Abstract
The logic of proofs is known to be complete for the semantics of proofs in Peano Arithmetic PA. In this paper we present a refinement of this theorem, we will show that we can assure that all the operations on proofs can be realized by feasible, that is PTIME-computable, functions. In particular we will show that the logic of proofs is complete for the semantics of proofs in Buss' bounded arithmetic S$^{1}_{2}$ . In view of recent applications of the Logic of Proofs in epistemology this result shows that explicit knowledge in the propositional framework can be made computationally feasible.
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