Geometry of Resource Interaction – A Minimalist Approach

Abstract

The Resource λ -calculus is a variation of the λ -calculus where arguments can be superposed and must be linearly used. Hence it is a model for linear and non-deterministic programming languages, and the target language of Taylor-Ehrhard expansion of λ -terms. In a strictly typed restriction of the Resource λ -calculus, we study the notion of path persistence, and we define a Geometry of Interaction that characterises it, is invariant under reduction, and counts addends in normal forms.

Geometry of resource interaction and Taylor–Ehrhard–Regnier expansion: a minimalist approach

“…This paper extends with Sections 6-8 a previous work (Solieri 2015) by the author, who is grateful to Michele Pagani and Stefano Guerrini for their advice, to Lionel Vaux and Laurent Regnier for interesting discussions and to anonymous reviewers of this paper for numerous comments and suggestions.…”

“…For simply typed terms, the aforementioned Theorem 4 of this paper generalises such commutation property to any reduction sequence and to any choice of strategy, and is more directly formulated and proved on paths. This paper extends with Section 5-7 a previous work (Solieri, 2015) by the author, who is grateful to Michele Pagani and Stefano Guerrini for their advice, to Lionel Vaux and Laurent Regnier for interesting discussions, and to anonymous reviewers of this paper for numerous comments and suggestions.…”

“…The quest for an effective semantic characterisation of persistence separately produced three notions of paths: regularity, defined by a dynamic algebra (Girard 1989;Danos and Regnier 1995), legality, formulated by topological conditions cycles (Asperti and Laneve 1995), consistency, expressed as traces of a token-machine execution (Gonthier et al 1992) and developed to study the implementation (Lamping 1989;Kathail 1990) of Lévy-optimal reduction (Lévy 1978). These notions are equivalent (Asperti et al 1994), and their common core idea -describing computation by local and asynchronous conditions on routing of paths -inspired the design of efficient parallel abstract machines (Mackie 1995;Danos et al 1997;Laurent 2001;Pinto 2001;Pedicini and Quaglia 2007;Dal Lago et al 2014, 2015Pedicini et al Unpublished, amongst others). More recently, the geometry of interaction (GoI) approach has been fruitfully employed for semantic investigations which characterised quantitative properties of programmes, with respect to both time (Dal Lago 2009;Perrinel 2014;Aubert et al 2016) and space complexity (Aubert and Seiller 2016a,b;Mazza 2015b;Mazza and Terui 2015).…”