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The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
The Computational Meaning of Probabilistic Coherence Spaces
"... Abstract—We study the probabilistic coherent spaces — a denotational semantics interpreting programs by power series with non negative real coefficients. We prove that this semantics is adequate for a probabilistic extension of the untyped λcalculus: the probability that a term reduces to a head no ..."
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Abstract—We study the probabilistic coherent spaces — a denotational semantics interpreting programs by power series with non negative real coefficients. We prove that this semantics is adequate for a probabilistic extension of the untyped λcalculus: the probability that a term reduces to a head normal form is equal to its denotation computed on a suitable set of values. The result gives, in a probabilistic setting, a quantitative refinement to the adequacy of Scott’s model for untyped λcalculus. I.
Nonlinearity as the Metric Completion of Linearity
"... Abstract. We summarize some recent results showing how the lambdacalculus may be obtained by considering the metric completion (with respect to a suitable notion of distance) of a space of affine lambdaterms, i.e., lambdaterms in which abstractions bind variables appearing at most once. This forma ..."
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Abstract. We summarize some recent results showing how the lambdacalculus may be obtained by considering the metric completion (with respect to a suitable notion of distance) of a space of affine lambdaterms, i.e., lambdaterms in which abstractions bind variables appearing at most once. This formalizes the intuitive idea that multiplicative additive linear logic is “dense ” in full linear logic (in fact, a prooftheoretic version of the abovementioned construction is also possible). We argue that thinking of nonlinearity as the “limit ” of linearity gives an interesting point of view on wellknown properties of the lambdacalculus and its relationship to computational complexity (through lambdacalculi whose normalization is timebounded). 1 Linearity and Approximations The concept of linearity in logic and computer science, introduced over two decades ago [12], has now entered firmly into the “toolbox ” of proof theorists and functional programming language theorists. It is present, in one way or another,