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23
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 46 (11 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.
Enumeration reducibility, nondeterministic computations and relative computability of partial functions
 in Recursion Theory Week, Proceedings Oberwolfach
, 1989
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Undefinable classes and definable elements in models of set theory and arithmetic
 Proc. Amer. Math. Soc
, 1988
"... ABSTRACT. Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class " X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2 " ma ..."
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Cited by 2 (0 self)
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ABSTRACT. Every countable model M of PA or ZFC, by a theorem of S. Simpson, has a "class " X which has the curious property: Every element of the expanded structure (M, X) is definable. Here we prove: THEOREM A. Every completion T of PA has a countable model M (indeed there are 2 " many such M 's for each T) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class X to M. THEOREM B. Let M 1 = ZF + "V = HOD " be a wellfounded model of any cardinality. There exists an undefinable class X such that the definable points of M and (M, X) coincide. THEOREM C. Let M t = PA or ZF +"V = HOD". There exists an undefinable class X such that the definable points of M and (M, X) coincide if one of the conditions below is satisfied. (A) The definable elements o/M are cofinal in M. (B) M is recursively saturated and cf (M) = uj. Let M be a model of Peano arithmetic PA (or ZermeloFraenkel set theory ZF).
Automatic Forcing and Genericity: On the Diagonalization Strength of Finite Automata
 In Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science
, 2003
"... Algorithmic and resourcebounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension f ..."
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Cited by 2 (0 self)
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Algorithmic and resourcebounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension functions computable by nite automata which are tailored for capturing diagonalizations over regular sets and functions. We show that the generic sets obtained either by the partial regular extension functions of any given xed constant length or by all total regular extension of constant length are just the sets with saturated (also called disjunctive) characteristic sequences. Here a sequence is saturated if every string occurs in as a substring. We also show that these automatic generic sets are not regular but may be context free. Furthermore, we introduce stronger automatic genericity notions based on regular extension functions of nonconstant length and we show that the corresponding generic sets are biimmune for the classes of regular and context free languages.
Rigidity and biinterpretability in the hyperdegrees
, 2005
"... Slaman and Woodin have developed and used settheoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In par ..."
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Cited by 1 (0 self)
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Slaman and Woodin have developed and used settheoretic methods to prove some remarkable theorems about automorphisms of, and de…nability in, the Turing degrees. Their methods apply to other coarser degree structures as well and, as they point out, give even stronger results for some of them. In particular, their methods can be used to show that the hyperarithmetic degrees are rigid and biinterpretable with second order arithmetic. We give a direct proof using only older coding style arguments to prove these results without any appeal to settheoretic or metamathematical considerations. Our methods also apply to various coarser reducibilities.
Recognizing Tautology by a Deterministic Algorithm Whose Whileloop’s Execution Time is Bounded by Forcing
, 1997
"... By Bennet and Gill [4], it is shown that if A is a random oracle then TAUTA / ∈ PA with probability 1, where TAUTA denotes the collection of all tautologies relative to A. Extending Dowd’s work [6], we present a forcing argument to bound execution time of a whileloop of a deterministic algorithm, b ..."
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Cited by 1 (1 self)
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By Bennet and Gill [4], it is shown that if A is a random oracle then TAUTA / ∈ PA with probability 1, where TAUTA denotes the collection of all tautologies relative to A. Extending Dowd’s work [6], we present a forcing argument to bound execution time of a whileloop of a deterministic algorithm, by which we show that for each positive integer r, if A is an rgeneric oracle in the sense of Dowd then rTAUTA≡PT TAUT ⊕ A, where rTAUTA denotes the collection of all rquery tautologies with respect to A. As a consequence, the following two assertions are equivalent: (i) if A is a random oracle then rTAUTA / ∈ PA with probability 1, (ii) R 6 = NP. 1
Proving Induction
, 2011
"... Abstract: The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in zfc, states that a p ..."
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Abstract: The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in zfc, states that a predictive function M exists with the following property: whatever world we live in, M correctly predicts the world’s present state given its previous states at all times apart from a wellordered subset. On the usual model of time a wellordered subset is small relative to the set of all times. M’s existence therefore seems to provide a solution to the hard problem. My paper argues for two conclusions. First, the theorem does not solve the hard problem of induction. More positively though, it solves a version of the problem in which the structure of time is given modulo our choice of set theory. 1 the hard problem Call the task of persuading an inductive sceptic that inductive inference to new conclusions is truthconducive the hard problem of induction. Inductive
BOOLEAN ALGEBRAS AND LOGIC
, 809
"... Abstract. In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will ..."
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Abstract. In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone’s theorem from Boolean algebra. We mention here that the method we used can also be extended to first order logic, yet we will not go for it in this paper. 1.
Zermelo's WellOrdering Theorem in Type Theory
"... Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, wi ..."
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Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1