{"product_id":"non-classical-aspects-in-proof-complexity-von-olaf-beyersdorff","title":"Non-classical Aspects in Proof Complexity","description":"\u003cp\u003eProof complexity focuses on the complexity of theorem proving procedures, a\u003c\/p\u003e\u003cp\u003etopic which is tightly linked to questions from computational complexity (the\u003c\/p\u003e\u003cp\u003eseparation of complexity classes), first-order arithmetic theories (bounded arithmetic),\u003c\/p\u003e\u003cp\u003eand practical questions as automated theorem proving. One fascinating\u003c\/p\u003e\u003cp\u003equestion in proof complexity is whether powerful computational resources as randomness\u003c\/p\u003e\u003cp\u003eor oracle access can shorten proofs or speed up proof search. In this\u003c\/p\u003e\u003cp\u003edissertation we investigated these questions for proof systems that use a limited\u003c\/p\u003e\u003cp\u003eamount of non-uniform information (advice). This model is very interesting as¿-\u003c\/p\u003e\u003cp\u003ein contrast to the classical setting¿-it admits an optimal proof system as recently\u003c\/p\u003e\u003cp\u003eshown by Cook and Krajícek. We give a complete complexity-theoretic classification\u003c\/p\u003e\u003cp\u003eof all languages admitting polynomially bounded proof systems with advice\u003c\/p\u003e\u003cp\u003eand explore whether the advice can be simplified or even eliminated while still\u003c\/p\u003e\u003cp\u003epreserving the power of the system.\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003ePropositional proof systems enjoy a close connection to bounded arithmetic.\u003c\/p\u003e\u003cp\u003eCook and Krajícek (JSL¿07) use the correspondence between proof systems with\u003c\/p\u003e\u003cp\u003eadvice and arithmetic theories to obtain a very strong Karp-Lipton collapse result\u003c\/p\u003e\u003cp\u003ein bounded arithmetic: if SAT has polynomial-size Boolean circuits, then the\u003c\/p\u003e\u003cp\u003epolynomial hierarchy collapses to the Boolean hierarchy. Here we show that this\u003c\/p\u003e\u003cp\u003ecollapse consequence is in fact optimal with respect to the theory PV, thereby\u003c\/p\u003e\u003cp\u003eanswering a question of Cook and Krajícek.\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThe second main topic of this dissertation is parameterized proof complexity, a\u003c\/p\u003e\u003cp\u003eparadigm developed by Dantchev, Martin, and Szeider (FOCS¿07) which transfers\u003c\/p\u003e\u003cp\u003ethe highly successful approach of parameterized complexity to the study of proof\u003c\/p\u003e\u003cp\u003elengths. In this thesis we introduce a powerful two player game to model and\u003c\/p\u003e\u003cp\u003estudy the complexity of proofs in a tree-like Resolution system in a setting arising\u003c\/p\u003e\u003cp\u003efrom parameterized complexity. This game is also applicable to show strong\u003c\/p\u003e\u003cp\u003elower bounds in other tree-like proof systems. Moreover, we obtain the first lower\u003c\/p\u003e\u003cp\u003ebound to the general dag-like Parameterized Resolution system for the pigeonhole\u003c\/p\u003e\u003cp\u003eprinciple and study a variant of the DPLL algorithm in the parameterized setting.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783954040360\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9783954040360","offer_id":39458816983133,"sku":"9783954040360","price":19.95,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/4727f7b5-cb93-45a8-aec7-550bf25bb785.jpg?v=1776497187","url":"https:\/\/shop.autorenwelt.de\/en\/products\/non-classical-aspects-in-proof-complexity-von-olaf-beyersdorff","provider":"Autorenwelt Shop","version":"1.0","type":"link"}