Confluence of Pure Differential Nets with Promotion

Abstract

Abstract. We study the confluence of Ehrhard and Regnier's differential nets with exponential promotion, in a pure setting. Confluence fails with promotion and codereliction in absence of associativity of (co)contractions. We thus introduce it as a necessary equivalence, together with other optional ones. We then prove that pure differential nets are Church-Rosser modulo such equivalences. This result generalizes to linear logic regular proof nets, where the same notion of equivalence was already studied in the literature, but only with respect to the problem of normalization in a typed setting. Our proof uses a result of finiteness of developments, which in this setting is given by strong normalization when blocking a suitable notion of "new" cuts.

An introduction to differential linear logic: proof-nets, models and antiderivatives

“…For confluence, one needs to introduce an equivalence relation on proof-structures which expresses typically that contraction is associative, see[Tra09]. For normalization, some conditions have to be satisfied by k; typically, it holds if one assumes that k = N but difficulties arise if k has additive inverses.…”

Strong normalization property for second order linear logic

“…Let us stress that the str-measure is really sharp and it can be generalized to other notions of net (like for example differential nets [25,26]). 15 One has to refuse ( /cc) steps, essentially because they are not ''local''; for the same reason ( /cc) and (ccad) steps of [7] and [24] can be performed only in presence of correctness, and of a much stronger notion of correctness then AC.…”

“…Furthermore, the str-measure defined in Section 3 is rather sharp and can be generalized to other notions of net (like for example differential nets [25,26]); it should be a useful tool also for λ-calculus with explicit substitutions.…”

The conservation theorem for differential nets

“…The reason for taking them into consideration is that cut-elimination in DiLL fails to give a confluent rewriting, not even locally. Indeed we cannot ignore associativity of (co)contractions and neutrality of (co)weakening over (co)contraction (see [Tra09a] and the discussion of subsection 2.3). This prompts us to introduce the former as an equivalence and the latter as a reduction.…”