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In typed functional programming there is often the need for combining pure and monadic (or other effectful) computations, but the required lifting must be done manually by the programmer and may result in cluttered code. This thesis explores ways to allow the compiler to perform this task automat- ically. Several possible approaches are described, where the final one reduces the task to solving a system of linear diophantine equations. Apart from monads, the described method is also considered for the case of applicative functors as another abstraction to represent effectful operations. 1

This thesis considers several instances of abstraction that arose in the design and implementation of the web programming language Links. The first concerns user interfaces, specified using HTML forms. We wish to construct forms from existing form fragments without introducing dependencies on the implementation details of those fragments. Surprisingly, many existing web systems do not support this simple scenario. We present a library which captures the essence of form abstraction, and extend it with more practical facilities, such as validation of the HTML a program produces and of the input a user submits. An important part of our library is a simple semantics, given as the composition of three primitive “idioms”, an interface to computation introduced by McBride and Paterson. In order to justify this approach we present a comparison of idioms with the related notions of monads and arrows, refining the informal claims in the literature. Our library forms part of the Links framework for stateless web interactions. We describe a related aspect of this system, a preprocessor that derives generic instances of functions, which we use to serialise server state between client requests. The abstraction in this case involves the shape of datatypes: the serialisation operation is specified independently of the particular types involved. Our final instance of abstraction involves abstract types. Functional programming languages typically offer one of two styles of abstract type: the abstraction boundary may be drawn using a private data constructor, or using a type signature. We show that there is a pair of semantics-preserving translations between these two styles. In the light of this, we revisit the decision of the Haskell designers to offer the constructor style, and define a library that supports signature-style definitions in Haskell by translation into the constructor style.

Algebraic theories describe mathematical structures that are defined in terms of operations and equations, and are extremely important throughout mathematics. Many generalisations of the classical notion of an algebraic theory have sprung up for use in different mathematical contexts; some examples include Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. The purpose of an algebraic theory is to describe its models, which are structures in which each of the abstract operations of the theory is given a concrete interpretation such that the equations of the theory hold. The process of going from a theory to its models is called semantics, and is encapsulated in a semantics functor. In order to define a model of a theory in a given category, it is necessary to have some structure that relates the arities of the operations in the theory with the objects of the category. This leads to the second central notion of this thesis, that of an interpretation of arities, or aritation for short. We show that any aritation gives rise to a semantics functor from the appropriate category of proto-theories, and that this functor has a left adjoint called the structure functor, giving rise to a structure{semantics adjunction. Furthermore, we show that the usual semantics for many existing notions of algebraic theory arises in this way by choosing an appropriate aritation. Another aim of this thesis is to find a convenient category of monads in the following sense. Every right adjoint into a category gives rise to a monad on that category, and in fact some functors that are not right adjoints do too, namely their codensity monads. This is the structure part of the structure{semantics adjunction for monads. However, the fact that not every functor has a codensity monad means that the structure functor is not defined on the category of all functors into the base category, but only on a full subcategory of it. This deficiency is solved when passing to general proto-theories with a canonical choice of aritation whose structure{semantics adjunction restricts to the usual one for monads. However, this comes at a cost: the semantics functor for general proto-theories is not full and faithful, unlike the one for monads. The condition that a semantics functor be full and faithful can be thought of as a kind of completeness theorem | it says that no information is lost when passing from a theory to its models. It is therefore desirable to retain this property of the semantics of monads if possible. The goal then, is to find a notion of algebraic theory that generalises monads for which the semantics functor is full and faithful with a left adjoint; equivalently the semantics functor should exhibit the category of theories as a re ective subcategory of the category of all functors into the base category. We achieve this (for well-behaved base categories) with a special kind of proto-theory enriched in topological spaces, which we call a complete topological proto-theory. We also pursue an analogy between the theory of proto-theories and that of groups. Under this analogy, monads correspond to finite groups, and complete topological proto-theories correspond to profinite groups. We give several characterisations of complete topological proto-theories in terms of monads, mirroring characterisations of profinite groups in terms of finite groups.