Theory of Finite or Infinite Trees Revisited

TL;DR

This paper introduces a first-order axiomatization of a theory of finite and infinite trees, providing a decision procedure and a constraint solver with explicit solutions, along with an implementation and performance comparison.

Contribution

It presents a complete first-order theory of trees with a decision procedure and a constraint solver, including an implementation in CHR and performance analysis.

Findings

The theory $T$ has at least one model and is complete.

The constraint solver transforms constraints into simple, explicit solutions.

Implementation in CHR performs competitively with other methods.

Abstract

We present in this paper a first-order axiomatization of an extended theory $T$ of finite or infinite trees, built on a signature containing an infinite set of function symbols and a relation $\fini(t)$ which enables to distinguish between finite or infinite trees. We show that $T$ has at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver which gives clear and explicit solutions for any first-order constraint satisfaction problem in $T$. The solver is given in the form of 16 rewriting rules which transform any first-order constraint $\phi$ into an equivalent disjunction $\phi$ of simple formulas such that $\phi$ is either the formula $\true$ or the formula $\false$ or a formula having at least one free variable, being equivalent neither to $\true$ nor to $\false$ and where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness of $T$. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.