A tier-based typed programming language characterizing Feasible Functionals

TL;DR

This paper introduces a tier-based typed programming language that characterizes the class of Basic Feasible Functionals, linking higher-order complexity with programming theory through a decidable type system.

Contribution

It provides the first programming language with a type system that guarantees the complexity bounds of BFF$_2$, using an imperative language with oracle calls and a decidable inference algorithm.

Findings

Type inference runs in cubic time.

Language accurately characterizes BFF$_2$.

Decidable type inference contrasts with previous intractable methods.

Abstract

The class of Basic Feasible Functionals BFF$_2$ is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF$_2$ based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not overly constrain the expressive power of the language.