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Polarised Intermediate Representation of Lambda Calculus with Sums

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Abstract

The theory of the λ-calculus with extensional sums is more complex than with only pairs and functions. We propose an untyped representation—an intermediate calculus—for the λ-calculus with sums, based on the following principles: 1) Computation is described as the reduction of pairs of an expression and a context; the context must be represented inside-out, 2) Operations are represented abstractly by their transition rule, 3) Positive and negative expressions are respectively eager and lazy; this polarity is an approximation of the type. We offer an introduction from the ground up to our approach, and we review the benefits. A structure of alternating phases naturally emerges through the study of normal forms, offering a reconstruction of focusing. Considering further purity assumption, we obtain maximal multi-focusing. As an application, we can deduce a syntax-directed algorithm to decide the equivalence of normal forms in the simply-typed λ-calculus with sums, and justify it with our intermediate calculus.
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Dates and versions

hal-01160579 , version 1 (05-06-2015)
hal-01160579 , version 2 (03-12-2015)

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Guillaume Munch-Maccagnoni, Gabriel Scherer. Polarised Intermediate Representation of Lambda Calculus with Sums. Thirtieth Annual ACM/IEEE Symposium on Logic In Computer Science (LICS 2015), Jul 2015, Kyoto, Japan. ⟨10.1109/LICS.2015.22⟩. ⟨hal-01160579v2⟩

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