Pure Subtype Systems: a Type Theory for Extensible Software
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Structured Recursion for Non-Uniform Data-Types
Structured recursion for non-uniform data-types by Paul Alexander Blampied, B.Sc. Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy, March 2000 Contents Acknowledgements .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 Chapter 1. Introduction .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 1.1. Non-uniform data-types .. .. .. .. .. .. .. .. .. .. .. .. 3 1.2. Program calculation .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1.3. Maps and folds in Squiggol .. .. .. .. .. .. .. .. .. .. .. 11 1.4. Related work .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 14 1.5. Purpose and contributions of this thesis .. .. .. .. .. .. .. 15 Chapter 2. Categorical data-types .. .. .. .. .. .. .. .. .. .. .. .. 18 2.1. Notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19 2.2. Data-types as initial algebras .. .. .. .. .. .. .. .. .. .. 21 2.3. Parameterized data-types .. .. .. .. .. .. .. .. .. .. .. 29 2.4. Calculation properties of catamorphisms .. .. .. .. .. .. .. 33 2.5. Existence of initial algebras .. .. .. .. .. .. .. .. .. .. .. 37 2.6. Algebraic data-types in functional programming .. .. .. .. 57 2.7. Chapter summary .. .. .. .. .. .. .. .. .. .. .. .. .. 61 Chapter 3. Folding in functor categories .. .. .. .. .. .. .. .. .. .. 62 3.1. Natural transformations and functor categories .. .. .. .. .. 63 3.2. Initial algebras in functor categories .. .. .. .. .. .. .. .. 68 3.3. Examples and non-examples .. .. .. .. .. .. .. .. .. .. 77 3.4. Existence of initial algebras in functor categories .. .. .. .. 82 3.5. -
First Class Overloading Via Insersection Type Parameters⋆
First Class Overloading via Insersection Type Parameters? Elton Cardoso2, Carlos Camar~ao1, and Lucilia Figueiredo2 1 Universidade Federal de Minas Gerais, [email protected] 2 Universidade Federal de Ouro Preto [email protected], [email protected] Abstract The Hindley-Milner type system imposes the restriction that function parameters must have monomorphic types. Lifting this restric- tion and providing system F “first class" polymorphism is clearly desir- able, but comes with the difficulty that inference of types for system F is undecidable. More practical type systems incorporating types of higher- rank have recently been proposed, that rely on system F but require type annotations for the definition of functions with polymorphic type parameters. However, these type annotations inevitably disallow some possible uses of higher-rank functions. To avoid this problem and to pro- mote code reuse, we explore using intersection types for specifying the types of function parameters that are used polymorphically inside the function body, allowing a flexible use of such functions, on applications to both polymorphic or overloaded arguments. 1 Introduction The Hindley-Milner type system [9] (HM) has been successfuly used as the basis for type systems of modern functional programming languages, such as Haskell [23] and ML [20]. This is due to its remarkable properties that a compiler can in- fer the principal type for any language expression, without any help from the programmer, and the type inference algorithm [5] is relatively simple. This is achieved, however, by imposing some restrictions, a major one being that func- tion parameters must have monomorphic types. For example, the following definition is not allowed in the HM type system: foo g = (g [True,False], g ['a','b','c']) Since parameter g is used with distinct types in the function's body (being applied to both a list of booleans and a list of characters), its type cannot be monomorphic, and this definition of foo cannot thus be typed in HM. -
Disjoint Polymorphism
Disjoint Polymorphism João Alpuim, Bruno C. d. S. Oliveira, and Zhiyuan Shi The University of Hong Kong {alpuim,bruno,zyshi}@cs.hku.hk Abstract. The combination of intersection types, a merge operator and parametric polymorphism enables important applications for program- ming. However, such combination makes it hard to achieve the desirable property of a coherent semantics: all valid reductions for the same expres- sion should have the same value. Recent work proposed disjoint inter- sections types as a means to ensure coherence in a simply typed setting. However, the addition of parametric polymorphism was not studied. This paper presents Fi: a calculus with disjoint intersection types, a vari- ant of parametric polymorphism and a merge operator. Fi is both type- safe and coherent. The key difficulty in adding polymorphism is that, when a type variable occurs in an intersection type, it is not statically known whether the instantiated type will be disjoint to other compo- nents of the intersection. To address this problem we propose disjoint polymorphism: a constrained form of parametric polymorphism, which allows disjointness constraints for type variables. With disjoint polymor- phism the calculus remains very flexible in terms of programs that can be written, while retaining coherence. 1 Introduction Intersection types [20,43] are a popular language feature for modern languages, such as Microsoft’s TypeScript [4], Redhat’s Ceylon [1], Facebook’s Flow [3] and Scala [37]. In those languages a typical use of intersection types, which has been known for a long time [19], is to model the subtyping aspects of OO-style multiple inheritance. -
