Semantic Selection of Premisses for Automated Theorem Proving

We develop and implement a novel algorithm for discovering the optimal sets of premisses for proving and disproving conjectures in first-order logic. The algorithm uses interpretations to semantically analyze the conjectures and the set of premisses of the given theory to find the optimal subsets of the premisses. For each given conjecture the algorithm repeatedly constructs interpretations using an automated model finder, uses the interpretations to compute the optimal subset of premisses (based on the knowledge it has at the point) and tries to prove the conjecture using an automated theorem prover. 1 Importance of selecting appropriate premisses in automated theorem proving A proper set of premisses1 can be essential for proving a conjecture by an automated theorem prover. Clearly, the larger the number of the initial premisses the larger the number of the inferred formulae. And as for the most proving techniques the number of inferred formulae is in general super-exponential in the number of input formulae, the impact on the performance of an automated theorem prover is quite significant. Removing even a single superfluous premiss can make the difference between being or not being able to prove the conjecture. This major problem is even more serious when reasoning within large mathematical theories, which can contain hundreds of premisses and thousands of conjectures. In such cases, selecting proper premisses for proving the conjectures becomes a necessity. In our previous work [Pud06a, Pud06b] we described a method for compacting proofs of conjectures. By constructing lemmas from the proofs and by syntactically analyzing both the lemmas and the proofs, we constructed sets of premisses that produced shorter proofs of given conjectures, or allowed to construct proofs much faster. However, the assumption was that for each conjecture we already know some set of premisses from which we were able to prove the conjecture (although possibly redundant and inefficient for constructing a proof). If we were not able to prove the conjecture at all, the situation was more complicated. Using the notion axioms for the formulae the prover uses as assumptions sometimes causes confusion, therefore we shall prefer the notion premisses instead. See section 2.1. One possibility is to syntactically analyze the formulae and/or use an AI algorithm for guessing the proper premisses. Successful examples of such procedures are Josef Urban’s tools for the Mizar Project [Urb06a, Urb06b], reducing axiom sets in software verification [RS98] or filtering of axioms for machine-generated problems [MP06]. Although even simple syntactic heuristics can be very effective, the syntactic approach is generally restricted to the cases when the (syntactic) structure of formulae well reflects their semantics. Clearly, this is not always the case. Moreover, the syntactic analysis is usually only a heuristic procedure that tries to learn what premisses could be needed but is likely to fail on a new kind of problem. Syntactic filters are also often incomplete in the sense that they can eventually remove too many premisses. The procedure we shall describe in this article uses semantic analysis. By observing which formulae are true in which interpretations, we can get a deeper insight into the nature of a conjecture and compute such a set of premisses that is proper for proving the conjecture. As far as we are aware, this is a novel approach, which has not been researched before. We shall focus on two interconnected goals: • Determine which sets of premisses are sufficient for proving the conjecture; • among those sets, choose such a set that contains no redundant premisses and that is minimal with respect to some criterion. The criterion can be just a simple one – to minimize the number of premisses, or a more complex one, for example, to avoid premisses of a certain kind that complicate the proving process. 2 Semantic analysis of problem using interpretations