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A BottomUp Approach to Foundations of Mathematics
"... this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject ..."
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this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject
Higher complexity search problems for bounded arithmetic and
, 2010
"... a formalized nogap theorem ..."
Dynamic ordinals – universal measures for implicit computational complexity
, 2002
"... We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordin ..."
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We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordinals as universal measures for implicit computational complexity. I.e., we describe the connections between generalised dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories. In particular, through the determination of generalised dynamic ordinals we reobtain wellknown independence results for relativised bounded arithmetic theories.
Bounded Arithmetic
, 2008
"... Definable functions Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Nonlogical symbols: {0,1,+, ·, ≤} + {.,#,...} x  = length of binary representation of x x#y = 2 x·y  produces polynomial growth rate Arnold Beckmann (joint work with Klaus ..."
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Definable functions Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Nonlogical symbols: {0,1,+, ·, ≤} + {.,#,...} x  = length of binary representation of x x#y = 2 x·y  produces polynomial growth rate Arnold Beckmann (joint work with Klaus Aehlig)
Prague
"... We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S1 2 proves NP ⊆ P/poly iff S12 proves coNP ⊆ proves that NP/O(log n). (iv) If S1 2 proves NP ⊆ ..."
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We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S1 2 proves NP ⊆ P/poly iff S12 proves coNP ⊆ proves that NP/O(log n). (iv) If S1 2 proves NP ⊆ P/poly then S12 the Polynomial Hierarchy collapses to PNP [log n]. (v) If S2 2 proves NP ⊆ P/poly then S2 2 proves that the Polynomial Hierarchy collapses to P NP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them. 1
Abstract Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic
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unknown title
, 2005
"... Some open problems in bounded arithmetic and propositional proof complexity (research proposal paper) $Id: alanorp.tex,v 1.5 2002/12/10 04:57:36 alan Exp $ LATEX’d on January 3, 2005 ..."
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Some open problems in bounded arithmetic and propositional proof complexity (research proposal paper) $Id: alanorp.tex,v 1.5 2002/12/10 04:57:36 alan Exp $ LATEX’d on January 3, 2005
or
, 2005
"... The broad relevance and importance of bounded arithmetic and propositional proof complexity are well appreciated; these subjects are two of three which are interconnected in ..."
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The broad relevance and importance of bounded arithmetic and propositional proof complexity are well appreciated; these subjects are two of three which are interconnected in