Results 1  10
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12
On Initial Segments of Computable Linear Orders
, 1997
"... We show there is a computable linear order with a # 0 2 initial segment that is not isomorphic to any computable linear order. ..."
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We show there is a computable linear order with a # 0 2 initial segment that is not isomorphic to any computable linear order.
The isomorphism problem for torsionfree abelian groups is analytic complete
 JOURNAL OF ALGEBRA
, 2008
"... We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete. ..."
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Cited by 7 (5 self)
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We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete.
EMBEDDING JUMP UPPER SEMILATTICES INTO THE TURING DEGREES
"... We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential ..."
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Cited by 5 (0 self)
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We prove that every countable jump upper semilattice can be embedded in D, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and D is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′ 〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1type of jusl with 0 is realized in D. On the other hand, we show that every quantifier free 1type of jump partial ordering (jpo) with 0 is realized in D. Moreover, we show that if every quantifier free type, p(x1,..., xn), of jpo with 0, which contains the formula x1 ≤ 0 (m) &... & xn ≤ 0 (m) for some m, is realized in D, then every every quantifier free type of jpo with 0 is realized in D. We also study the question of whether every jusl with the c.p.p. and size κ ≤ 2 ℵ0 is embeddable in D. We show that for κ = 2 ℵ0 the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)
Model Theoretic Complexity of Automatic Structures
 PROC. TAMC ’08, LNCS 4978
, 2008
"... We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any autom ..."
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Cited by 4 (2 self)
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We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic wellfounded partial order is bounded by ωω; 2) The ordinal heights of automatic wellfounded relations are unbounded below ωCK 1, the first noncomputable ordinal; 3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ωCK 1, ωCK 1 +1; 4) For any computable ordinal α, there is an automatic successor tree of CantorBendixson rank α.
BARWISE: INFINITARY LOGIC AND ADMISSIBLE SETS
"... 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3 ..."
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1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3
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. ∗The first and the second authors acknowledge the generous support of the FWF through
ON AUTOMORPHIC TUPLES OF ELEMENTS IN COMPUTABLE MODELS
"... Abstract: A criterion is obtained for existence of two isomorphic but not hyperarithmetically isomorphic tuples in a hyperarithmetical model. This criterion is used to show that such a situation occurs in the models of wellknown classes. ..."
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Abstract: A criterion is obtained for existence of two isomorphic but not hyperarithmetically isomorphic tuples in a hyperarithmetical model. This criterion is used to show that such a situation occurs in the models of wellknown classes.
ON THE PIONEONE SEPARATION PRINCIPLE
, 2007
"... We study the proof theoretic strength of the Π 1 1separation axiom scheme. We show that Π 1 1separation lies strictly in between the ∆ 1 1comprehension and Σ 1 1choice axiom schemes over RCA0. ..."
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We study the proof theoretic strength of the Π 1 1separation axiom scheme. We show that Π 1 1separation lies strictly in between the ∆ 1 1comprehension and Σ 1 1choice axiom schemes over RCA0.
Models with High Scott Rank
, 2008
"... Scott rank is a measure of modeltheoretic complexity; the Scott rank of a structure A in the language L is the least ordinal β for which A is prime in its Lωβ,ωtheory. By a result of Nadel, the Scott rank of a structure A is at most ωA 1 + 1, where ωA 1 is the least ordinal not recursive in A. We ..."
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Scott rank is a measure of modeltheoretic complexity; the Scott rank of a structure A in the language L is the least ordinal β for which A is prime in its Lωβ,ωtheory. By a result of Nadel, the Scott rank of a structure A is at most ωA 1 + 1, where ωA 1 is the least ordinal not recursive in A. We say that the Scott rank of A is high if it is at least ωA 1. Let α be a Σ1 admissible ordinal. A structure A of high Scott rank (and for which ω A 1 = α) will have Scott rank α + 1 if it realizes a nonprincipal Lα,ωtype, and Scott rank α otherwise. For α = ω CK 1, the least nonrecursive ordinal, several sorts of constructions are known. The Harrison ordering ω CK 1 (1 + η), where η is the ordertype of the rationals, has Scott rank ω CK 1 + 1. Makkai constructs a model with Scott rank ω CK 1 whose L ω CK 1,ωtheory is ℵ0categorical. Millar and Sacks produce a model A with Scott rank ω CK 1 (in which ω A 1 = ω CK 1) but whose L ω CK 1,ωtheory is not