# Geometry of Interaction 1: Interpretation of System F

TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.

Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.

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01 Jan 1989

TL;DR: In this paper, the Curry-Howard isomorphism and the normalisation theorem of a natural deduction system T coherence spaces have been studied in the context of linear logic and linear logic semantics.

Abstract: Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system F coherence semantics of the sum cut elimination (Hauptsatz) strong normalisation for F representation theorem semantics of System F what is linear logic?

1,771 citations

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TL;DR: This paper describes the syntax and semantics of a simple quantum programming language with high-level features such as loops, recursive procedures, and structured data types, and has an interesting denotational semantics in terms of complete partial orders of superoperators.

Abstract: We propose the design of a programming language for quantum computing. Traditionally, quantum algorithms are frequently expressed at the hardware level, for instance in terms of the quantum circuit model or quantum Turing machines. These approaches do not encourage structured programming or abstractions such as data types. In this paper, we describe the syntax and semantics of a simple quantum programming language with high-level features such as loops, recursive procedures, and structured data types. The language is functional in nature, statically typed, free of run-time errors, and has an interesting denotational semantics in terms of complete partial orders of superoperators.

510 citations

### Cites methods from "Geometry of Interaction 1: Interpre..."

...(1) This formula is similar to the execution formula of Girard’s Geometry of Interaction (Girard 1989)....

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01 Apr 1996TL;DR: Traced monoidal categories are introduced, a structure theorem is proved for them, and an example is provided where the structure theorem has application as discussed by the authors. But this is not the case for all categories.

Abstract: Traced monoidal categories are introduced, a structure theorem is proved for them, and an example is provided where the structure theorem has application.

488 citations

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TL;DR: Girard's linear logic is studied from the point of view of giving a concrete computational interpretation of the logic, based on the Curry—Howard isomorphism, which opens up a promising new approach to the parallel implementation of functional programming languages.

Abstract: We study Girard's linear logic from the point of view of giving a concrete computational interpretation of the logic, based on the Curry—Howard isomorphism. In the case of Intuitionistic linear logic, this leads to a refinement of the lambda calculus, giving finer control over order of evaluation and storage allocation, while maintaining the logical content of programs as proofs, and computation as cut-elimination. In the classical case, it leads to a concurrent process paradigm with an operational semantics in the style of Berry and Boudol's chemical abstract machine. This opens up a promising new approach to the parallel implementation of functional programming languages; and offers the prospect of typed concurrent programming in which correctness is guaranteed by the typing.

471 citations

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19 Apr 1994

TL;DR: An intensional model for the programming language PCF is described, in which the types of PCF are interpreted by games, and the terms by certain "history-free" strategies are interpreted.

469 citations

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TL;DR: In this article, the authors considered the C*-algebra generated by n≧2 isometries S 1S 1+1+snS * n = 1.

Abstract: We consider theC*-algebra\(\mathcal{O}_n \) generated byn≧2 isometriesS1,...,Sn on an infinite-dimensional Hilbert space, with the property thatS1S*1+...+SnS*n=1. It turns out that\(\mathcal{O}_n \) has the structure of a crossed product of a finite simpleC*-algebra ℱ by a single endomorphism scaling the trace of ℱ by 1/n. Thus,\(\mathcal{O}_n \) is a separableC*-algebra sharing many of the properties of a factor of typeIIIλ with λ=1/n. As a consequence we show that\(\mathcal{O}_n \) is simple and that its isomorphism type does not depend on the choice ofS1,...,Sn.

1,353 citations