We review the origins of structural operational semantics. The main publication ‘A Structural Approach to Operational Semantics, ’ also known as the ‘Aarhus Notes, ’ appeared in 1981 [G.D. Plotkin, A structural approach to operational semantics, DAIMI FN-19, Computer Science Department, Aarhus University, 1981]. The development of the ideas dates back to the early 1970s, involving many people and building on previous work on programming languages and logic. The former included abstract syntax, the SECD machine, and the abstract interpreting machines of the Vienna school; the latter included the λ-calculus and formal systems. The initial development of structural operational semantics was for simple functional languages, more or less variations of the λ-calculus; after that the ideas were gradually extended to include languages with parallel features, such as Milner’s CCS. This experience set the ground for a more systematic exposition, the subject of an invited course of lectures at Aarhus University; some of these appeared in print as the 1981 Notes. We discuss the content of these lectures and some related considerations such as ‘small state’ versus ‘grand state, ’ structural versus compositional semantics, the influence of the Scott–Strachey approach to denotational semantics, the treatment of recursion and jumps, and static semantics. We next discuss relations with other work and some immediate further development. We conclude with an account of an old, previously unpublished, idea: an alternative, perhaps more readable, graphical presentation of systems of rules for operational semantics.

An equality relation on the terms of the A-calculus is an equivalence relation closed under the (syntactical) operations of application and A-abstraction. We may distinguish between syntactic and semantic ways of introducing equality relations, /^-equality is introduced syntactically; it is the least equality relation satisfying the

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and -calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in -calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) -calculus. Note: This article is based on a lecture given by Jan Willem Klop at RTA '98 held in Tsukuba, Japan. Contents 1 Introduction 2 2 Early views on descendants 3 3 Preliminaries 5 3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Reduction . . . . . . . . . . . . . . . . . . . . . ....

. In this paper we discuss usability, and propose to take that notion as a formalisation of (un)definedness in typed lambda calculus, especially in calculi based on PCF. We discuss some important properties that make usability attractive as a formalisation of (un)definedness. There is a remarkable difference between usability and solvability: in the untyped lambda calculus the solvable terms are precisely the terms with a head normal form, whereas in typed lambda calculus the usable terms are "between" the terms with a normal form and the terms with a (weak) head normal form. 1 Introduction The elementary form of undefinedness arises on the level of natural numbers, when the evaluation of a (closed) term M of type Nat does not terminate, i.e., when M does not have a normal form. Such a term is also often called meaningless. However, for higher types it is not so evident which terms should be called meaningless. Analogous to the situation for ground types, it is often felt to be attrac...

The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories ( = equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and -calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in -calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) -calculus. Note: This article is based on a lecture given by Jan Willem Klop at RTA '98 held in Tsukuba, Japan. Contents 1 Introduction 2 2 Early views on descendants 3 3 Preliminaries 5 3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Reduction . . . . . . . . . . . . . . . . . . . . . ....

Abstract. A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable pre-order on types which allows to derive τ for M. Simple easiness implies easiness. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a nonempty co-r.e. (complement of a recursively enumerable) set of easy, but non simple easy, λ-terms. Key words: Lambda calculus, easy terms, simple easy terms, filter models 1