Decidability properties for fragments of CHR

TL;DR

This paper investigates the decidability of termination in two restricted dialects of Constraint Handling Rules (CHR), showing that both are less expressive than Turing Machines and establishing decidability results for their termination properties.

Contribution

The paper introduces two restricted CHR dialects and proves their termination and infinite computation decidability, respectively, demonstrating their limited expressiveness compared to Turing Machines.

Findings

Existence of infinite computation is decidable in the first dialect.

Existence of terminating computation is decidable in the second dialect.

Both dialects are strictly less expressive than Turing Machines.

Abstract

We study the decidability of termination for two CHR dialects which, similarly to the Datalog like languages, are defined by using a signature which does not allow function symbols (of arity >0). Both languages allow the use of the = built-in in the body of rules, thus are built on a host language that supports unification. However each imposes one further restriction. The first CHR dialect allows only range-restricted rules, that is, it does not allow the use of variables in the body or in the guard of a rule if they do not appear in the head. We show that the existence of an infinite computation is decidable for this dialect. The second dialect instead limits the number of atoms in the head of rules to one. We prove that in this case, the existence of a terminating computation is decidable. These results show that both dialects are strictly less expressive than Turing Machines. It is worth noting that the language (without function symbols) without these restrictions is as expressive as Turing Machines.