Section: New Results

Disproving Non-Theorems with Saturating Search

Participants : Taus Brock-Nannestad, Kaustuv Chaudhuri.

High-performance automated reasoning techniques such as resolution and the inverse method are well suited for proving true conjectures, but are ill-behaved for false conjectures. For example, for a simple theory of even numbers that states that 0 is even and that $n+2$ is even whenever $n$ is even, it is obviously the case that the conjecture “3 is even” is unprovable, but the algorithm would loop forever proving “0 is even”, “2 is even”, “4 is even”, etc. This behavior is observed even in the best saturation-based (i.e., forward-reasoning) theorem provers.

In [25] (TABLEAUX'15), we show how to finitely constrain the search space of saturation-based theorem provers by the use of unsound extensions of the goal query. These unsound extensions, when combined with forward subsumption, guarantee that only a finite number of consequences would ever be constructed based on any goal query, so the proof search procedure is guaranteed to terminate. If a proof is found among them that does not use the unsound extensions, then we can can succeed with that proof. If no proof is found, then we can soundly assert that the original goal query was also unprovable, since even a weakened version of it was unprovable. The only other possibility is that a proof is found using the unsound extension; in this case, we use the particular instance of unsoundness to refine the original unsound goal to prevent it from being found again, while maintaining the invariant that the search space is finite, and rerun the search. Since first-order logic is undecidable, we may need to repeat the refinement procedure indefinitely, but for many kinds of domains, particularly those arising from typed signatures (such as the even numbers example above), we do eventually find a saturating approximation that guarantees that the conjecture has no proof.

This algorithm has been implemented as part of the Mætning theorem prover explained in the section on Software above. We plan to extend it in the future with various automatic refinement heuristics.