On the Dependencies of Logical Rules

Abstract

Abstract. Many correctness criteria have been proposed since linear logic was introduced and it is not clear how they relate to each other. In this paper, we study proof-nets and their correctness criteria from the perspective of dependency, as introduced by Mogbil and Jacobé de Naurois. We introduce a new correctness criterion, called DepGraph, and show that together with Danos' contractibility criterion and Mogbil and Naurois criterion, they form the three faces of a notion of dependency which is crucial for correctness of proof-structures. Finally, we study the logical meaning of the dependency relation and show that it allows to recover and characterize some constraints on the ordering of inferences which are implicit in the proof-net.

Proof nets for multiplicative cyclic linear logic and Lambek calculus

“…Similarly to the standard (commutative) MLL case, the crucial point of the sequentialization procedure is given by the Splitting Lemma 5. Actually, as observed in [Bagnol et al 2015] in the MLL case, there exists an alternative way (straightforward, indeed, that skips the Splitting Lemma) to sequentialize McyLL proof nets, based on the following "contraction as parsing" strategy: if an APS is contractile then, by convergence of Σ, there exists a "contraction strategy" which starts by contracting the axioms links and whose retraction steps can be interpreted as instances of inference rules of a (possibly open) sequential proof; in the case of success, the retraction sequence ends up with a collapsed graph (as usual, a •-node) labeled by a closed sequent proof (see details in the appendix A.1).…”

“…In spite of other syntaxes based on correction graphs (set of tests) like "switchings" or "trips", retraction gives a direct (simpler, indeed) sequentialization procedure of correct proof nets without passing through a Splitting Lemma. The genuine idea [Bagnol et al 2015 (ii) by the following inference rules:…”

Infinets: The Parallel Syntax for Non-wellfounded Proof-Theory

“…To prove that S is deeply lock-free, we show that < D(S) ⊆< O(π) where n 1 < O(π) n 2 if n 1 is introduced below n 2 in π (see prop. 5 in [5]). To prove that S is widely lock-free, we will define a wait function, f .…”

Causality in Linear Logic

“…The causal structs σ 2 and σ 3 are very close to proof nets, and it is easy to see that σ 2 represents a correct proof net while σ 3 does not. In particular, there exists a proof P such that Tr (P ) ⊆ σ 2 but there are no such proof Q for σ 3 . Clearly, σ 3 should not be acyclic.…”

“…The notion of dependency between logical rules has also been studied in [3] in the case of MLL. From a proof net R, they build a partial order D`, ⊗ (R) which we believe is very related to P where P is a sequentialisation of R. Indeed, in the case of MLL without MIX a partial order is enough to capture the dependency between rules.…”