Polymorphic Intersection Type Assignment for Rewrite Systems with Abstraction and -Rule Extended Abstract
Polymorphic Intersection Type Assignment for Rewrite Systems with Abstraction and -rule Extended Abstract Steffen van Bakel , Franco Barbanera , and Maribel Fernandez´ Department of Computing, Imperial College, 180 Queen’s Gate, London SW7 2BZ. [email protected] Dipartimento di Matematica, Universita` degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italia. [email protected] LIENS (CNRS URA 8548), Ecole Normale Superieure,´ 45, rue d’Ulm, 75005 Paris, France. [email protected] Abstract. We define two type assignment systems for first-order rewriting ex- tended with application, -abstraction, and -reduction (TRS ). The types used in these systems are a combination of ( -free) intersection and polymorphic types. The first system is the general one, for which we prove a subject reduction theorem and show that all typeable terms are strongly normalisable. The second is a decidable subsystem of the first, by restricting types to Rank 2. For this sys- tem we define, using an extended notion of unification, a notion of principal type, and show that type assignment is decidable. Introduction The combination of -calculus (LC) and term rewriting systems (TRS) has attracted attention not only from the area of programming language design, but also from the rapidly evolving field of theorem provers. It is well-known by now that type disciplines provide an environment in which rewrite rules and -reduction can be combined with- out loss of their useful properties. This is supported by a number of results for a broad range of type systems [11, 12, 20, 7, 8, 5]. In this paper we study the combination of LC and TRS as a basis for the design of a programming language. -
Binary Search Trees
Introduction Recursive data types Binary Trees Binary Search Trees Organizing information Sum-of-Product data types Theory of Programming Languages Computer Science Department Wellesley College Introduction Recursive data types Binary Trees Binary Search Trees Table of contents Introduction Recursive data types Binary Trees Binary Search Trees Introduction Recursive data types Binary Trees Binary Search Trees Sum-of-product data types Every general-purpose programming language must allow the • processing of values with different structure that are nevertheless considered to have the same “type”. For example, in the processing of simple geometric figures, we • want a notion of a “figure type” that includes circles with a radius, rectangles with a width and height, and triangles with three sides: The name in the oval is a tag that indicates which kind of • figure the value is, and the branches leading down from the oval indicate the components of the value. Such types are known as sum-of-product data types because they consist of a sum of tagged types, each of which holds on to a product of components. Introduction Recursive data types Binary Trees Binary Search Trees Declaring the new figure type in Ocaml In Ocaml we can declare a new figure type that represents these sorts of geometric figures as follows: type figure = Circ of float (* radius *) | Rect of float * float (* width, height *) | Tri of float * float * float (* side1, side2, side3 *) Such a declaration is known as a data type declaration. It consists of a series of |-separated clauses of the form constructor-name of component-types, where constructor-name must be capitalized. -
Recursive Type Generativity
Recursive Type Generativity Derek Dreyer Toyota Technological Institute at Chicago [email protected] Abstract 1. Introduction Existential types provide a simple and elegant foundation for un- Recursive modules are one of the most frequently requested exten- derstanding generative abstract data types, of the kind supported by sions to the ML languages. After all, the ability to have cyclic de- the Standard ML module system. However, in attempting to extend pendencies between different files is a feature that is commonplace ML with support for recursive modules, we have found that the tra- in mainstream languages like C and Java. To the novice program- ditional existential account of type generativity does not work well mer especially, it seems very strange that the ML module system in the presence of mutually recursive module definitions. The key should provide such powerful mechanisms for data abstraction and problem is that, in recursive modules, one may wish to define an code reuse as functors and translucent signatures, and yet not allow abstract type in a context where a name for the type already exists, mutually recursive functions and data types to be broken into sepa- but the existential type mechanism does not allow one to do so. rate modules. Certainly, for simple examples of recursive modules, We propose a novel account of recursive type generativity that it is difficult to convincingly argue why ML could not be extended resolves this problem. The basic idea is to separate the act of gener- in some ad hoc way to allow them. However, when one considers ating a name for an abstract type from the act of defining its under- the semantics of a general recursive module mechanism, one runs lying representation. -
