On the Reconstruction of Proofs in Distributed Theorem Proving: a Modified Clause-Diffusion Method

Proof reconstruction is the operation of extracting the computed prooffrom the trace of a theorem-proving run. We study the problem of proof reconstruction indistributed theorem provingbecause of the distributed nature of the derivation and especially because of deletions of clauses bycontraction,it may happen that a deductive process generates the empty clause, but does not have all the necessary information to reconstruct the proof. We analyse this problem and we present a method for distributed theorem proving, calledModified Clause-Diffusion, which guarantees that the deductive process that generates the empty clause will be able to reconstruct the distributed proof. This result is obtained without imposing a centralized control on the deductive processes or resorting to a round of post-processing with ad hoc communication. We prove that Modified Clause-Diffusion is fair (hence complete) and guarantees proof reconstruction. First we define a set of conditions, next we prove that they are sufficient for proof reconstruction, then we show that Modified Clause-Diffusion satisfies them. Fairness is proved in the same way, which has the advantage that the sufficient conditions provide a treatment of the problem relevant for distributed theorem proving in general.