Law and Order in Algorithmics

An algorithm is the input-output effect of a computer program; mathematically, the notion of algorithm comes close to the notion of function. Just as arithmetic is the theory and practice of calculating with numbers, so is ALGORITHMICS the theory and practice of calculating with algorithms. Just as a law in arithmetic is an equation between numbers, like a(b+c) = ab + ac, so is a LAW in algorithmics an equation between algorithms. The goal of the research done is: (extending algorithmics by) the systematic detection and use of laws for algorithms. To this end category theory (a branch of mathematics) is used to formalise the notion of algorithm, and to formally prove theorems and laws about algorithms.\ud \ud The underlying motivation for the research is the conviction that algorithmics may be of help in the construction of computer programs, just as arithmetic is of help in solving numeric problems. In particular, algorithmics provides the means to derive computer programs by calculation, from a given specification of the input-output effect.\ud \ud In Chapter 2 the systematic detection and use of laws is applied to category theory itself. The result is a way to conduct and present proofs in category theory, that is an alternative to the conventional way (diagram chasing).\ud \ud In Chapter 3--4 several laws are formally derived in a systematic fashion. These laws facilitate to calculate with those algorithms that are defined by induction on their input, or on their output. Technically, initial algebras and terminal co-algebras play an crucial role here.\ud \ud In Chapter 5 a category theoretic formalisation of the notion of law itself is derived and investigated. This result provides a tool to formulate and prove theorems about laws-in-general, and, more specifically, about equationally specified datatypes.\ud \ud Finally, in Chapter 6 laws are derived for arbitrary recursive algorithms. Here the notion of ORDER plays a crucial role. The results are relevant for current functional programming languages

Aspects of predicative algebraic set theory II:

Abstract This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative category with small maps" which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive set theory this leads to, as well as ideas for future work

On the ∞\infty-topos semantics of homotopy type theory

Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course delivered at the Logic and Higher Structures workshop at CIRM-Luminy, attempt to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any $\infty$-topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.Comment: These lecture notes were written to accompany a 4.5 hour mini-course delivered at the workshop Logique et structures sup\'erieures held at CIRM - Luminy from 21-25 February 2022. Video is available at https://www.carmin.tv/en/collections/logic-and-higher-structures-logique-et-structures-superieure

Combinatorial Species and Labelled Structures

The theory of combinatorial species was developed in the 1980s as part of the mathematical subfield of enumerative combinatorics, unifying and putting on a firmer theoretical basis a collection of techniques centered around generating functions. The theory of algebraic data types was developed, around the same time, in functional programming languages such as Hope and Miranda, and is still used today in languages such as Haskell, the ML family, and Scala. Despite their disparate origins, the two theories have striking similarities. In particular, both constitute algebraic frameworks in which to construct structures of interest. Though the similarity has not gone unnoticed, a link between combinatorial species and algebraic data types has never been systematically explored. This dissertation lays the theoretical groundwork for a precise—and, hopefully, useful—bridge bewteen the two theories. One of the key contributions is to port the theory of species from a classical, untyped set theory to a constructive type theory. This porting process is nontrivial, and involves fundamental issues related to equality and finiteness; the recently developed homotopy type theory is put to good use formalizing these issues in a satisfactory way. In conjunction with this port, species as general functor categories are considered, systematically analyzing the categorical properties necessary to define each standard species operation. Another key contribution is to clarify the role of species as labelled shapes, not containing any data, and to use the theory of analytic functors to model labelled data structures, which have both labelled shapes and data associated to the labels. Finally, some novel species variants are considered, which may prove to be of use in explicitly modelling the memory layout used to store labelled data structures

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Categorical Term Rewriting: Monads and Modularity

Laboratory for Foundations of Computer ScienceTerm rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, that is, if the components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntax-oriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (non-collapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results

Classification and homological invariants of compact quantum groups of combinatorial type

Compact quantum groups can be found by solving certain combinatorics problems, as first shown by Banica and Speicher. Any system of partitions of finite sets which is closed under reflection and two kinds of concatenation gives rise to a quantum subgroup of the free orthogonal quantum group. Later Freslon, Tarrago and Weber extended this construction to colored partitions. Only recently, Mančinska and Roberson generalized this from finite sets to finite graphs. The present thesis contributes to the classification programs for quantum groups induced by two-colored partitions in Chapter 1 and those induced by uncolored graphs in Chapter 2. While these constructions produce numerous quantum groups, little is known about which of those are actually new and not isomorphic to others. In an effort to elucidate this, Chapter 3 shows that any such quantum group interpolating the unitary group and the free unitary quantum group can be written as a quotient of a wreath graph product of one of the two. Another way of making distinctions between such quantum groups of combinatorial type is to study quantum group invariants, such as cohomology. Chapter 4 computes the first order with trivial coefficients for the discrete duals of all of Tarrago and Weber’s quantum groups. For a handful of those Chapter 5 computes the L²-Betti numbers following Bichon, Kyed and Raum’s method. Chapter 6 proposes a common categorial framework covering all the aforementioned constructions for the first time.Durch das Lösen gewisser Kombinatorikrätsel lassen sich kompakte Quantengruppen finden, wie von Banica und Speicher gezeigt. Jede Sammlung von Partitionen endlicher Mengen, die unter Spiegelung und zwei Arten Konkatenierung abgeschlossen ist, ergibt eine Unterquantengruppe der freien orthogonalen Quantengruppe. Freslon, Tarrago und Weber erweiterten dies auf “gefärbte Partionen”. Erst kürzlich ersetzten Mancinska und Roberson die endlichen Mengen durch endliche Graphen. Die Dissertation trägt zu zwei entsprechenden Klassifikationsvorhaben bei: zweifarbige Partitionen in Kapitel 1, ungefärbte Graphen in Kapitel 2. Zwar ergeben sich viele Quantengruppen. Doch ist nur wenig darüber bekannt, welche davon tatsächlich neu sind. Um dieser Frage nachzugehen, wird in Kapitel 3 bewiesen, dass jede solche Quantengruppe zwischen der unitären Gruppe und der freien unitären Quantengruppe Quotient eines Kranzgraphprodukts einer dieser beiden ist. Eine andere Möglichkeit, solche Quantengruppen kombinatorischen Typs von einander zu unterscheiden bieten Invarianten wie Kohomologie. Von letzterer, mit trivialen Koeffizienten, wird in Kapitel 4 die erste Ordnung berechnet, und zwar für die diskreten Dualen aller von Tarrago und Webers Quantengruppen. Für eine handvoll davon werden in Kapitel 5 noch nach der Methode von Bichon, Kyed und Raum die L2-Betti-Zahlen bestimmt. Kapitel 6 enthält den Vorschlag eines gemeinsamen Rahmens für erstmals alle zuvor genannten Konstruktionen von Quantengruppen.Deutsche Forschungsgemeinschaft, SFB-TRR 195, IRTG scholarshi