On the Expressive Power of Multiple Heads in CHR

TL;DR

This paper formally analyzes the expressive power of multi-headed rules in Constraint Handling Rules (CHR), proving that multiple heads increase the language's expressive capabilities, especially in relation to Turing completeness and encoding limitations.

Contribution

It provides the first formal proof that multiple heads in CHR enhance its expressive power and characterizes conditions under which single-headed CHR is Turing complete.

Findings

Multi-headed CHR is Turing complete with powerful enough constraint theories.

Single-headed CHR is not Turing complete without function symbols.

Multiple heads increase the expressive power and encoding capabilities of CHR.

Abstract

Constraint Handling Rules (CHR) is a committed-choice declarative language which has been originally designed for writing constraint solvers and which is nowadays a general purpose language. CHR programs consist of multi-headed guarded rules which allow to rewrite constraints into simpler ones until a solved form is reached. Many empirical evidences suggest that multiple heads augment the expressive power of the language, however no formal result in this direction has been proved, so far. In the first part of this paper we analyze the Turing completeness of CHR with respect to the underneath constraint theory. We prove that if the constraint theory is powerful enough then restricting to single head rules does not affect the Turing completeness of the language. On the other hand, differently from the case of the multi-headed language, the single head CHR language is not Turing powerful when the underlying signature (for the constraint theory) does not contain function symbols. In the second part we prove that, no matter which constraint theory is considered, under some reasonable assumptions it is not possible to encode the CHR language (with multi-headed rules) into a single headed language while preserving the semantics of the programs. We also show that, under some stronger assumptions, considering an increasing number of atoms in the head of a rule augments the expressive power of the language. These results provide a formal proof for the claim that multiple heads augment the expressive power of the CHR language.