Safe Recursion on Notation into a Light Logic by Levels

TL;DR

This paper embeds Safe Recursion on Notation into Light Affine Logic by Levels, enabling polynomial-size proof net representations that simulate SRN terms with bounded argument sizes, bridging recursive and proof theoretic principles.

Contribution

It introduces a novel embedding of SRN into LALL using recursive Scott numerals and fuzzy paragraph boxes, linking recursive complexity with light linear logic.

Findings

Proof nets simulate SRN terms with size polynomial in argument bound l

Embedding captures the structural complexity of SRN in proof net size

Bridges the gap between recursive principles and stratified proof theoretic systems

Abstract

We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic L4. 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 proof nets |t|^l in LALL. Every proof net |t|^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 proof 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 L4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of |t|^l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of |t|^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 principles SRN relies on, and the proof theoretic one, called /stratification/, at the base of Light Linear Logic.