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 ML4. LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of nets (|7~|ʹ)/ ^ in LALL. Every net |7~|ʹ in the family simulates t on arguments whose bit length is bounded by the integer I. 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 ML4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of |7]ʹ depends on the number of composition and recursion schemes used to define t. namely the structural complexity of t. Moreover, the size of \t]ʹ is a polynomial in /, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called stratification, at the base of Light Linear Logic.