## Section: New Results

### Proof Nets and the Linear Substitution Calculus

Participant : Beniamino Accattoli.

This work [21] belongs to line of work *Cost Models and Abstract Machines for Functional Programs*, supported by the ANR project COCA HOLA, and it has been published in the proceedings of the international conference ICTAC 2018.

The *Linear Substitution Calculus* (LSC) is a refinement of the $\lambda $-calculus that is crucial for the study of cost models for functional programs, as it enables a sharp and yet simple decomposition of the evaluation of $\lambda $-terms, and it is employed in the proof of various results about cost models in the literature.

In this work we show that the LSC is isomorphic to the linear logic representation of the $\lambda $-calculus. More precisely, it is isomorphic to the *proof nets* presentation of such a fragment of linear logic. Proof nets are a graphical formalism, which—as most graphical formalisms—is handy for intuitions but not prone to formal reasoning. The result is relevant because it allows to manipulate formally a graphical formalism (proof nets) by means of an ordinary term syntax (the LSC).