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Several variants of linear logic have been proposed to characterizecomplexity classes in the proofs-as-programs correspondence. Light linear logic(LLL) ensures a polynomial bound on reduction time, and characterizes in thisway polynomial time (Ptime). In this paper we study the complexity of linearlogic proof-nets and propose three semantic criteria based on contextsemantics: stratification, dependence control and nesting. Stratification aloneentails an elementary time bound, the three criteria entail together apolynomial time bound.  These criteria can be used to prove the complexity soundness of severalexisting variants of linear logic. We define a decidable syntactic subsystem oflinear logic: SDNLL. We prove that the proof-nets of SDNLL satisfy the threecriteria, which implies that SDNLL is sound for Ptime. Several previoussubsystems of linear logic characterizing polynomial time (LLL, mL^4, maximalsystem of MS) are embedded in SDNLL, proving its Ptime completeness.

Paths-based criteria and application to linear logic subsystems characterizing polynomial time