# On the Elementary Affine Lambda-Calculus with and Without Fixed Points

## TL;DR

This paper investigates the elementary affine lambda-calculus, demonstrating that removing type fixpoints limits its expressive power to regular languages, and proposes improvements for characterizing complexity classes with recursive types.

## Contribution

It shows that without recursive types, the calculus characterizes regular languages, and introduces an aesthetic enhancement for representing FP and k-FEXPTIME with recursive types.

## Key findings

- Without fixpoints, the calculus characterizes regular languages.
- Semantic evaluation method proves the necessity of fixpoints for complexity classes.
- Improved characterization of FP and k-FEXPTIME with recursive types.

## Abstract

The elementary affine lambda-calculus was introduced as a polyvalent setting for implicit computational complexity, allowing for characterizations of polynomial time and hyperexponential time predicates. But these results rely on type fixpoints (a.k.a. recursive types), and it was unknown whether this feature of the type system was really necessary. We give a positive answer by showing that without type fixpoints, we get a characterization of regular languages instead of polynomial time. The proof uses the semantic evaluation method. We also propose an aesthetic improvement on the characterization of the function classes FP and k-FEXPTIME in the presence of recursive types.