On the Complexity of Propositional Proof Systems

In the thesis we have investigated the complexity of proofs in several propositional proof systems. Our main motivation has been to contribute to the line of research started by Cook and Reckhow to obtain how much knowledge as possible about the complexity of di erent proof systems to show that there is no super proof system. We also have been motivated by more applied questions concerning the automatic generation of Theorems. The results presented in the thesis were obtained in the papers [15, 14, 16] Regarding the complexity of proofs we prove both lower and upper bounds for the size of the proofs in several proof systems (resolution and some of its restrictions, Cutting Planes, Polynomial Calculus, Frege systems). Our results give better or new separations between such proof systems. On the other hand our work also concerns with automated theorem proving questions in resolution and Polynomial Calculus. Some of our results imply that restricting the search space to seek for resolution refutations is not always a good strategy. We show that a recently proposed algorithm that nds resolution proofs cannot have a good performance. We also compare it with the Grobner basis algorithm used to nd proofs in Polynomial Calculus, a proof systems based on polynomials.