Results 1  10
of
28
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 ..."
Abstract

Cited by 34 (13 self)
 Add to MetaCart
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...
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
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 ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
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...
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. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
$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
Abstract

Cited by 6 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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...
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.