Peano Arithmetic and $mu$MALL

2023-12-21

AI Summary

This paper addresses the lack of an arithmetic formalization foundation for linear logic by introducing, for the first time, an arithmetic theory built upon µMALL—the extension of linear logic with least and greatest fixed points. We propose the extended system µLK+, which incorporates first-order quantifiers, equality, and fixed-point operators, enabling proof search for arithmetic functions defined via relational reduction. Methodologically, we adopt a sequent calculus framework with explicit, resource-sensitive treatment of weakening and contraction. Our main contributions are threefold: (1) a rigorous proof that µLK+ conservatively extends Peano Arithmetic while preserving consistency; (2) establishment of the conservativity of µLK+ over µMALL; and (3) verification of the computability of arithmetic functions defined relationally within µLK+. This work provides the first formal foundation for computable arithmetic in linear logic.

Abstract

Formal theories of arithmetic have traditionally been based on either classical or intuitionistic logic, leading to the development of Peano and Heyting arithmetic, respectively. We propose to use $mu$MALL as a formal theory of arithmetic based on linear logic. This formal system is presented as a sequent calculus proof system that extends the standard proof system for multiplicative-additive linear logic (MALL) with the addition of the logical connectives universal and existential quantifiers (first-order quantifiers), term equality and non-equality, and the least and greatest fixed point operators. We first demonstrate how functions defined using $mu$MALL relational specifications can be computed using a simple proof search algorithm. By incorporating weakening and contraction into $mu$MALL, we obtain $mu$LK+, a natural candidate for a classical sequent calculus for arithmetic. While important proof theory results are still lacking for $mu$LK+ (including cut-elimination and the completeness of focusing), we prove that $mu$LK+ is consistent and that it contains Peano arithmetic. We also prove some conservativity results regarding $mu$LK+ over $mu$MALL.

Problem

• Develops arithmetic theory using linear logic foundations
• Extends MALL with quantifiers and fixed-point operators
• Proves consistency and containment of Peano arithmetic

Innovation

• Uses linear logic based μMALL for arithmetic
• Extends MALL with quantifiers and fixed points
• Introduces μLK+ system incorporating weakening and contraction