Time-complexity semantics for feasible affine recursions (extended abstract)

TL;DR

This paper extends the ATR programming formalism to include a broader class of affine recursions, enabling the expression of algorithms like insertion and selection sort within a feasible complexity framework.

Contribution

It introduces an extension to ATR that allows direct expression of affine recursions, overcoming previous limitations and simplifying the time-complexity semantics for higher-type recursions.

Findings

ATR programs run in type-2 polynomial-time

Classic sorting algorithms are expressible in the extended ATR

The extended semantics facilitate analysis of higher-type recurrences

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 extending and simplifying the original time-complexity semantics for ATR to develop a set of tools for extracting and solving the higher-type recurrences arising from feasible affine recursions.