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

The λ-calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from values to values ) that may jeopardise the applicability of theoretical results. In this paper we introduce calculi, based on a categorical semantics for computations , that provide a correct basis for proving equivalence of programs for a wide range of notions of computation .

Notions of computation and monads

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

Universal coalgebra: a theory of systems

In the semantics of programming, finite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with infinite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages (Milner, 1980; Park, 1981). Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras (Aczel and Mendler, 1989). Thus the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to: coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken as the basic ingredients of a theory called universal coalgebra. Some standard results from universal algebra are reformulated (using the afore mentioned correspondence) and proved for a large class of coalgebras, leading to a series of results on, e.g., the lattices of subcoalgebras and bisimulations, simple coalgebras and coinduction, and a covariety theorem for coalgebras similar to Birkhoff's variety theorem.

The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of `name-abstraction' and `fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.

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A New Approach to Abstract Syntax with Variable Binding

This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving lexically scoped binding constructs. It introduces Nominal Logic, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping, for freshness of names, and for name-binding. Its axioms express properties of these constructs satisfied by the FM-sets model of syntax involving binding, which was recently introduced by the author and M.J. Gabbay and makes use of the Fraenkel-Mostowski permutation model of set theory. Nominal Logic serves as a vehicle for making two general points. First, name-swapping has much nicer logical properties than more general, non-bijective forms of renaming while at the same time providing a sufficient foundation for a theory of structural induction/recursion for syntax modulo α-equivalence. Secondly, it is useful for the practice of operational semantics to make explicit the equivariance property of assertions about syntax - namely that their validity is invariant under name-swapping.

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Nominal Logic: A First Order Theory of Names and Binding

The Fraenkel-Mostowski permutation model of set theory with atoms (FM-sets) can serve as the semantic basis of meta-logics for specifying and reasoning about formal systems involving name binding, /spl alpha/-conversion, capture avoiding substitution, and so on. We show that in FM-set theory one can express statements quantifying over 'fresh' names and we use this to give a novel set-theoretic interpretation of name abstraction. Inductively defined FM-sets involving this name abstraction set former (together with cartesian product and disjoint union) can correctly encode object-level syntax module e-conversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice.

/pdf/a-new-approach-to-abstract-syntax-involving-binders-5254s5ylv0.pdf

A new approach to abstract syntax involving binders

Nominal sets provide a promising new mathematical analysis of names in formal languages based upon symmetry, with many applications to the syntax and semantics of programming language constructs that involve binding, or localising names. Part I provides an introduction to the basic theory of nominal sets. In Part II, the author surveys some of the applications that have developed in programming language semantics (both operational and denotational), functional programming and logic programming. As the first book to give a detailed account of the theory of nominal sets, it will be welcomed by researchers and graduate students in theoretical computer science.

Andrew M. Pitts

Papers

A New Approach to Abstract Syntax with Variable Binding

Nominal Logic: A First Order Theory of Names and Binding

A new approach to abstract syntax involving binders

Nominal Sets: Names and Symmetry in Computer Science

Nominal unification