Bøger af Olaf Beyersdorff

Proof complexity focuses on the complexity of theorem proving procedures, atopic which is tightly linked to questions from computational complexity (theseparation of complexity classes), first-order arithmetic theories (bounded arithmetic),and practical questions as automated theorem proving. One fascinatingquestion in proof complexity is whether powerful computational resources as randomnessor oracle access can shorten proofs or speed up proof search. In thisdissertation we investigated these questions for proof systems that use a limitedamount of non-uniform information (advice). This model is very interesting as¿-in contrast to the classical setting¿-it admits an optimal proof system as recentlyshown by Cook and Krajícek. We give a complete complexity-theoretic classificationof all languages admitting polynomially bounded proof systems with adviceand explore whether the advice can be simplified or even eliminated while stillpreserving the power of the system.Propositional proof systems enjoy a close connection to bounded arithmetic.Cook and Krajícek (JSL¿07) use the correspondence between proof systems withadvice and arithmetic theories to obtain a very strong Karp-Lipton collapse resultin bounded arithmetic: if SAT has polynomial-size Boolean circuits, then thepolynomial hierarchy collapses to the Boolean hierarchy. Here we show that thiscollapse consequence is in fact optimal with respect to the theory PV, therebyanswering a question of Cook and Krajícek.The second main topic of this dissertation is parameterized proof complexity, aparadigm developed by Dantchev, Martin, and Szeider (FOCS¿07) which transfersthe highly successful approach of parameterized complexity to the study of prooflengths. In this thesis we introduce a powerful two player game to model andstudy the complexity of proofs in a tree-like Resolution system in a setting arisingfrom parameterized complexity. This game is also applicable to show stronglower bounds in other tree-like proof systems. Moreover, we obtain the first lowerbound to the general dag-like Parameterized Resolution system for the pigeonholeprinciple and study a variant of the DPLL algorithm in the parameterized setting.