Proof-Net as Graph, Taylor Expansion as Pullback

Abstract

We introduce a new graphical representation for multiplicative and exponential linear logic proof-structures, based only on standard labelled oriented graphs and standard notions of graph theory. The inductive structure of boxes is handled by means of a box-tree. Our proof-structures are canonical and allows for an elegant definition of their Taylor expansion by means of pullbacks.

Gluing resource proof-structures: inhabitation and inverting the Taylor expansion

“…Following [GPT19], we adopt an alternative non-inductive approach, which strongly refines [GPT16]: the Taylor expansion is defined pointwise (see Example 5.2 and Figure 5). Indeed, proof-structures have a tree structure made explicit by their box-function.…”

“…As the study of cut-elimination is left to future work, our interest for DiLL is just to have an unitary syntax subsuming both MELL and DiLL 0 : this is why, unlike [Pag09,Tra11], our DiLL proof-structures are not allowed to contain a set of DiLL proof-structures inside a box. We reuse the syntax of proof-structures given in [GPT19], based on the graph notions introduced in Section 3. Note that, unlike [PT09], we allow the presence of cuts (vertices of type cut).…”

“…In fact, as somehow anticipated by Boudes [Bou09], such a Taylor expansion operation can be carried on any tree-like structure. This primitive, abstract, notion of Taylor expansion can then be pulled back to the structure of interest, as shown in [GPT19] and put forth again here.…”

“…(1) Following [GPT19], we introduce rigorous definitions of all the notions of graph theory (Section 3) involved in the definition of proof-structures and Taylor expansion. In this way, we can present here a purely graphical definition of MELL proof-structures (Section 4), so as to keep Girard's original intuition of a proof-structure as a graph even in MELL and to avoid ad hoc technicalities to identify the border and the content of a box.…”

A Deep Inference System for Differential Linear Logic

“…What is also interesting is that LL promotion rule can be encoded in DiLL 0 through the notion of syntactic Taylor expansion [8,10,26,29,15,3,16,17]: a proof in LL can be decomposed into a possibly infinite set of (promotion-free) proofs in DiLL 0 . Given a proof in LL with exactly one promotion rule !p, the idea is to replace !p (which makes the resource π available at will) with an infinite set of "differential" proofs in DiLL 0 , each of them taking n ∈ N copies of π so as to make the resource π available exactly n times.…”