Safe Recursion on Notation into a Light Logic by Levels

We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels ( LALL), derived from the logic ML 4 . LALL is an intuitionistic deductive system, with a polynomial ti me cut elimination strategy. The embedding allows to represent every term t of SRN as a family of nets hdte l il2N in LALL. Every net dte l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the “fuzzy” borders of paragraph boxes that LALL inherits from ML 4 to “freely” duplicate the arguments, especially the safe ones, of t. Finally, the type of dte l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of dte l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principle s SRN relies on, and the proof theoretic one, called stratification, at the base of Light Linear Logic.