Light logics and optimal reduction: Completeness and complexity

Abstract

International audienceTyping of lambda-terms in elementary and light affine logic (EAL and LAL, respectively) has been studied for two different reasons: on the one hand the evaluation of typed terms using LAL (EAL, respectively) proof-nets admits a guaranteed polynomial (elementary, respectively) bound; on the other hand these terms can also be evaluated by optimal reduction using the abstract version of Lamping's algorithm. The first reduction is global while the second one is local and asynchronous. We prove that for LAL (EAL, respectively) typed terms, Lamping's abstract algorithm also admits a polynomial (elementary, respectively) bound. We also give a proof of its soundness and completeness (for EAL and LAL with type fixpoints), by using a simple geometry of interaction model (context semantics)

On context semantics and interaction nets

“…These bounds were previously known, but Dal Lago's proofs were much shorter. Then, his tool was used to prove strong bounds which were previously unknown [1,20]. We hope that our tool will lead to similar results.…”

Plugging-in proof development environments usingLocksinLF

“…In this section we give a shallow encoding of Elementary Affine Logic as presented in (Baillot et al 2007). This example will exemplify how locks can be used to deal with global syntactic constraints as in the promotion rule of Elementary Affine Logic.…”

“…Definition 4.5 (Elementary Affine Logic (Baillot et al 2007)). Elementary Affine Logic can be specified by the following rules:…”

“…The predicates involved in the locks are defined as follows: A few remarks are mandatory. The promotion rule in (Baillot et al 2007) is in effect a family of natural deduction rules with a growing number of assumptions. Our encoding achieves this via the auxiliary judgement V(·), whose effect is to allow for the user to mimick progressively the application of the promotion rule, as it is intuitively illustrated by the following steps (for the sake of readability and simplicity we drop the lock notation):…”

“…First, we encode in CLLF P Fitch-Prawitz consistent Set-Theory (FPST), as presented in (Prawitz 1965), and we illustrate its expressive power by showing, by way of example, how it can type all strongly normalizing terms. Then, we give signatures in CLLF P of a weakly normalizing λ-calculus and a system of Light Linear Logic (Baillot et al 2007). Finally, in Subsection 4.5, we show how to encode partial functions in CLLF P?…”