Does Advice Help to Prove Propositional Tautologies?

One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [6], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajíček [5] have recently started the investigation of proof systems which are computed by poly-time functions using advice. While this yields a more powerful model, it is also less directly applicable in practice. In this note we investigate the question whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. In addition, we show that if advice is ”not very helpful” for proving tautologies, then there exists an optimal propositional proof system without advice. In our main result, we prove that advice can be transferred from the proof to the formula, leading to an easier computational model. We obtain this result by employing a recent technique by Buhrman and Hitchcock [4].