in order to fit a broader notion of what physical computing
systems really are. Recent studies have proposed a formalism that can be used
to carve out a more proper notion of physical computing. In this paper we
present a framework which capture such results in a very natural way via some
basic constructions in Category Theory. Furthermore, we show that, within our
framework, the compositional nature of physical computing systems is naturally
formalized, and that it can be organized in coherent structures by the means of
their relational nature
Dinaturality Meets Genericity: A Game Semantics of Bounded Polymorphism
We study subtyping and parametric polymorphism, with the aim of providing direct and tractable semantic representations of type systems with these expressive features. The liveness order uses the Player-Opponent duality of game semantics to give a simple representation of subtyping: we generalize it to include graphs extracted directly from second-order intuitionistic types, and use the resulting complete lattice to interpret bounded polymorphic types in the style of System F_<:, but with a more tractable subtyping relation.
To extend this to a semantics of terms, we use the type-derived graphs as arenas, on which strategies correspond to dinatural transformations with respect to the canonical coercions ("on the nose" copycats) induced by the liveness ordering. This relationship between the interpretation of generic and subtype polymorphism thus provides the basis of the semantics of our type system
Category Theory for Autonomous Robots: The Marathon 2 Use Case
Model-based systems engineering (MBSE) is a methodology that exploits system
representation during the entire system life-cycle. The use of formal models
has gained momentum in robotics engineering over the past few years. Models
play a crucial role in robot design; they serve as the basis for achieving
holistic properties, such as functional reliability or adaptive resilience, and
facilitate the automated production of modules. We propose the use of formal
conceptualizations beyond the engineering phase, providing accurate models that
can be leveraged at runtime. This paper explores the use of Category Theory, a
mathematical framework for describing abstractions, as a formal language to
produce such robot models. To showcase its practical application, we present a
concrete example based on the Marathon 2 experiment. Here, we illustrate the
potential of formalizing systems -- including their recovery mechanisms --
which allows engineers to design more trustworthy autonomous robots. This, in
turn, enhances their dependability and performance
Bootstrapping extensionality
Intuitionistic type theory is a formal system designed by Per Martin-Loef to be a full-fledged foundation in which to develop constructive mathematics. One particular variant, intensional type theory (ITT), features nice computational properties like decidable type-checking, making it especially suitable for computer implementation. However, as traditionally defined, ITT lacks many vital extensionality principles, such as function extensionality. We would like to extend ITT with the desired extensionality principles while retaining its convenient computational behaviour. To do so, we must first understand the extent of its expressive power, from its strengths to its limitations.
The contents of this thesis are an investigation into intensional type theory, and in particular into its power to express extensional concepts. We begin, in the first part, by developing an extension to the strict setoid model of type theory with a universe of setoids. The model construction is carried out in a minimal intensional type theoretic metatheory, thus providing a way to bootstrap extensionality by ``compiling'' it down to a few building blocks such as inductive families and proof-irrelevance.
In the second part of the thesis we explore inductive-inductive types (ITTs) and their relation to simpler forms of induction in an intensional setting. We develop a general method to reduce a subclass of infinitary IITs to inductive families, via an encoding that can be expressed in ITT without any extensionality besides proof-irrelevance. Our results contribute to further understand IITs and the expressive power of intensional type theory, and can be of practical use when formalizing mathematics in proof assistants that do not natively support induction-induction
Frobenius Structures in Star-Autonomous Categories
It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear
Paranatural Category Theory
We establish and advocate for a novel branch of category theory, centered
around strong dinatural transformations (herein known as "paranatural
transformations"). Paranatural transformations generalize natural
transformations to mixed-variant difunctors, but, unlike other such
generalizations, are composable and exceptionally well-behaved. We define the
category of difunctors and paranatural transformations, prove a novel "diYoneda
Lemma" for this category, and explore some of the category-theoretic
implications.
We also develop three compelling uses for paranatural category theory:
parametric polymorphism, impredicative encodings of (co)inductive types, and
difunctor models of type theory. Paranatural transformations capture the
essence of parametricity, with their "paranaturality condition" coinciding
exactly with the "free theorem" of the corresponding polymorphic type; the
paranatural analogue of the (co)end calculus provides an elegant and general
framework for reasoning about initial algebras, terminal coalgebras,
bisimulations, and representation independence; and "diYoneda reasoning"
facilitates the lifting of Grothendieck universes into difunctor models of type
theory. We develop these topics and propose further avenues of research
Enriched universal algebra
Following the classical approach of Birkhoff, we introduce enriched universal
algebra. Given a suitable base of enrichment , we define a language
to be a collection of -ary function symbols whose arities
are taken among the objects of . The class of -terms is
constructed recursively from the symbols of , the morphisms in
, and by incorporating the monoidal structure of .
Then, -structures and interpretations of terms are defined, leading
to enriched equational theories. In this framework we prove several fundamental
theorems of universal algebra, including the characterization of algebras for
finitary monads on as models of an equational theories, and
several Birkhoff-type theorems
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A Categorical Framework for Program Semantics and Semantic Abstraction
Categorical semantics of type theories are often characterized as
structure-preserving functors. This is because in category theory both the
syntax and the domain of interpretation are uniformly treated as structured
categories, so that we can express interpretations as structure-preserving
functors between them. This mathematical characterization of semantics makes it
convenient to manipulate and to reason about relationships between
interpretations. Motivated by this success of functorial semantics, we address
the question of finding a functorial analogue in abstract interpretation, a
general framework for comparing semantics, so that we can bring similar
benefits of functorial semantics to semantic abstractions used in abstract
interpretation. Major differences concern the notion of interpretation that is
being considered. Indeed, conventional semantics are value-based whereas
abstract interpretation typically deals with more complex properties. In this
paper, we propose a functorial approach to abstract interpretation and study
associated fundamental concepts therein. In our approach, interpretations are
expressed as oplax functors in the category of posets, and abstraction
relations between interpretations are expressed as lax natural transformations
representing concretizations. We present examples of these formal concepts from
monadic semantics of programming languages and discuss soundness.Comment: MFPS 202
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