Survey on Dowing Graphs and Graph-Theoretic Data Structures |
However, invalid formulas (those thatare not entailed by agiven theory), cannot always be recognized. In addition, a consistent formaltheory that contains the first-order theory of the naturalnumbers (thus having certain "proper axioms"), by Gödel's incompleteness theorem, containstrue statements which cannot be proven. In these cases, an automated theoremprover may fail to terminate while searching for a proof. Despite thesetheoretical limits, in practice, theorem provers can solve many hard problems,even in these undecidable logics. Asimpler, but related, problem is proofverification, where an existing proof for a theorem is certified valid.For this, it is generally required that each individual proof step can beverified by a primitive recursive function orprogram, and hence the problem is always decidable. Sincethe proofs generated by automated theorem provers are typically very large, theproblem of proof compression is crucial and varioustechniques aiming at making the prover's output smaller, and consequently moreeasily understandable and checkable, have been developed.