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

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

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Implementing Mathematics with The Nuprl Proof Development System

The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed l-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lo¨f's system of arities. The treatment of rules and proofs focuses on his notion of a judgment. Logics are represented in LF via a new principle, the judgments as types principle, whereby each judgment is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgments and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools, such as proof editors and proof checkers, can be constructed.

A Framework for Defining Logics

Universal coalgebra: a theory of systems

Most of us have learnt to live with the fact that some singular terms do not denote. It would be fun were it otherwise, but no jolly fat man at the North Pole brings presents to good children at Yuletide; the name "Santa Claus" does not denote. Sets and classes may first have presented themselves to us as extensions of predicates, but it seems now the consensus that the paradoxes of set theory show that not all predicates have sets as extensions. Ramsey notwithstanding, many agree with the early Russell, and with Poincare, in seeing an affinity between Russell's paradox and the paradox of the liar. So if the lesson of the paradoxes of set theory is that a predicate need no more have a set as extension than the name "Santa Claus" need denote someone, then perhaps the lesson of the liar paradox is that nothing answers to a liar sentence.i To try this idea out systematically, we want some categories. Some singular terms denote, and "object" is a general term for the sort of thing singular terms succeed in denoting. We may say that there is no object denoted by the name "Santa Claus".

Non-well-founded sets

Publisher Summary Inductive definitions of sets are often informally presented by giving some rules for generating elements of the set and then adding that an object is to be in the set only if it has been generated according to the rules. An equivalent formulation is to characterize the set as the smallest set closed under the rules. This chapter discusses monotone induction and its role in extensions of recursion theory. The chapter reviews some of the work on non-monotone induction and outlines the separate motivation that has led to its development. The chapter briefly considers inductive definitions in a more general context.

An Introduction to Inductive Definitions

We prove that every set-based functor on the category of classes has a final coalgebra. This result strengthens the final coalgebra theorem announced in the book “Non-well-founded Sets”, by the first author.

A Final Coalgebra Theorem

By adding to Martin-LSf's intuitionistic theory of types a ‘type of sets’ we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets. This interpretation is a

The Type Theoretic Interpretation of Constructive Set Theory

The notion of Frege structure is introduced and shown to give a coherent context for the rigorous development of Frege's logical notion of set and an explanation of Russell's paradox.

Peter Aczel

Papers

Non-well-founded sets

An Introduction to Inductive Definitions

A Final Coalgebra Theorem

The Type Theoretic Interpretation of Constructive Set Theory

Frege Structures and the notions of proposition, truth and set