Unsound Theorem Proving

Abstract

We discuss the benefits of complete unsound inference procedures for efficient methods of disproof. We give a framework for converting a sound and complete saturation-based inference procedure into successive unsound and complete procedures, that serve as successive approximations to the theory. The idea is to successively add new statements in such a way that the inference procedure will halt. Then the satisfiability is evaluated over a stronger theory. This gives an over-approximation to the given theory. We show how to successively compute better over-approximations. Similarly, a sound an incomplete theorem prover will give an under-approximation. In our framework, we succesively compute better over and under-approximations in this way.

We illustrate this framework with Knuth-Bendix Completion, and show that in some theories this method becomes a decision procedure. Then we illustrate the framework with a new method for the (nonground) word problem, based on Congruence Closure. We show a class where this becomes a decision procedure. Also, we show that this new inference system is interesting in its own right. Given a particular goal, in many cases we can halt the procedure at some point and say that all the equations for solving the goal have been generated already. This is generally not possible in Knuth-Bendix Completion.