Cross-Platform Language Design
Cross-Platform Language Design THIS IS A TEMPORARY TITLE PAGE It will be replaced for the final print by a version provided by the service academique. Thèse n. 1234 2011 présentée le 11 Juin 2018 à la Faculté Informatique et Communications Laboratoire de Méthodes de Programmation 1 programme doctoral en Informatique et Communications École Polytechnique Fédérale de Lausanne pour l’obtention du grade de Docteur ès Sciences par Sébastien Doeraene acceptée sur proposition du jury: Prof James Larus, président du jury Prof Martin Odersky, directeur de thèse Prof Edouard Bugnion, rapporteur Dr Andreas Rossberg, rapporteur Prof Peter Van Roy, rapporteur Lausanne, EPFL, 2018 It is better to repent a sin than regret the loss of a pleasure. — Oscar Wilde Acknowledgments Although there is only one name written in a large font on the front page, there are many people without which this thesis would never have happened, or would not have been quite the same. Five years is a long time, during which I had the privilege to work, discuss, sing, learn and have fun with many people. I am afraid to make a list, for I am sure I will forget some. Nevertheless, I will try my best. First, I would like to thank my advisor, Martin Odersky, for giving me the opportunity to fulfill a dream, that of being part of the design and development team of my favorite programming language. Many thanks for letting me explore the design of Scala.js in my own way, while at the same time always being there when I needed him. -
Blossom: a Language Built to Grow
Macalester College DigitalCommons@Macalester College Mathematics, Statistics, and Computer Science Honors Projects Mathematics, Statistics, and Computer Science 4-26-2016 Blossom: A Language Built to Grow Jeffrey Lyman Macalester College Follow this and additional works at: https://digitalcommons.macalester.edu/mathcs_honors Part of the Computer Sciences Commons, Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Lyman, Jeffrey, "Blossom: A Language Built to Grow" (2016). Mathematics, Statistics, and Computer Science Honors Projects. 45. https://digitalcommons.macalester.edu/mathcs_honors/45 This Honors Project - Open Access is brought to you for free and open access by the Mathematics, Statistics, and Computer Science at DigitalCommons@Macalester College. It has been accepted for inclusion in Mathematics, Statistics, and Computer Science Honors Projects by an authorized administrator of DigitalCommons@Macalester College. For more information, please contact [email protected]. In Memory of Daniel Schanus Macalester College Department of Mathematics, Statistics, and Computer Science Blossom A Language Built to Grow Jeffrey Lyman April 26, 2016 Advisor Libby Shoop Readers Paul Cantrell, Brett Jackson, Libby Shoop Contents 1 Introduction 4 1.1 Blossom . .4 2 Theoretic Basis 6 2.1 The Potential of Types . .6 2.2 Type basics . .6 2.3 Subtyping . .7 2.4 Duck Typing . .8 2.5 Hindley Milner Typing . .9 2.6 Typeclasses . 10 2.7 Type Level Operators . 11 2.8 Dependent types . 11 2.9 Hoare Types . 12 2.10 Success Types . 13 2.11 Gradual Typing . 14 2.12 Synthesis . 14 3 Language Description 16 3.1 Design goals . 16 3.2 Type System . 17 3.3 Hello World . -
Rank 2 Type Systems and Recursive De Nitions
Rank 2 typ e systems and recursive de nitions Technical Memorandum MIT/LCS/TM{531 Trevor Jim Lab oratory for Computer Science Massachusetts Institute of Technology August 1995; revised Novemb er 1995 Abstract We demonstrate an equivalence b etween the rank 2 fragments of the p olymorphic lamb da calculus System F and the intersection typ e dis- cipline: exactly the same terms are typable in each system. An imme- diate consequence is that typability in the rank 2 intersection system is DEXPTIME-complete. Weintro duce a rank 2 system combining intersections and p olymorphism, and prove that it typ es exactly the same terms as the other rank 2 systems. The combined system sug- gests a new rule for typing recursive de nitions. The result is a rank 2 typ e system with decidable typ e inference that can typ e some inter- esting examples of p olymorphic recursion. Finally,we discuss some applications of the typ e system in data representation optimizations suchasunboxing and overloading. Keywords: Rank 2 typ es, intersection typ es, p olymorphic recursion, boxing/unboxing, overloading. 1 Intro duction In the past decade, Milner's typ e inference algorithm for ML has b ecome phenomenally successful. As the basis of p opular programming languages like Standard ML and Haskell, Milner's algorithm is the preferred metho d of typ e inference among language implementors. And in the theoretical 545 Technology Square, Cambridge, MA 02139, [email protected]. Supp orted by NSF grants CCR{9113196 and CCR{9417382, and ONR Contract N00014{92{J{1310. -
Data Structures Are Ways to Organize Data (Informa- Tion). Examples
CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 1 ] What are Data Structures? Data structures are ways to organize data (informa- tion). Examples: simple variables — primitive types objects — collection of data items of various types arrays — collection of data items of the same type, stored contiguously linked lists — sequence of data items, each one points to the next one Typically, algorithms go with the data structures to manipulate the data (e.g., the methods of a class). This course will cover some more complicated data structures: how to implement them efficiently what they are good for CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 2 ] Abstract Data Types An abstract data type (ADT) defines a state of an object and operations that act on the object, possibly changing the state. Similar to a Java class. This course will cover specifications of several common ADTs pros and cons of different implementations of the ADTs (e.g., array or linked list? sorted or unsorted?) how the ADT can be used to solve other problems CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 3 ] Specific ADTs The ADTs to be studied (and some sample applica- tions) are: stack evaluate arithmetic expressions queue simulate complex systems, such as traffic general list AI systems, including the LISP language tree simple and fast sorting table database applications, with quick look-up CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 4 ] How Does C Fit In? Although data structures are universal (can be imple- mented in any programming language), this course will use Java and C: non-object-oriented parts of Java are based on C C is not object-oriented We will learn how to gain the advantages of data ab- straction and modularity in C, by using self-discipline to achieve what Java forces you to do. -
A Facet-Oriented Modelling
A Facet-oriented modelling JUAN DE LARA, Universidad Autónoma de Madrid (Spain) ESTHER GUERRA, Universidad Autónoma de Madrid (Spain) JÖRG KIENZLE, McGill University (Canada) Models are the central assets in model-driven engineering (MDE), as they are actively used in all phases of software development. Models are built using metamodel-based languages, and so, objects in models are typed by a metamodel class. This typing is static, established at creation time, and cannot be changed later. Therefore, objects in MDE are closed and fixed with respect to the class they conform to, the fields they have, and the wellformedness constraints they must comply with. This hampers many MDE activities, like the reuse of model-related artefacts such as transformations, the opportunistic or dynamic combination of metamodels, or the dynamic reconfiguration of models. To alleviate this rigidity, we propose making model objects open so that they can acquire or drop so-called facets. These contribute with a type, fields and constraints to the objects holding them. Facets are defined by regular metamodels, hence being a lightweight extension of standard metamodelling. Facet metamodels may declare usage interfaces, as well as laws that govern the assignment of facets to objects (or classes). This paper describes our proposal, reporting on a theory, analysis techniques and an implementation. The benefits of the approach are validated on the basis of five case studies dealing with annotation models, transformation reuse, multi-view modelling, multi-level modelling and language product lines. Categories and Subject Descriptors: [Software and its engineering]: Model-driven software engineering; Domain specific languages; Design languages; Software design engineering Additional Key Words and Phrases: Metamodelling, Flexible Modelling, Role-Based Modelling, METADEPTH ACM Reference Format: Juan de Lara, Esther Guerra, Jörg Kienzle. -
Approximations, Fibrations and Intersection Type Systems
Approximations, Fibrations and Intersection Type Systems Damiano Mazza*, Luc Pellissier† & Pierre Vial‡ June 16, 2017 Introduction The discovery of linear logic [7] has introduced the notionof linearity in computer science and proof theory. A remarkable fact of linear logic lies in its approximation theorem, stating that an arbitrary proof (not necessarily linear, that is, using its premisses any number of times) can be approximated arbitrarily well by a linear proof. This notion of approximation has then been explored in different directions [4, 10,14]. Approximations are known to be related to relational models, which in turn are related to intersection types [2, 12, 13]. In this work, we investigate approximations in the “type-systems as functors” perspective pioneered by [11]. After recasting fundamental properties of type systems, such as subject reduction and expansion in this framework, we give an intersection type system framework for linear logic, whose derivations are simply-typed approximations. Any calculus that translates meaningfully to linear logic is then endowed by a intersection type system, computed by pulling back one of these intersection type systems for linear logic, and which inherit its properties. All standard intersection type systems (idempotent, such as in [6, 1] or not, such as in [3]) for call-by-name and call-by-value λ-calculus, characterizing weak, strong, and head normalization, fit in this picture, thus justifying the equation: Simply-typed approximations = intersection types derivations. We moreover obtain new type systems, by considering other translations and reductions. 1 What is a type system? The starting point of this research is the work of [11].