Jean-Yves Girard. Towards a geometry of interaction. In Categories in Computer Science, volume 92 of Contemporary Mathematics, pages 69 -- 108. AMS, June 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in which linear logic can be fruitfully exploited to address aspects of logic programming. Girard modeled the difference between the classical, external logic of Horn clauses and the internal logic of Prolog that arises from the use of depthfirst search using a non commutative linear logic [11]. Cerrito appplied classical linear logic to the problem of formalizing finite failure for certain kinds of Horn clause programs where negations are permitted in the body of clauses [5] Linear logic has been used to extend the basic design of logic programming languages in at least two papers ....

Jean-Yves Girard. Towards a geometry of interaction. In Categories in Computer Science, volume 92 of Contemporary Mathematics, pages 69 -- 108. AMS, June 1987.

....A. The stream description seems more natural for the cigarette buying example than for computational interpretations involving stored data. If A represents having a dollar, then A could represent having an unlimited supply of dollars, rather than having one, arbitrarily reusable dollar. Girard [13] mentions that the proof rules for do not uniquely determine this modality. That is, if one added to linear logic a second modal operator # subject to the same inference rules as , one could not deduce that the two operators are equivalent in the sense that A # # A and vice versa. The ....

Jean-Yves Girard. Toward a geometry of interaction. In Gray and Scedrov [15], pages 69--108.

....coming from categorical models of the calculus; and SKI combinators coming from combinatory logic. In [9] the second author gave a different kind of compilation of a simple calculus based functional programming language coming directly from Girard s Geometry of Interaction semantics [7, 6]; a semantics of computation capturing the actual reduction process. One way of understanding the Geometry of Interaction is to think of a program represented as a graph. The meaning of a program is then given by the set of paths in the graph. Of course not all paths, but some particular ones ....

Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andr'e Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989. 15

....between game semantics and concurrency semantics, and [Abr94] for other aspects. We now describe composition in terms of the functions inducing strategies. Say we have oe f : A B; oe g : B C. We want to find h such that oe f ; oe g = oe h . We shall compute h by the execution formula [Gir89b, Gir89a, Gir88]. Before giving the formal definition, let us explain the idea, which is rather simple. We want to hook the strategies up so that Player s moves in B under oe get turned into Opponent s moves in B for , and vice versa. Consider the following picture: oe Delta Delta Delta Delta Delta ....

Jean-Yves Girard. Towards a geometry of interaction. In J. W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.

.... involving C algebras: Mulvey s Quantales [Mulvey, 1986, Borceux et al. 1989, Rosenthal, 1990] are an abstraction of the lattice of closed right ideals of a C algebra; in Girard s Geometry of Interaction (linear) proofs are represented and characterised within a C algebra setting [Girard, 1989b, Girard, 1989a, Girard, 1988, Girard, 1994, Girard et al. 1995] and finally there exists an interesting characterisation of (the Lindenbaum algebras, i.e. MV algebras, of) multi valued logics by Mundici based on the K Theory of C algebras [Mundici, 1986, Mundici, 1989] C algebras are ....

....but we can in principle approximate it as close as desired (although this might also not be a numerically favourable or stable way to it) 5.2. Proof Normalisation. In a series of articles J. Y. Girard gave a semantics of proofs in linear logic which is based on such a Hilbert space construction [Girard, 1989b, Girard, 1989a, Girard, 1988, Girard, 1994] Actually he started top down, namely not by considering the space of statements but by establishing a mathematical model for (the algebra of) proof steps. Suppose we require, that a linear proof or a linear conclusion , transforming one Hilbert ....

Jean-Yves Girard. Towards a Geometry of Interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 69--108, American Mathematical Society, 1989.

....model of quantum computation. 4.2 Linear Logic Another interesting example of an abstract Hilbert machine can be found in Girard s Geometry of Interaction . In a series of articles J. Y. Girard gives a semantics of proofs in linear logic which is based on a Hilbert space construction [16, 15, 14, 17]. Its quite simple to embed the common logical descriptions of facts or states into a Hilbert space structure. Let us fix a certain (countable) set of qualities or predicates p i . A predicate formula F e.g. in disjunctive normal form, i.e. F = f 1 f 2 : f k with f i = p i 1 p i 2 : ....

....approximate it as close as desired (even if this might not be a numerically favourable or stable way to do it) 5. 2 Proof Normalisation Let us look more concretely at Girard s Geometry of Interaction we introduced above, where every proof (procedure) is encoded as an abstract C operator [16, 15, 14, 17]. Proof normalisation is as said implemented by the so called execution formula EX(U; oe) U P (oeU) n . In the light of what we said about the iterative (approximate) solution to linear equations, this formula can be seen as constructing the solution of linear equation, namely: ....

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Jean-Yves Girard. Towards a Geometry of Interaction. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 69--108. American Mathematical Society, 1989.

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