On Sharing, Memoization, and Polynomial Time (Long Version)

TL;DR

This paper investigates how sharing and memoization in evaluation mechanisms influence the class of polynomial-time computable functions, establishing invariance and confirming the soundness of ramified recurrence for polynomial time.

Contribution

It proves the invariance of a cost model with sharing and memoization and confirms ramified recurrence as sound for polynomial-time computation, solving an open problem.

Findings

Cost model with sharing is invariant

Ramified recurrence is sound for polynomial time

Open problem in implicit computational complexity settled

Abstract

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.