Incompleteness in the Finite Domain

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Khaniki, Erfan | 2019

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 52287 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ardeshir, Mohammad
Abstract:
In this thesis, we study proof complexity conjectures and also introduce their mathematical logic equivalents in terms of provability and unprovability in strong enough first-order arithmetical theories. One of the most important conjectures in this theory is the following conjecture. The non-existence of an optimal proof system for propositional tautologies: In general, a proof system is a computable function in polynomial time such that its range is exactly the set of tautologies. We say proof system P, polynomially simulates proof system Q if and only if there exists a polynomial h such that for all tautologies such as A and for all proofs like a, if Qpaq A, then there exists a proof like b such that |b| ď hp|a|q and Ppbq A. We say proof system P is optimal if and only if it polynomially simulates all proof systems. The important conjecture about this definition is that the set of propositional tautologies does not have an optimal proof system. Considering the importance of this conjecture, we construct relativized worlds to show the independence of this conjecture from some other conjectures
Keywords:
Mathematical Logic ; Proof Complenity ; Proof System ; Finite Consistency ; Search Problems

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