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**11 - 14**of**14**### Realizability Models for Sequential Computation

, 1998

"... We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of seq ..."

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We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...

### Optimising Parallel Pattern-matching

"... Parallel pattern-matching (PPM) provides true commutative implementation of functions defined by cases in functional languages, because no argument is given precedence over any other. However, the requirement for concurrency (in general) to support these semantics means that current implementations ..."

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Parallel pattern-matching (PPM) provides true commutative implementation of functions defined by cases in functional languages, because no argument is given precedence over any other. However, the requirement for concurrency (in general) to support these semantics means that current implementations incur a significant performance penalty over simple, traditional left-to-right semantics. We describe a source-level program transformation scheme that analyses a PPM definition and is often able to generate an equivalent definition that can be executed without concurrency. Where sequential implementation is not possible, the scheme is sometimes able to generate an equivalent definition that reduces the number of concurrent threads required to execute a definition. This transformation scheme promises to deliver a major improvement in the performance of PPM implementations.

### Lazy Least Fixed Points in ML

"... In this paper, we present an algorithm for computing the least solution of a system of monotone equations. This algorithm can be viewed as an effective form of the following well-known fixed point theorem: Theorem Let V be a finite set of variables. Let (P, ≤, ⊥) be a partially ordered set ..."

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In this paper, we present an algorithm for computing the least solution of a system of monotone equations. This algorithm can be viewed as an effective form of the following well-known fixed point theorem: Theorem Let V be a finite set of variables. Let (P, ≤, ⊥) be a partially ordered set

### Continuity of Gödel’s system T definable functionals via effectful forcing

"... It is well-known that the Gödel’s system T definable functions (N → N) → N are continuous, and that their restrictions from the Baire type (N → N) to the Cantor type (N → 2) are uniformly continuous. We offer a new, relatively short and self-contained proof. The main technical idea is a concrete no ..."

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It is well-known that the Gödel’s system T definable functions (N → N) → N are continuous, and that their restrictions from the Baire type (N → N) to the Cantor type (N → 2) are uniformly continuous. We offer a new, relatively short and self-contained proof. The main technical idea is a concrete notion of generic element that doesn’t rely on forcing, Kripke semantics or sheaves, which seems to be related to generic effects in programming. The proof uses standard techniques from programming language semantics, such as dialogues, monads, and logical relations. We write this proof in intensional Martin-Löf type theory (MLTT), in Agda notation. Because MLTT has a computational interpretation and Agda can be seen as a programming language, we can run our proof to compute moduli of (uniform) continuity of T-definable functions.