On Paths-Based Criteria for Polynomial Time Complexity in Proof-Nets

Several variants of linear logic have been proposed to characterize complexity classes in the proofs-as-programs correspondence. Light linear logic (LLL) ensures a polynomial bound on reduction time, and characterizes in this way the class Ptime. In this paper we study the complexity of linear logic proof-nets and propose two sufficient criteria, one for elementary time soundness and the other one for Ptime soundness, based on the study of paths inside the proof-net. These criteria offer several benefits: they provide a bound for any reduction sequence and they can be used to prove the complexity soundness of several variants of linear logic. As an example we show with our criteria a strong polytime bound for the system L-4 (light linear logic by levels).