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36
Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definabl ..."
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Cited by 34 (13 self)
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The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
The Bounded Injury Priority Method and the Learnability of Unions of Rectangles
, 1994
"... We develop a bounded version of the finite injury priority method in recursion theory. We use this to study the learnability of unions of rectangles over the domain f0; : : : ; n \Gamma 1g d with only equivalence queries. Applying this method, we show three main results: (1) The class of unions of ..."
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Cited by 18 (2 self)
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We develop a bounded version of the finite injury priority method in recursion theory. We use this to study the learnability of unions of rectangles over the domain f0; : : : ; n \Gamma 1g d with only equivalence queries. Applying this method, we show three main results: (1) The class of unions of rectangles is polynomial time learnable for constant dimension d. (2) The class of unions of rectangles whose projections at some unknown dimension are pairwisedisjoint is polynomial time learnable. (3) The class of unions of two disjoint rectangles is polynomial time learnable with unions of two rectangles as hypotheses. The preliminary version of this paper is partially contained in [BCH]. y Computer Science Department, Boston University, Boston, MA 02215; zchen@cs.bu.edu. The author was supported by NSF grant CCR9103055 and by a Boston University Presidential Graduate Fellowship. z Computer Science Department, Boston University, Boston, MA 02215; homer@cs.bu.edu. The author was su...
On the Cutting Edge of Relativization: The Resource Bounded Injury Method
, 1994
"... In this report we present a new method of diagonalization that is a refinement of the wellknown finite injury priority method discovered independently by Friedberg and Muchnik in 1957. In the resource bounded injury method , it is necessary in addition to proving that the number injuries for a give ..."
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Cited by 8 (0 self)
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In this report we present a new method of diagonalization that is a refinement of the wellknown finite injury priority method discovered independently by Friedberg and Muchnik in 1957. In the resource bounded injury method , it is necessary in addition to proving that the number injuries for a given requirement is finite to carefully count these injuries and prove that this number does not exceed a bound given by the index of the requirement. The method is used to construct an oracle relative to which the polynomial time hierarchy collapses to an extent that the second level of this hierarchy (P NP A ) captures nondeterministic exponential time. This oracle is an answer to an open problem posed by Heller in 1984 that has thus far resisted existing methods and that has recently regained interest by work of Fu et. al. and by work of Homer and Mocas. Moreover, our oracle provides a constructive counterexample to Sewelson's conjecture that does not make use of information theoretical l...
An explicit solution to Post’s Problem over the reals
, 2008
"... In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete ..."
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Cited by 6 (3 self)
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In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Then we strengthen the above result and show as well the existence of an uncountable number of incomparable semidecidable Turing degrees below the real Halting Problem in the BSS model. Again, our proof will give concrete such problems representing these different degrees. Finally we show the corresponding result for the linear BSS model, that is over (R, +, −,<)rather than (R, +, −, ×, ÷,<).
Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
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Cited by 6 (2 self)
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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
1997], There is no degree invariant halfjump
 Proc. Am. Math. Soc
, 1997
"... We prove that there is no degree invariant solution to Post’s problem that always gives an intermediate degree. In fact, assuming definable determinacy, if W is any definable operator on degrees such that a
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Cited by 6 (5 self)
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We prove that there is no degree invariant solution to Post’s problem that always gives an intermediate degree. In fact, assuming definable determinacy, if W is any definable operator on degrees such that a <W(a) < a ′ on a cone then W is low2 or high2 on a cone of degrees, i. e. there is a degree c such that W(a) ′ ′ = a ′ ′ for every a ≥ c or W(a) ′ ′ = a ′′ ′ for every a ≥ c.
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
Two Oracles that Force a Big Crunch
, 1999
"... The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new ty ..."
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Cited by 5 (1 self)
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The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new type of injury argument that we call \resource bounded injury." In the special case of the construction of this oracle, a tree method can be used to transform unbounded search into exponentially bounded, hence recursive, search. This transformation of the construction can be interleaved with another construction so that relative to the new combined oracle also P = UP = NP\coNP. This leads to the curious situation where LOW(NP) = P, but LOW(P NP ) = NEXP, and the complete p m degree for P NP collapses to a single pisomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constru...
Conjectures and Questions from Gerald Sacks’s Degrees of Unsolvability
 Archive for Mathematical Logic
, 1993
"... We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particular ..."
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Cited by 4 (1 self)
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We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals