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.Comment: v.6 Minor changes, new formulation of definitions, changed some
details in the proof