Proof Nets and the Linear Substitution Calculus

Since the very beginning of the theory of linear logic it is known how to represent the $\lambda$-calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of sharing---the exponentials---and a micro-step operational semantics, while the $\lambda$-calculus has no sharing and a small-step operational semantics. Here we show that the \emph{linear substitution calculus}, a simple refinement of the $\lambda$-calculus with sharing, is isomorphic to proof nets at the operational level. Nonetheless, two different terms with sharing can still have the same proof nets representation---a further result is the characterisation of the equality induced by proof nets over terms with sharing. Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature.