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An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
On the Use of Impredicative Reasoning to Construct a Class of Partial Models of ZF Within ZF PRELIMINARY UNPUBLISHED DRAFT
, 2008
"... Gödel’s incompleteness results show that if ZF is consistent, it is impossible to construct within ZF itself a single object (set) that represents a complete and precise semantic model of ZF. Nevertheless, it has also become clear since then that many kinds of partial models can be constructed withi ..."
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Gödel’s incompleteness results show that if ZF is consistent, it is impossible to construct within ZF itself a single object (set) that represents a complete and precise semantic model of ZF. Nevertheless, it has also become clear since then that many kinds of partial models can be constructed within ZF that reveal interesting characteristics of ZF and related formal systems. We develop here one particular class of partial models of ZF, which rely on an extreme form of impredicative reasoning that nevertheless follow the accepted rules of set theory and firstorder logic and are constructible within exactly the same variant of ZF as the one being modelled. Some study of these partial models yields interesting results. 1
unknown title
, 2009
"... The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning ..."
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The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning
unknown title
, 2009
"... The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning ..."
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The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning
unknown title
, 2009
"... The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning ..."
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The significance of Nathanson’s boss factor in legitimising Aristotle’s particularisation Why we need to revise current interpretations of Cantor’s, Gödel’s, Turing’s and Tarski’s formal reasoning
Extracting Herbrand trees in classical realizability using forcing ∗
, 2013
"... Krivine presented in [9] a methodology to combine Cohen’s forcing with the theory of classical realizability and showed that the forcing condition can be seen as a reference that is not subject to backtracks. The underlying classical program transformation was then analyzed by Miquel [11] in a fully ..."
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Krivine presented in [9] a methodology to combine Cohen’s forcing with the theory of classical realizability and showed that the forcing condition can be seen as a reference that is not subject to backtracks. The underlying classical program transformation was then analyzed by Miquel [11] in a fully typed setting in classical higherorder arithmetic (PAω +). As a case study of this methodology, we present a method to extract a Herbrand tree from a classical realizer of inconsistency, following the ideas underlying the compactness theorem and the proof of Herbrand’s theorem. Unlike the traditional proof based on Kőnig’s lemma (using a fixed enumeration of atomic formulas), our method is based on the introduction of a particular Cohen real. It is formalized as a proof in PAω +, making explicit the construction of generic sets in this framework in the particular case where the set of forcing conditions is arithmetical. We then analyze the algorithmic content of this proof.