Two algorithms in search of a type system

TL;DR

This paper extends the ATR programming formalism to include a broader class of affine recursions, enabling natural expression of sorting algorithms while maintaining polynomial-time complexity guarantees.

Contribution

It introduces an extension to ATR that allows direct expression of affine recursions, including sorting algorithms, without losing feasibility guarantees.

Findings

ATR programs run in type-2 polynomial-time

Extended ATR can express insertion- and selection-sort algorithms

Refined semantics ensure new recursions remain feasible

Abstract

The authors' ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable (ATR types are confined to levels 0, 1, and 2). A limitation of the original version of ATR is that the only directly expressible recursions are tail-recursions. Here we extend ATR so that a broad range of affine recursions are directly expressible. In particular, the revised ATR can fairly naturally express the classic insertion- and selection-sort algorithms, thus overcoming a sticking point of most prior implicit-complexity-based formalisms. The paper's main work is in refining the original time-complexity semantics for ATR to show that these new recursion schemes do not lead out of the realm of feasibility.