Results 1  10
of
11
EFFECTIVELY CATEGORICAL ABELIAN GROUPS
"... We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finit ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆0 ncategoricity in this class of groups, for n> 1. We introduce a new algebraic concept of Sindependence which is a generalization of the wellknown notion of pindependence. We develop the theory of Pindependent sets. We apply these techniques to show that every homogeneous completely decomposable group is ∆0 3categorical. We prove that a homogeneous completely decomposable group of infinite rank is ∆0 2categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semilow complement (within the set of all primes). Finally, we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.
GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY
"... Abstract. Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function which agrees with the characteristic function of A ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function which agrees with the characteristic function of A on its domain D, and furthermore D has density 1, i.e. limn→ ∞ {k < n: k ∈ D}/n = 1. A set A is called coarsely computable if there is a computable set R such that the symmetric difference of A and R has density 0. We prove that there is a c.e. set which is generically computable but not coarsely computable and vice versa. We show that every nonzero Turing degree contains a set which is not generically computable and also a set which is not coarsely computable. We prove that there is a c.e. set of density 1 which has no computable subset of density 1. Finally, we define and study generic reducibility. 1.
Asymptotic density and the Ershov hierarchy, in preparation
"... Abstract. We classify the asymptotic densities of the ∆02 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n ≥ 2, a real r ∈ [0, 1] is the density of an nc.e. set if and only if it is a difference of leftΠ02 reals. Further, we show that the densities of th ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We classify the asymptotic densities of the ∆02 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n ≥ 2, a real r ∈ [0, 1] is the density of an nc.e. set if and only if it is a difference of leftΠ02 reals. Further, we show that the densities of the ωc.e. sets coincide with the densities of the ∆02 sets, and there are ωc.e. sets whose density is not the density of an nc.e. set for any n ∈ ω. 1.
ASYMPTOTIC DENSITY AND THE COARSE COMPUTABILITY BOUND
"... Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibilit ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r ∈ (0, 1] there are sets A0, A1 such that γ(A0) = γ(A1) = r where A0 is coarsely computable at density r while A1 is not coarsely computable at density r. We show that a real r ∈ [0, 1] is equal to γ(A) for some c.e. set A if and only if r is leftΣ03. A surprising result is that if G is a ∆02 1generic set, and A 6T G with γ(A) = 1, then A is coarsely computable at density 1. 1.
Asymptotic density, computable traceability, and 1randomness, in preparation
"... Abstract. Let r be a real number in the unit interval [0, 1]. A set A ⊆ ω is said to be coarsely computable at density r if there is a computable function f such that {n  f(n) = A(n)} has lower density at least r. Our main results are that A is coarsely computable at density 1/2 if A is either com ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Let r be a real number in the unit interval [0, 1]. A set A ⊆ ω is said to be coarsely computable at density r if there is a computable function f such that {n  f(n) = A(n)} has lower density at least r. Our main results are that A is coarsely computable at density 1/2 if A is either computably traceable or truthtable reducible to a 1random set. In the other direction, we show that if a degree a is either hyperimmune or PA, then there is an acomputable set which is not coarsely computable at any positive density. 1.
COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS
, 2015
"... Abstract. A coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively ra ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1random and B is computable from every coarse description D of A, then B is Ktrivial, which implies that if A is in fact weakly 2random then B is computable. Our main tool is a kind of compactness theorem for coneavoiding descriptions, which also allows us to prove the same result for 1genericity in place of weak 2randomness. In the other direction, we show that if A 6T ∅ ′ is a 1random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all Ktrivial sets are computable from every coarse description of some 1random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2random sets, at least in the nonuniform coarse degrees.
THE DEGREES OF BIHYPERHYPERIMMUNE SETS
"... Abstract. We study the degrees of bihyperhyperimmune (bihhi) sets. Our main result characterizes these degrees as those that compute a function that is not dominated by any ∆02 function, and equivalently, those that compute a weak 2generic. These characterizations imply that the collection of bi ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We study the degrees of bihyperhyperimmune (bihhi) sets. Our main result characterizes these degrees as those that compute a function that is not dominated by any ∆02 function, and equivalently, those that compute a weak 2generic. These characterizations imply that the collection of bihhi Turing degrees is closed upwards. 1.