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18771号 / 理博第4029号 / 新制||理||1580(附属図書館) / 31722 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 長谷川 真人, 教授 玉川 安騎男, 准教授 照井 一成 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We present a general semantic account of Gifford-style type-and-effect systems. These type systems provide lightweight static analyses annotating program phrases with the sets of possible computational effects they may cause, such as memory access and modification, exception raising, and non-deterministic choice. The analyses are used, for example, to justify the program transformations typically used in optimising compilers, such as code reordering and inlining. Despite their existence for over two decades, there is no prior comprehensive theory of type-and-effect systems accounting for their syntax and semantics, and justifying their use in effect-dependent program transformation. We achieve this generality by recourse to the theory of algebraic effects, a development of Moggi’s monadic theory of computational effects that emphasises the operations causing the effects at hand and their equational theory. The key observation is that annotation effects can be identified with the effect operations. Our first main contribution is the uniform construction of semantic models for typeand- effect analysis by a process we call conservative restriction. Our construction requires an algebraic model of the unannotated programming language and a relevant notion of predicate. It then generates a model for Gifford-style type-and-effect analysis. This uniform construction subsumes existing ad-hoc models for type-and-effect systems, and is applicable in all cases in which the semantics can be given via enriched Lawvere theories. Our second main contribution is a demonstration that our theory accounts for the various aspects of Gifford-style effect systems. We begin with a version of Levy’s Callby- push-value that includes algebraic effects. We add effect annotations, and design a general type-and-effect system for such call-by-push-value variants. The annotated language can be thought of as an intermediate representation used for program optimisation. We relate the unannotated semantics to the conservative restriction semantics, and establish the soundness of program transformations based on this effect analysis. We develop and classify a range of validated transformations, generalising many existing ones and adding some new ones. We also give modularly-checkable sufficient conditions for the validity of these optimisations. In the final part of this thesis, we demonstrate our theory by analysing a simple example language involving global state with multiple regions, exceptions, and nondeterminism. We give decision procedures for the applicability of the various effect-dependent transformations, and establish their soundness and completeness.

Aspect-oriented programming (AOP) aims to enhance modularity and reusability in software systems by offering an abstraction mechanism to deal with crosscutting concerns. But, in most general-purpose aspect languages aspects have almost unrestricted power, eventually conflicting with these goals. This work presents Effective Aspects: a novel approach to embed the pointcut/advice model of AOP in a statically-typed functional programming language like Haskell; along two main contributions. First, we define a monadic embedding of the full pointcut/advicemodel of AOP. Type soundness is guaranteed by exploiting the underlying type system, in particular phantom types and a new anti-unification type class. In this model aspects are first-class, can be deployed dynamically, and the pointcut language is extensible, therefore combining the flexibility of dynamically-typed aspect languages with the guarantees of a static type system. Monads enable us to directly reason about computational effects both in aspects and base programs using traditional monadic techniques. Using this we extend the notion of Open Modules with effects, and also with protected pointcut interfaces to external advising. These restrictions are enforced statically using the type system. Also, we adapt the techniques of EffectiveAdvice to reason about and enforce control flow properties as well as to control effect interference. We show that the parametricity-based approach to effect interference falls short in the presence of multiple aspects and propose a different approach using monad views, a novel technique for handling the monad stack, developed by Schrijvers and Oliveira. Then, we exploit the properties of our model to enable the modular construction of new semantics for aspect scoping and weaving. Our second contribution builds upon a powerful model to reason about mixin-based composition of effectful components and their interference, based on equational reasoning, parametricity, and algebraic laws about monadic effects. Our contribution is to show how to reason about interference in the presence of unrestricted quantification through pointcuts. We show that global reasoning can be compositional, which is key for the scalability of the approach in the face of large and evolving systems. We prove a general equivalence theorem that is based on a few conditions that can be established, reused, and adapted separately as the system evolves. The theorem is defined for an abstract monadic AOP model; we illustrate its use with a simple version of the model just described. This work brings type-based reasoning about effects for the first time in the pointcut/advice model, in a framework that is expressive, extensible and well-suited for development of robust aspect-oriented systems as well as a research tool for new aspect semantics.

