Coinductive Soundness of Corecursive Type Class Resolution

TL;DR

This paper proves that corecursive type class resolution in Haskell is coinductively sound with respect to greatest Herbrand models, but not inductively sound, and discusses related incompleteness results.

Contribution

It establishes the coinductive soundness of corecursive type class resolution and introduces new theoretical insights into its limitations and properties.

Findings

Corecursive resolution is coinductively sound for greatest Herbrand models.

It is not inductively sound for least Herbrand models.

Incompleteness results are established for various proof system fragments.

Abstract

Horn clauses and first-order resolution are commonly used to implement type classes in Haskell. Several corecursive extensions to type class resolution have recently been proposed, with the goal of allowing (co)recursive dictionary construction where resolution does not termi- nate. This paper shows, for the first time, that corecursive type class resolution and its extensions are coinductively sound with respect to the greatest Herbrand models of logic programs and that they are induc- tively unsound with respect to the least Herbrand models. We establish incompleteness results for various fragments of the proof system.