Generalised species of rigid resource terms

Abstract

This paper introduces a variant of the resource calculus, the rigid resource calculus, in which a permutation of elements in a bag is distinct from but isomorphic to the original bag. It is designed so that the Taylor expansion within it coincides with the interpretation by generalised species of Fiore et al., which generalises both Joyal's combinatorial species and Girard's normal functors, and which can be seen as a proofrelevant extension of the relational model. As an application, we prove the commutation between computing Böhm trees and (standard) Taylor expansions for a particular nondeterministic calculus.

On the Taylor expansion of $\lambda$-terms and the groupoid structure of their rigid approximants

“…Step 4: apart from this change of focus, the general architecture of our approach does not depart much from that of Ehrhard and Regnier, but we believe the obtained combinatorial results are closer to the original intuition behind the definition of m. In fact, a notable intermediate result (Lemma 5.11, p.24) is that the function that maps each permutation term to the permutation it induces on the occurrences of a fixed variable is functorial: one might understand Ehrhard and Regnier's proof of Step 4 as the image of ours through that functor. Moreover, our study suggests interesting connections with otherwise independent approaches to denotational semantics based on generalized species of structures [FGHW08,TAO17] and rigid intersection type systems [MPV18].…”

“…By contrast, for our purposes, it is essential to keep ⊕ as a free binary operator: following Tsukada, Asada and Ong [TAO17], we keep track of the branching structure of choices along the reduction. This information will be reflected in the Taylor expansion to be introduced in Section 4: this is the key to recover the uniformity property while allowing for nondeterministic superpositions of terms.…”

“…It is indeed most natural to compare our proposals to the line of work of Tsukada, Asada and Ong [TAO17,TAO18]. On the one hand, Tsukada et al thrive to develop an abstract understanding of reduction paths in a nondeterministic λ-calculus.…”

“…As stated before, our proposal (ii) to restore uniformity in a nondeterministic setting is valid only because the resource calculus keeps a syntactic track of choices. The corresponding constructors are exactly those used by Tsukada, Asada and Ong [TAO17] who were interested in identifying equivalent execution paths of nondeterministic programs, but those authors do not mention, nor rely upon any coherence property: this forbids Steps 1 to 4 and, instead, they depend on infinite sums of arbitrary coefficients to be well defined. By contrast, Dal Lago and Leventis have independently proposed nearly the same solution as ours [LL19, Section 2.2], with only a minor technical difference in the case of sums.…”

The (In)Efficiency of interaction

“…Our study does not really depend on their sequential structure, we only need to use bijections between multi sets, to describe the SIAM, and these bijections are just more easily managed using sequences rather than multi sets. This rigid approach has been already used fruitfully by Tsukada et al [2017] and Mazza et al [2018].…”