Polynomial Time and Dependent Types

TL;DR

This paper integrates dependent types with linear type systems to ensure polynomial time computation, extending to full dependent types with resource guarantees, aiming to unify resource reasoning with type theory.

Contribution

It introduces two systems combining dependent types with polynomial time guarantees, extending them via Quantitative Type Theory, and proves their soundness using realisability techniques.

Findings

Two polynomial time systems with dependent types are proposed.

Both systems are extended to full dependent types with resource guarantees.

Soundness of the systems is formally proven.

Abstract

We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one, based on the LFPL system of Martin Hofmann, that controls construction via a payment method. Both of these are extended to full dependent types via Quantitative Type Theory, allowing for arbitrary computation in types alongside guaranteed polynomial time computation in terms. We prove the soundness of the systems using a realisability technique due to Dal Lago and Hofmann. Our long-term goal is to combine the extensional reasoning of type theory with intensional reasoning about the resources intrinsically consumed by programs. This paper is a step along this path, which we hope will lead both to practical systems for reasoning about programs' resource usage, and to theoretical use as a form of synthetic computational complexity theory.