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41
Post's Program and incomplete recursively enumerable sets
, 1991
"... : A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a first order property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing i ..."
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Cited by 21 (4 self)
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: A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a first order property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete; and (2) there exists an r.e. set A satisfying Q(A). This resolves a long open question stemming from Post's Program of 1944, and it sheds new light on the fundamental problem of the relationship between the algebraic structure of an r.e. set A and the (Turing) degree of information which A encodes. Recursively enumerable (r.e.) sets have been a central topic in mathematical logic, in recursion theory (i.e. computability theory), and in undecidable problems. They are the next most effective type of set beyond recursive (i.e. computable) sets, and they occur naturally in many branches of mathematics. This together with the existence of nonrecursive r.e. sets has enabled them to pl...
Effective Categoricity of Equivalence Structures
 Annals of Pure and Applied Logic 141 (2006
, 2005
"... We investigate effective categoricity of computable equivalence structures A. We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are inf ..."
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Cited by 13 (9 self)
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We investigate effective categoricity of computable equivalence structures A. We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively ∆ 0 3 categorical, we further investigate when they are ∆ 0 2 categorical. We also obtain results on the index sets of computable equivalence structures. ∗ The authors would like to thank the anonymous referee for his comments and suggestions. † Calvert was partially supported by the NSF grants DMS9970452, DMS0139626, and DMS0353748, Harizanov by the NSF grant DMS0502499, and the last three authors by the NSF binational grant DMS0075899. Harizanov and Morozov also gratefully acknowledge the
Arrow’s Theorem, Countably Many Agents, and More Visible Invisible Dictators
"... For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow’s conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a co ..."
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Cited by 9 (4 self)
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For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow’s conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain “oracle ” resolves Mihara’s impossibility result (1997) about computability of social welfare functions.
On the computational power of circuits of spiking neurons
 J. of Physiology (Paris
, 2003
"... It is quite difficult to construct circuits of spiking neurons that can carry out complex computational tasks. On the other hand even randomly connected circuits of spiking neurons can in principle be used for complex computational tasks such as timewarp invariant speech recognition. This is possib ..."
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Cited by 8 (0 self)
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It is quite difficult to construct circuits of spiking neurons that can carry out complex computational tasks. On the other hand even randomly connected circuits of spiking neurons can in principle be used for complex computational tasks such as timewarp invariant speech recognition. This is possible because such circuits have an inherent tendency to integrate incoming information in such a way that simple linear readouts can be trained to transform the current circuit activity into the target output for a very large number of computational tasks. Consequently we propose to analyze circuits of spiking neurons in terms of their roles as analog fading memory and nonlinear kernels, rather than as implementations of specific computational operations and algorithms. This article is a sequel to [31], and contains new results about the performance of generic neural microcircuit models for the recognition of speech that is subject to linear
1998], The Π3theory of the computably enumerable Turing degrees is undecidable
 Trans. Amer. Math. Soc
, 1998
"... Abstract. We show the undecidability of the Π3theory of the partial order of computably enumerable Turing degrees. Recursively enumerable (henceforth called computably enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets o ..."
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Cited by 7 (2 self)
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Abstract. We show the undecidability of the Π3theory of the partial order of computably enumerable Turing degrees. Recursively enumerable (henceforth called computably enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the computably enumerable sets
On the Turing degrees of weakly computable real numbers
 Journal of Logic and Computation
, 1986
"... The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others ..."
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Cited by 6 (3 self)
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The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semicomputable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1
Embedding Lattices with Top Preserved Below NonGL2 Degrees
, 1997
"... this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0 ..."
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Cited by 5 (1 self)
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this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0
Decidability Of The TwoQuantifier Theory Of The Recursively Enumerable Weak TruthTable Degrees And Other Distributive Upper SemiLattices
 Journal of Symbolic Logic
, 1996
"... . We give a decision procedure for the 89theory of the weak truthtable (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wttdegrees by a map which preserves the least and greatest e ..."
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Cited by 4 (0 self)
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. We give a decision procedure for the 89theory of the weak truthtable (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wttdegrees by a map which preserves the least and greatest elements: A finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semilattice has a decidable twoquantifier theory. These criteria are applied not only to the weak truthtable degrees of the recursively enumerable sets but also to various substructures of the polynomial manyone (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexitytheoretic results. The fact that the pmdegrees of the recursive sets have a decidable twoquantifier theor...