General Ramified Recurrence and Polynomial-time Completeness

TL;DR

This paper presents a formal framework that precisely characterizes functions computable in polynomial time using structural recursion over inductively defined data structures like lists and trees, ensuring both soundness and completeness.

Contribution

It introduces a sound and complete implicit-complexity formalism for polynomial-time functions defined by structural recursion over inductive data structures.

Findings

The formalism is sound, guaranteeing feasible run times for programs.

The formalism is complete, capturing all polynomial-time functions defined by structural recursion.

Applicable to data structures like lists and trees with data sharing.

Abstract

We exhibit a sound and complete implicit-complexity formalism for functions feasibly computable by structural recursions over inductively defined data structures. Feasibly computable here means that the structural-recursive definition runs in time polynomial in the size of the representation of the inputs where these representations may make use of data sharing. Inductively defined data structures here includes lists and trees. Soundness here means that the programs within the implicit-complexity formalism have feasible run times. Completeness here means that each function computed by a feasible structural recursion has a program in the implicit-complexity formalism. This paper is a follow up on the work of Avanzini, Dal Lago, Martini, and Zorzi who focused on the soundness of such formalisms but did not consider the question of completeness.