No abstract

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 λ-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Löf's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements 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.

No abstract

No abstract

No abstract

I am delighted to see my Aarhus notes [59] on SOS, Structural Operational Semantics, published as part of this special issue. The notes already contain some historical remarks, but the reader may be interested to know more of the personal intellectual context in which they arose. I must straightaway admit that at this distance in time I do not claim total accuracy or completeness: what I write should rather be considered as a reconstruction, based on (possibly faulty) memory, papers, old notes and consultations with colleagues.As a postgraduate I learnt the untyped λ-calculus from Rod Burstall. I was further deeply impressed by the work of Peter Landin on the semantics of programming languages [34][35][36][37] which includes his abstract SECD machine. One should also single out John McCarthy's contributions [45][46][47][48], which include his 1962 introduction of abstract syntax, an essential tool, then and since, for all approaches to the semantics of programming languages. The IBM Vienna school [42,41] were interested in specifying real programming languages, and, in particular, worked on an abstract interpreting machine for PL/I using VDL, their Vienna Definition Language; they were influenced by the ideas of McCarthy, Landin and Elgot [18].I remember attending a seminar at Edinburgh where the intricacies of their PL/I abstract machine were explained. The states of these machines are tuples of various kinds of complex trees and there is also a stack of environments; the transition rules involve much tree traversal to access syntactical control points, handle jumps, and to manage concurrency. I recall not much liking this way of doing operational semantics. It seemed far too complex, burying essential semantical ideas in masses of detail; further, the machine states were too big. The lesson I took from this was that abstract interpreting machines do not scale up well when used as a human-oriented method of specification for real languages (but see below for further comment).Rules play a central rôle in SOS. I recall two contributions in particular. The first is Smullyan's elegant work on formal systems [66]. These are systems of rules of the form:where the antecedents and consequents are atoms of the form P (t 1 , . . . , t n ) and where, in turn, P is a predicate symbol of a given arity n and the t j are terms, taken to be strings of constants and variables. These systems enable the inductive specification of sets of strings; for example it will be clear how to regard Post production systems as formal systems with a single unary predicate symbol

We present a categorical theory of 'well-behaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets the following for free: an operational model satisfying the rules and a canonical, internally fully abstract denotational model which satisfies the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of well-behaved rules for structural operational semantics, such as GSOS.

We model notions of computation using algebraic operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad. We focus on semantics for global and local state, showing that taking operations and equations as primitive yields a mathematical relationship that reflects their computational relationship.