A type system is a syntactic method for automatically checking the absence of certain erroneous behaviors by classifying program phrases according to the kinds of values they compute. The study of type systems -- and of programming languages from a type-theoretic perspective -- has important applications in software engineering, language design, high-performance compilers, and security.This text provides a comprehensive introduction both to type systems in computer science and to the basic theory of programming languages. The approach is pragmatic and operational; each new concept is motivated by programming examples and the more theoretical sections are driven by the needs of implementations. Each chapter is accompanied by numerous exercises and solutions, as well as a running implementation, available via the Web. Dependencies between chapters are explicitly identified, allowing readers to choose a variety of paths through the material.The core topics include the untyped lambda-calculus, simple type systems, type reconstruction, universal and existential polymorphism, subtyping, bounded quantification, recursive types, kinds, and type operators. Extended case studies develop a variety of approaches to modeling the features of object-oriented languages.

https://www.seas.upenn.edu/~bcpierce/tapl/contents.pdf

Types and Programming Languages

From the Publisher:
Standard ML is general-purpose programming language designed for large projects. This book provides a formal definition of Standard ML for the benefit of all concerned with the language, including users and implementers. Because computer programs are increasingly required to withstand rigorous analysis, it is all the more important that the language in which they are written be defined with full rigor. The authors have defined their semantic objects in mathematical notation that is completely independent of Standard ML.

The Definition of Standard ML

Our objective is to understand the notion of type in programming languages, present a model of typed, polymorphic programming languages that reflects recent research in type theory, and examine the relevance of recent research to the design of practical programming languages.Object-oriented languages provide both a framework and a motivation for exploring the interaction among the concepts of type, data abstraction, and polymorphism, since they extend the notion of type to data abstraction and since type inheritance is an important form of polymorphism. We develop a l-calculus-based model for type systems that allows us to explore these interactions in a simple setting, unencumbered by complexities of production programming languages.The evolution of languages from untyped universes to monomorphic and then polymorphic type systems is reviewed. Mechanisms for polymorphism such as overloading, coercion, subtyping, and parameterization are examined. A unifying framework for polymorphic type systems is developed in terms of the typed l-calculus augmented to include binding of types by quantification as well as binding of values by abstraction.The typed l-calculus is augmented by universal quantification to model generic functions with type parameters, existential quantification and packaging (information hiding) to model abstract data types, and bounded quantification to model subtypes and type inheritance. In this way we obtain a simple and precise characterization of a powerful type system that includes abstract data types, parametric polymorphism, and multiple inheritance in a single consistent framework. The mechanisms for type checking for the augmented l-calculus are discussed.The augmented typed l-calculus is used as a programming language for a variety of illustrative examples. We christen this language Fun because fun instead of l is the functional abstraction keyword and because it is pleasant to deal with.Fun is mathematically simple and can serve as a basis for the design and implementation of real programming languages with type facilities that are more powerful and expressive than those of existing programming languages. In particular, it provides a basis for the design of strongly typed object-oriented languages.

/pdf/on-understanding-types-data-abstraction-and-polymorphism-1a53y4aipq.pdf

On understanding types, data abstraction, and polymorphism

Conditional rewriting logic as a unified model of concurrency

bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

The ls-calculus is a refinement of the l-calculus where substitutions are manipulated explicitly. The ls-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical l-calculus and concrete implementations.

/pdf/explicit-substitutions-4m66dv87cc.pdf

Explicit substitutions

We present the μ -calculus, a syntax for λ-calculus + control operators exhibiting symmetries such as program/context and call-by-name/call-by-value. This calculus is derived from implicational Gentzen's sequent calculus LK, a key classical logical system in proof theory. Under the Curry-Howard correspondence between proofs and programs, we can see LK, or more precisely a formulation called LKμ , as a syntax-directed system of simple types for μ -calculus. For μ -calculus, choosing a call-by-name or call-by-value discipline for reduction amounts to choosing one of the two possible symmetric orientations of a critical pair. Our analysis leads us to revisit the question of what is a natural syntax for call-by-value functional computation. We define a translation of λμ-calculus into μ -calculus and two dual translations back to λ-calculus, and we recover known CPS translations by composing these translations.

/pdf/the-duality-of-computation-256c2q3h32.pdf

The duality of computation

This book is a thoroughly revised edition of a monograph that presents an approach to the design and implementation of sequential programming languages based on the relationship between lambda-calculus and category theory The foundations of a new "categorical" combinatory logic are laid down Compilation and evaluation techniques are investigated A simple abstract machine, called the Categorical Abstract Machine, is presented: it has served as the core of the implementation of the language CAML, of the ML family, developed at INRIA-Roquencourt and Ecole Normale Superieure, and first released in 1987 The main characteristics of this approach are conceptual simplicity and compactness, with bearings on portability, efficiency, and correctness proofs A mathematical semantics of sequentiality is proposed, in which "sequential algorithms" rather than functions are used to interpret procedures The theoretical investigation has led to the development of a programming language, CDSO, in which basic and functional types are not differentiated The evaluation framework is a demand-driven data flow network The model of sequential algorithms is fully abstract with respect to this language: two procedures have the same denotation if and only if they have the same behaviour Background on full abstraction is given The new edition covers new results, and introduces new connections, as suggested by the following non-exhaustive fist of keywords: confluence properties of categorical combinators, explicit substitutions, control operations, linear logic, geometry of interaction, strong stability

Categorical Combinators, Sequential Algorithms and Functional Programming

This book describes the mathematical aspects of the semantics of programming languages. The main goals are to provide formal tools to assess the meaning of programming constructs in both a language-independent and a machine-independent way, and to prove properties about programs, such as whether they terminate, or whether their result is a solution of the problem they are supposed to solve. In order to achieve this the authors first present, in an elementary and unified way, the theory of certain topological spaces that have proved of use in the modelling of various families of typed lambda calculi considered as core programming languages and as meta-languages for denotational semantics. This theory is known as Domain Theory, and was founded as a subject by Scott and Plotkin. One of the main concerns is to establish links between mathematical structures and more syntactic approaches to semantics, often referred to as operational semantics, which is also described. This dual approach has the double advantage of motivating computer scientists to do some mathematics and of interesting mathematicians in unfamiliar application areas from computer science.

Pierre-Louis Curien

Papers

Explicit substitutions

The duality of computation

Categorical Combinators, Sequential Algorithms and Functional Programming

Domains and Lambda-Calculi

Sequential algorithms on concrete data structures