Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide.
We show that the normal form of the Taylor expansion of a $\lambda$-term is
isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof
along three independent directions. First, we simplify the final step of the
proof by following the left reduction strategy directly in the resource
calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a
groupoid of permutations of copies of arguments in a rigid variant of the
resource calculus, and relate the coefficients of Taylor expansion with this
structure, while Ehrhard and Regnier worked with groups of permutations of
occurrences of variables. Finally, we extend all the results to a
nondeterministic setting: by contrast with previous attempts, we show that the
uniformity property that was crucial in Ehrhard and Regnier's approach can be
preserved in this setting.
The aim of this work is to characterize three fundamental normalization proprieties in lambda-calculus trough the Taylor expansion. The general proof strategy consists in stating the dependence of ordinary reduction strategies on their resource counterparts and in finding a convenient resource term in the support of the Taylor expansion that behaves well under the considered kind of reduction.
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