On the Deduction Theorem and Complete Disjoint NP-Pairs

In this paper we ask the question whether the extended Frege proof system EF satisfies a weak version of the deduction theorem. We prove that if this is the case, then complete disjoint NP-pairs exist. On the other hand, if EF is an optimal proof system, then the weak deduction theorem holds for EF . Hence the weak deduction property for EF is a natural intermediate condition between the optimality of EF and the completeness of its canonical pair. We also exhibit two conditions that imply the completeness of the canonical pair of Frege systems.