The logic of linear processes

We present the logic of linear processes (i.e. LLP), a calculus for the connectives of multiplicative additive linear logic (without ⊕) that defines the meaning of connectives in the style of a process calculus. 1 A transition-based view of linear logic Because we want to represent states as linear (and, more generally, as substructural) contexts, we want to define a logic in such a way that the idea of state transition is fundamental. In other words: A proposition defined by the effect it has on state. This is a justification for logic that is primarily pragmatist, as opposed to Martin Löf’s verificationist basis for logic.1 This is opposed to the large amount of previous work that “reads off” a state transition logic from a sequent calculus presentation of the logic. Cervesato and Scedrov’s journal paper on ω is both a prime example of this technique and a thorough review of related work in the area [CS09]. 1.1 Transitions We take states to be either individual resources A res, the empty resource ·, or combinations of resources (∆1,∆2), where the comma is commutative, associative, and has the unit ·. Resources have structure — in particular, they are all the connectives of multiplicative additive linear logic save for ⊕ — and the meaning of those connectives is given by transition rules. Multiplicative and 1In particular, the “cheats” are analogous. The fundamentally pragmatist connective of implication is handled by Martin Löf by treating its “verification” in terms of the pragmatist notion of hypothetical reasoning, and we handle the fundamentally verificationist notion of multiplicative conjunction by treating its “effect” in terms of the verificationist notion of resource combination.