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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 ..."
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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.
Isomorphism and BiEmbeddability Relations on Computable Structures
, 2010
"... We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and biembeddability. We use the notion of tcreducibility to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on h ..."
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We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and biembeddability. We use the notion of tcreducibility to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω. We also show that the biembeddability relation on an appropriate hyperarithmetical class of computable structures may have the same complexity as any given Σ 1 1 equivalence relation on ω.
Isomorphism Relations on Computable Structures ∗
, 2011
"... We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F Freducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ..."
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We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F Freducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω. 1
DECIDABILITY AND COMPUTABILITY OF CERTAIN TORSIONFREE ABELIAN GROUPS
"... Abstract. We study completely decomposable torsionfree abelian groups of the form GS: = ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ0 2 and has a computable copy if and only if S is Σ0 3. 1. ..."
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Abstract. We study completely decomposable torsionfree abelian groups of the form GS: = ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ0 2 and has a computable copy if and only if S is Σ0 3. 1.
JUMP DEGREES OF TORSIONFREE ABELIAN GROUPS
"... Abstract. We show, for each computable ordinal α and degree a> 0 (α) , the existence of a torsionfree abelian group with proper α th jump degree a. 1. ..."
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Abstract. We show, for each computable ordinal α and degree a> 0 (α) , the existence of a torsionfree abelian group with proper α th jump degree a. 1.
COMPUTABLE CATEGORICITY VERSUS RELATIVE COMPUTABLE CATEGORICITY
"... Abstract. We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1decidable computably categorical structure is relatively ∆0 2categorical. We study the complexit ..."
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Abstract. We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1decidable computably categorical structure is relatively ∆0 2categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations. 1.
THE COMPLEXITY OF COMPUTABLE CATEGORICITY
"... We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably ..."
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We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably categorical but not relatively ∆0αcategorical.
RESEARCH STATEMENT
, 2010
"... I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this c ..."
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I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this complexity and in measuring it. The motivations for this come from different areas. Form a foundational viewpoint, we want to know what assumptions we really need to do mathematics (ZF C is way much more than we usually use), and we are also interested in knowing what assumptions are used in the different areas of mathematics. Form a computational viewpoint, it is important to know what part of mathematics can be done by mechanical algorithms, and, even for the part that can’t be done mechanically, we want to know how constructive are the objects we deal with. Furthermore, it is sometimes the case that this computational analysis allows us to find connections between constructions in different areas of mathematics, and in many cases to obtain a deeper understanding of mathematical objects being analyzed. My work is quite diverse in terms of the techniques I have used, the approaches I have taken, and the areas of mathematics that I have analyzed. However, my background area is Computability Theory, and most of my work can be considered as part of this branch of Mathematical Logic.