La programmation orientée aspect (AOP) vise à améliorer la modularité et la réutilisation des couches logiciels en proposant un mécanisme d’abstraction pour faire face aux préoccupations transversales. Cependant, dans la plupart des langages d’aspects généralistes, les aspects ont un pouvoir presque illimité, rentrant éventuellement en conflit avec ces objectifs. Dans ce travail, nous présentons Effective Aspects : une nouvelle approche pour incorporer le modèle pointcut/advice de l’AOP dans un langage de programmation fonctionnel statiquement typé comme Haskell. Notre travail comprend deux contributions principales. Premièrement, nous définissons un plongement monadique du modèle pointcut/advice complet de l’AOP. La correction du typage est garantie par l’exploitation du système de type sous-jacent, en particulier les types fantômes et une nouvelle classe de type pour faire de l’anti-unification de types. Dans ce modèle, les aspects sont de première classe, peuvent être déployés de façon dynamique, et le langage de pointcuts est extensible, combinant donc la flexibilité des langages d’aspect typés dynamiquement avec les garanties d’un système de type statique. Les monades nous permettent de raisonner directement sur les effets du calcul à la fois dans les aspects et les programmes de base en utilisant des techniques monadiques traditionnelle. Avec ce système, nous étendons la notion de “open modules” avec des effets, et aussi avec les interfaces de pointcut protégés à l’extérieur d’un advice. Ces restrictions sont appliquées statiquement par le système de type. Aussi, nous adaptons les techniques de EffectiveAdvice afin de raisonner sur des propriétés du flot de contrôle. En outre, nous montrons comment contrôler l’interférence des effets en utilisant l’approche fondée sur la paramétricité de EffectiveAdvice. Nous montrons que cette approche n’est pas satisfaisante en présence de multiples aspects et proposons une approche différente en utilisant des vues monadiques, une nouvelle technique pour le traitement de la pile monadique, développée par Schrijvers et Oliveira. Ensuite, nous exploitons les propriétés de notre modèle pour permettre la construction modulaire de nouvelles sémantiques pour la portée d’aspects et le tissage. Notre deuxième contribution s’appuie sur un modèle puissant pour raisonner sur la composition de mixins avec effets et leur interférence, fondée sur un raisonnement équationnelle, paramétrique, et les lois algébriques sur les effets monadiques. Notre contribution est de montrer comment raisonner sur l’interférence en présence de quantification sans restriction pour les pointcuts. Nous montrons que le raisonnement global peut être compositionnelle, ce qui est essentiel pour le passage à l’échelle de l’approche face aux évolutions de grands systèmes. / Aspect-oriented programming (AOP) aims to enhance modularity and reusability in software systems by offering an abstraction mechanism to deal with crosscutting concerns. But, in most general-purpose aspect languages aspects have almost unrestricted power, eventually conflicting with these goals. This work presents Effective Aspects: a novel approach to embed the pointcut/advice model of AOP in a statically-typed functional programming language like Haskell; along two main contributions. First, we define a monadic embedding of the full pointcut/advicemodel of AOP. Type soundness is guaranteed by exploiting the underlying type system, in particular phantom types and a new anti-unification type class. In this model aspects are first-class, can be deployed dynamically, and the pointcut language is extensible, therefore combining the flexibility of dynamically-typed aspect languages with the guarantees of a static type system. Monads enable us to directly reason about computational effects both in aspects and base programs using traditional monadic techniques. Using this we extend the notion of Open Modules with effects, and also with protected pointcut interfaces to external advising. These restrictions are enforced statically using the type system. Also, we adapt the techniques of EffectiveAdvice to reason about and enforce control flow properties as well as to control effect interference. We show that the parametricity-based approach to effect interference falls short in the presence of multiple aspects and propose a different approach using monad views, a novel technique for handling the monad stack, developed by Schrijvers and Oliveira. Then, we exploit the properties of our model to enable the modular construction of new semantics for aspect scoping and weaving. Our second contribution builds upon a powerful model to reason about mixin-based composition of effectful components and their interference, based on equational reasoning, parametricity, and algebraic laws about monadic effects. Our contribution is to show how to reason about interference in the presence of unrestricted quantification through pointcuts. We show that global reasoning can be compositional, which is key for the scalability of the approach in the face of large and evolving systems. We prove a general equivalence theorem that is based on a few conditions that can be established, reused, and adapted separately as the system evolves. The theorem is defined for an abstract monadic AOP model; we illustrate its use with a simple version of the model just described. This work brings type-based reasoning about effects for the first time in the pointcut/advice model, in a framework that is expressive, extensible and well-suited for development of robust aspect-oriented systems as well as a research tool for new aspect semantics.

Functional reactive programming (FRP) is a paradigm extending functional languages with primitives which operate on state. Typical FRP systems contain many dozens of such primitives. This thesis aims to identify a minimal subset of primitives which captures the same set of behavior as these systems, and to provide an axiomatic semantics for them using first-order linear temporal logic, with the aim of utilizing these semantics in formal verification of FRP programs. Furthermore, we identify several important properties of these primitives and prove that they are satisfied using the Coq proof assistant.

