A nonstandard standardization theorem

Abstract

Standardization is a fundamental notion for connecting programming languages and rewriting calculi. Since both programming languages and calculi rely on substitution for defining their dynamics, explicit substitutions (ES) help further close the gap between theory and practice. This paper focuses on standardization for the linear substitution calculus, a calculus with ES capable of mimicking reduction in λ-calculus and linear logic proof-nets. For the latter, proof-nets can be formalized by means of a simple equational theory over the linear substitution calculus.Contrary to other extant calculi with ES, our system can be equipped with a residual theory in the sense of Lévy, which is used to prove a left-to-right standardization theorem for the calculus with ES but without the equational theory. Such a theorem, however, does not lift from the calculus with ES to proof-nets, because the notion of left-to-right derivation is not preserved by the equational theory. We then relax the notion of left-to-right standard derivation, based on a total order on redexes, to a more liberal notion of standard derivation based on partial orders.Our proofs rely on Gonthier, Lévy, and Melliès' axiomatic theory for standardization. However, we go beyond merely applying their framework, revisiting some of its key concepts: we obtain uniqueness (modulo) of standard derivations in an abstract way and we provide a coinductive characterization of their key abstract notion of external redex. This last point is then used to give a simple proof that linear head reduction-a nondeterministic strategy having a central role in the theory of linear logic-is standard.

The Negligible and Yet Subtle Cost of Pattern Matching

“…A more thorough study is left to an eventual longer version of this work. In particular, our results are proved using simple calculi with explicit substitutions (ES) inspired by the linear substitution calculus-a variation over a λ-calculus with ES by Robin Milner [32] developed by Accattoli and Kesner [2,8]-in which both the search of the redex and α-renaming are left to the meta-level. To be formal, we should make both tasks explicit in the form of an abstract machine.…”

“…More recently, also the search of the redex has been factored out, bringing it back to the implicit level, making ES act at a distance, without percolating through the term. The paradigmatic framework of this simpler, at a distance approach is the linear substitution calculus (LSC), a variation over a λ-calculus with ES by Robin Milner [32] developed by Accattoli and Kesner [2,8]-a LSC-like calculus is used in forthcoming Sect. 4.…”

Beta reduction is invariant, indeed

“…There is a natural candidate for extending the approach to reduction to normal form: just iterate the (linear) head strategy on the arguments, obtaining the (linear) LO strategy, that does compute normal forms [1]. As we will show, for linear LO derivations the subterm property holds.…”

“…What is left to do, then, is to find the strategy X for explicit substitutions, which is both mechanisable and a high-level implementation of → LO . Unfortunately, the linear LO strategy → LO (defined in [1]) is mechanisable but the pair (→ LO , → LO ) does not form a high-level implementation system, since → LO lacks the syntactic bound property. (Just consider the family {t n } n∈N of terms that evaluate in a linear number of → steps to an output u n of size exponential in n.)…”

“…Such a framework allows an easy management of sharing and, in contrast to previous approaches to explicit substitution, it admits a theory of standardization [1]. Indeed, the proof passes through a fine quantitative study of the relationship between standard reductions for the λ-calculus and some special standard reductions for the linear substitution calculus.…”