Continuous Functions on Final Comodels of Free Algebraic Theories

In 2009, Ghani, Hancock and Pattinson gave a tree-like representation of stream processors $A^{\mathbb{N}} \rightarrow B^{\mathbb{N}}$. In 2021, Garner showed that this representation can be established in terms of algebraic theory and comodels: the set of infinite streams $A^{\mathbb{N}}$ is the final comodel of the algebraic theory of $A$-valued input $\mathbb{T}_A$ and the set of stream processors $\mathit{Top}(A^{\mathbb{N}},B^{\mathbb{N}})$ can be seen as the final $\mathbb{T}_A$-$\mathbb{T}_B$-bimodel. In this paper, we generalize Garner's results to the case of free algebraic theories.Comment: 17 page

Continuous Functions on Final Comodels of Free Algebraic Theories

In 2009, Ghani, Hancock and Pattinson gave a tree-like representation of stream processors A[N] → B[N]. In 2021, Garner showed that this representation can be established in terms of algebraic theory and comodels: the set of infinite streams A[N] is the final comodel of the algebraic theory of A-valued input [T][A] and the set of stream processors Top(A[N] , B[N]) can be seen as the final [T][A]-[T][B]-bimodel. In this paper, we generalize Garner's results to the case of free algebraic theories

Cartesian closed bicategories: type theory and coherence

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. However, calculations in such bicategories quickly fall into a quagmire of coherence data. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call local coherence. I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called biclones. The result brings together two extensions of the simply-typed lambda calculus: a 2-dimensional type theory in the style of Hilken, which encodes the 2-dimensional nature of a bicategory, and a version of explicit substitution, which encodes a composition operation that is only associative and unital up to isomorphism. For products and exponentials I develop the theory of cartesian and cartesian closed biclones and pursue a connection with the representable multicategories of Hermida. Unlike preceding 2-categorical type theories, in which products and exponentials are encoded by postulating a unit and counit satisfying the triangle laws, the universal properties for products and exponentials are encoded using T. Fiore's biuniversal arrows. Because the type theory is extracted from the construction of a free biclone, its syntactic model satisfies a suitable 2-dimensional freeness universal property generalising the classical Curry--Howard--Lambek correspondence. One may therefore describe the type theory as an `internal language'. The relationship with the classical situation is made precise by a result establishing that the type theory I construct is the simply-typed lambda calculus up to isomorphism. This relationship is exploited for the proof of local coherence. It is has been known for some time that one may use the normalisation-by-evaluation strategy to prove the simply-typed lambda calculus is strongly normalising. Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed. Once these prerequisites have been met, the argument is remarkably similar to that in the categorical setting

Enriched Lawvere Theories for Operational Semantics

Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus.Comment: In Proceedings ACT 2019, arXiv:2009.0633

Language concepts and design patterns

Programming languages aim at the construction of simple but expressive programs. To achieve this, plenty of language concepts have arisen over time. Design patterns aim at the solution of common design problems. To achieve this, plenty of approved design concepts have been collected. We claim that language concepts and design patterns are essentially the same. Indeed, a language may offer a design pattern as a language concept; we call such patterns "language patterns". A design pattern can be implemented in terms of other design or language patterns. Since a concrete programming language only supports a subset of language patterns, every other pattern must be expressed in terms of this subset. We call such an implementation a "workaround". The specification of a workaround imposes proof obligations: it must be shown that a workaround simulates the pattern. Once proved correct, we can collect patterns and their workarounds in a trustworthy catalogue. This helps software developers to correctly apply patterns in any language and helps the language designer to decide which patterns to put into the language core. We demonstrate this pattern integration process with well-known design patterns and concepts of object-oriented languages. Additionally, we list important language patterns together with their workarounds

Profunctors, Open Maps and Bisimulation

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ``lifting'', and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation. But the results are likely to have more general applicability because of the ubiquitous nature of profunctors

Categorical combinators

Our main aim is to present the connection between λ-calculus and Cartesian closed categories both in an untyped and purely syntactic setting. More specifically we establish a syntactic equivalence theorem between what we call categorical combinatory logic and λ-calculus with explicit products and projections, with β and η-rules as well as with surjective pairing. “Combinatory logic” is of course inspired by Curry's combinatory logic, based on the well-known S, K, I. Our combinatory logic is “categorical” because its combinators and rules are obtained by extracting untyped information from Cartesian closed categories (looking at arrows only, thus forgetting about objects). Compiling λ-calculus into these combinators happens to be natural and provokes only n log n code expansion. Moreover categorical combinatory logic is entirely faithful to β-reduction where combinatory logic needs additional rather complex and unnatural axioms to be. The connection easily extends to the corresponding typed calculi, where typed categorical combinatory logic is a free Cartesian closed category where the notion of terminal object is replaced by the explicit manipulation of applying (a function to its argument) and coupling (arguments to build datas in products). Our syntactic equivalences induce equivalences at the model level. The paper is intended as a mathematical foundation for developing implementations of functional programming languages based on a “categorical abstract machine,” as developed in a companion paper (Cousineau, Curien, and Mauny, in “Proceedings, ACM Conf. on Functional Programming Languages and Computer Architecture,” Nancy, 1985)

Category-Theoretic Quantitative Compositional Distributional Models of Natural Language Semantics

This thesis is about the problem of compositionality in distributional semantics. Distributional semantics presupposes that the meanings of words are a function of their occurrences in textual contexts. It models words as distributions over these contexts and represents them as vectors in high dimensional spaces. The problem of compositionality for such models concerns itself with how to produce representations for larger units of text by composing the representations of smaller units of text. This thesis focuses on a particular approach to this compositionality problem, namely using the categorical framework developed by Coecke, Sadrzadeh, and Clark, which combines syntactic analysis formalisms with distributional semantic representations of meaning to produce syntactically motivated composition operations. This thesis shows how this approach can be theoretically extended and practically implemented to produce concrete compositional distributional models of natural language semantics. It furthermore demonstrates that such models can perform on par with, or better than, other competing approaches in the field of natural language processing. There are three principal contributions to computational linguistics in this thesis. The first is to extend the DisCoCat framework on the syntactic front and semantic front, incorporating a number of syntactic analysis formalisms and providing learning procedures allowing for the generation of concrete compositional distributional models. The second contribution is to evaluate the models developed from the procedures presented here, showing that they outperform other compositional distributional models present in the literature. The third contribution is to show how using category theory to solve linguistic problems forms a sound basis for research, illustrated by examples of work on this topic, that also suggest directions for future research.Comment: DPhil Thesis, University of Oxford, Submitted and accepted in 201