Computaçãao quântica pode ser entendida como transformação da informação codificada no estado de um sistema físico quântico. A idéia básica da computação quântica é codificar dados utilizando bits quânticos (qubits). Diferentemente do bit clássico, o qubit pode existir em uma superposição dos seus estados básicos permitindo o “paralelismo quântico”, o qual é uma característica importante da computação quântica visto que pode aumentar consideravelmente a velocidade de processamento dos algoritmos. Entretanto, tipos de dados quânticos são bastante poderosos não somente por causa da superposição de estados. Existem outras propriedades ímpares como medida e emaranhamento. Nesta tese, nós discutimos que um modelo realístico para computações quânticas deve ser geral com respeito a medidas, e completo com respeito a comunicação entre o mundo quântico e o mundo clássico. Nós, então, explicamos e estruturamos computações quânticas gerais e completas em Haskell utilizando construções conhecidas da área de semântica e linguagens de programação clássicas, como mônadas e setas. Em mais detalhes, esta tese se concentra nas seguintes contribuições. Mônadas e Setas. Paralelismo quântico, emaranhamento e medida quântica certamente vão além do escopo de linguagens funcionais “puras”. Nós mostramos que o paralelismo quântico pode ser modelado utilizando-se uma pequena generalização de mônadas, chamada mônadas indexadas ou estruturas Kleisli. Além disso, nós mostramos que a medida quântica pode ser explicada utilizando-se uma generalização mais radical de mônadas, as assim chamadas setas, mais especificamente, setas indexadas, as quais definimos nesta tese. Este resultado conecta características quânticas “genéricas” e “completas” `a construções semânticas de linguagens de programação bem fundamentadas. Entendendo as Interpretações da Mecânica Quântica como Efeitos Computacionais. Em um experimento hipotético, Einstein, Podolsky e Rosen demonstraram algumas consequências contra-intuitivas da mecânica quântica. A idéia básica é que duas partículas parecem sempre comunicar alguma informação mesmo estando separadas por uma distância arbitrariamente grande. Existe muito debate e muitos artigos sobre esse tópico, mas é interessante notar que, como proposto por Amr Sabry, essas características estranhas podem ser essencialmente modeladas por atribuições a variáveis globais. Baseados nesta idéia nós modelamos este comportamento estranho utilizando noções gerais de efeitos computacionais incorporados nas noções de mônadas e setas. Provando Propriedades de Programas Quânticos Utilizando Leis Algébricas. Nós desenvolvemos um trabalho preliminar para fazer provas equacionais sobre algoritmos quânticos escritos em uma sublinguagem pura de uma linguagem de programação funcional quântica, chamada QML. / Quantum computation can be understood as transformation of information encoded in the state of a quantum physical system. The basic idea behind quantum computation is to encode data using quantum bits (qubits). Differently from the classical bit, the qubit can be in a superposition of basic states leading to “quantum parallelism”, which is an important characteristic of quantum computation since it can greatly increase the speed processing of algorithms. However, quantum data types are computationally very powerful not only due to superposition. There are other odd properties like measurement and entangled. In this thesis we argue that a realistic model for quantum computations should be general with respect to measurements, and complete with respect to the information flow between the quantum and classical worlds. We thus explain and structure general and complete quantum programming in Haskell using well known constructions from classical semantics and programming languages, like monads and arrows. In more detail, this thesis focuses on the following contributions. Monads and Arrows. Quantum parallelism, entanglement, and measurement certainly go beyond “pure” functional programming. We have shown that quantum parallelism can be modelled using a slightly generalisation of monads called indexed monads, or Kleisli structures. We have also build on this insight and showed that quantum measurement can be explained using a more radical generalisation of monads, the so-called arrows, more specifically, indexed arrows, which we define in this thesis. This result connects “generic” and “complete” quantum features to well-founded semantics constructions and programming languages. Understanding of Interpretations of QuantumMechanics as Computational Effects. In a thought experiment, Einsten, Podolsky, and Rosen demonstrate some counter-intuitive consequences of quantum mechanics. The basic idea is that two entangled particles appear to always communicate some information even when they are separated by arbitrarily large distances. There has been endless debate and papers on this topic, but it is interesting that, as proposed by Amr Sabry, this strangeness can be essentially modelled by assignments to global variables. We build on that, and model this strangeness using the general notions of computational effects embodied in monads and arrows. Reasoning about Quantum Programs Using Algebraic Laws. We have developed a preliminary work to do equational reasoning about quantum algorithms written in a pure sublanguage of a functional quantum programming language, called QML.