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22
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
Effective model theory: the number of models and their complexity
 MODELS AND COMPUTABILITY
, 1999
"... Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems i ..."
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Cited by 18 (6 self)
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Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.
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 ..."
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Cited by 17 (6 self)
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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
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
Degree Spectra of Relations on Computable Structures
 J. Symbolic Logic
, 1999
"... Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can ..."
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Cited by 11 (5 self)
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Abstract We give some new examples of possible degree spectra of invariant relations on \Delta 02categorical computable structures that demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the kinds of sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a \Delta 02categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [22] to establish the same result for computable relations on computable linear orderings.
Ordered Groups: A Case Study In Reverse Mathematics
 Bulletin of Symbolic Logic
, 1999
"... this article, we will be concerned only with fully ordered groups and will use the term ordered group to mean fully ordered group. There are a number of group conditions which imply full orderability. The simplest is given by the following classical theorem. ..."
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Cited by 9 (2 self)
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this article, we will be concerned only with fully ordered groups and will use the term ordered group to mean fully ordered group. There are a number of group conditions which imply full orderability. The simplest is given by the following classical theorem.
Computable categoricity of trees of finite height
 Journal of Symbolic Logic
"... We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3conditi ..."
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Cited by 7 (1 self)
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We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0 n+1categorical but not ∆0 ncategorical.
The Complexity of Finding SUBSEQ(A)
"... Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sig ..."
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Cited by 3 (0 self)
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Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sigma 2hard). We also study the complexity of going from Ato SUBSEQ(A) for several representations of A and SUBSEQ(A).