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45
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 42 (3 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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Cited by 24 (16 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
Finite Model Theory and Descriptive Complexity
, 2002
"... This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the ..."
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Cited by 23 (7 self)
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This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the
UP TO EQUIMORPHISM, HYPERARITHMETIC IS RECURSIVE
"... Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω CK 1 if and only if it is equimorphic to a ..."
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Cited by 6 (3 self)
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Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω CK 1 if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity, is recursively presentable.
The isomorphism problem for computable abelian pgroups of bounded length
"... Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a ..."
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Cited by 6 (3 self)
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Abstract. Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are nonclassifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from nonclassifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian pgroups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new backandforth relations on such groups are calculated. 1.
On the nbackandforth types of Boolean algebras
 In preparation
"... Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean al ..."
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Cited by 5 (1 self)
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Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically Σ 0 n relations of Boolean algebras as existential formulas over a finite set of relations. 1.
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.)
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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Cited by 5 (1 self)
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 4 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.