Logarithmic Space and Permutations

TL;DR

This paper extends Girard's proof-based approach to computational complexity, demonstrating that operators on Hilbert Space can characterize the class L of logarithmic space languages, building on prior work on co-NL.

Contribution

It introduces a novel set of operators on Hilbert Space that characterize the class L, expanding the proof-as-programs framework to logarithmic space complexity.

Findings

Operators on Hilbert Space characterize L

Extension of Girard's framework to logarithmic space

Provides new proof-theoretic characterization of L

Abstract

In a recent work, Girard proposed a new and innovative approach to computational complexity based on the proofs-as-programs correspondence. In a previous paper, the authors showed how Girard proposal succeeds in obtaining a new characterization of co-NL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages.