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27
On the strength of Ramsey’s Theorem for pairs
 Journal of Symbolic Logic
, 2001
"... Abstract. We study the proof–theoretic strength and effective content denote Ramof the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable ..."
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Cited by 42 (9 self)
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Abstract. We study the proof–theoretic strength and effective content denote Ramof the infinite form of Ramsey’s theorem for pairs. Let RT n k sey’s theorem for k–colorings of n–element sets, and let RT n < ∞ denote (∀k)RTn k. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X ′ ′ ≤T 0 (n). Let I�n and B�n denote the �n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models is conservative of arithmetic enables us to show that RCA0 + I �2 + RT2 2 over RCA0 + I �2 for �1 1 statements and that RCA0 + I �3 + RT2 < ∞ is �1 1conservative over RCA0 + I �3. It follows that RCA0 + RT2 2 does not imply B �3. In contrast, J. Hirst showed that RCA0 + RT2 < ∞ does imply B �3, and we include a proof of a slightly strengthened version of this result. It follows that RT2 < ∞ is strictly stronger than RT2 2 over RC A0. 1.
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Cited by 3 (3 self)
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.
Model Theory of the Computably Enumerable ManyOne Degrees
"... We investigate model theoretic properties of Rm , the partial order of computably enumerable manyone degrees. We prove that all nontrivial final segments and the set of minimal degrees are automorphism bases, and that some proper half open initial segment is an elementary substructure of Rm \Gamma f ..."
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Cited by 3 (3 self)
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We investigate model theoretic properties of Rm , the partial order of computably enumerable manyone degrees. We prove that all nontrivial final segments and the set of minimal degrees are automorphism bases, and that some proper half open initial segment is an elementary substructure of Rm \Gamma f1g. This shows that Rm is not a minimal model. In an appendix, we show that the manyone degree of an rmaximal set is join irreducible. Keywords: Model theory, manyone degrees This article is dedicated to Mari Santos. 1 Introduction Manyone reducibility, introduced by Post [9], is a rather fine way to measure the relative complexity of subsets of !: X is manyone reducible to Y , written X m Y , if X = f \Gamma1 (Y ) for a computable function f (we assume that f can also assume the values TRUE and FALSE to avoid trivialities). However, it appears naturally in a wide variety of contexts, for instance interpretability of theories and word problems of subgroups. Let Dm and Rm denote t...
On the Query Complexity of Sets
, 1996
"... . There has been much research over the last eleven years that considers the number of queries needed to compute a function as a measure of its complexity. We are interested in the complexity of certain sets in this context. We study the sets ODD A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; ..."
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Cited by 3 (3 self)
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. There has been much research over the last eleven years that considers the number of queries needed to compute a function as a measure of its complexity. We are interested in the complexity of certain sets in this context. We study the sets ODD A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; xn gj is oddg and WMOD(m) A n = f(x1 ; : : : ; xn) : jA " fx1 ; : : : ; xn gj 6j 0 (mod m)g. If A = K or A is semirecursive, we obtain tight bounds on the query complexity of ODD A n and WMOD(m) A n . We obtain lower bounds for A r.e. The lower bounds for A r.e. are derived from the lower bounds for A semirecursive. We obtain that every ttdegree has a set A such that ODD A n requires n parallel queries to A, and a set B such that ODD B n can be decided with one query to B. Hence for boundedquery complexity, how information is packaged is more important than Turing degree. We investigate when extra queries add power. We show that, for several nonrecursive sets A, the more queries you can...
Parsimony Hierarchies for Inductive Inference
 JOURNAL OF SYMBOLIC LOGIC
, 2004
"... Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsi ..."
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Cited by 2 (1 self)
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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A limcomputable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "notsonearly" minimal size, e.g., to be within a limcomputable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are limcomputable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of limcomputability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the limcomputable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.
Global Properties of Degree Structures
"... We investigate degree structures induced by manyone reducibility and Turing reducibility on the computably enumerable (c.e.), the arithmetical all all subsets of N. We study which subsets of the degree structure automorphism bases: for instance the minimal degrees form an automorphism base for ..."
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We investigate degree structures induced by manyone reducibility and Turing reducibility on the computably enumerable (c.e.), the arithmetical all all subsets of N. We study which subsets of the degree structure automorphism bases: for instance the minimal degrees form an automorphism base for the c.e. many one degrees, but not for the other degree structures based on m . We develop a method to show a subset is an automorphism base and apply it to the c.e. mdegrees and to give a modified proof of a result of AmbosSpies that initial intervals of the c.e. degrees are automorphism bases. Also, we show that the arithmetical mdegrees form a prime model. A central topic of computability theory is the study of sets of natural numbers under a notion of relative computability. Specifying a reducibility and an appropriate class of sets gives rise to a degree structure which may have interesting "global" properties. This is the case for the degree structures induced by manyone redu...
Gems In The Field Of Bounded Queries
"... Let A be a set. Given {x1 , . . . , xn}, I may want to know (1) which elements of {x1 , . . . , xn} are in A, (2) how many elements of {x1 , . . . , xn} are in A, or (3) is {x1 , . . . , xn}#A  even. All of these can be determined with n queries to A. For which A,n can we get by with fe ..."
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Cited by 1 (1 self)
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Let A be a set. Given {x1 , . . . , xn}, I may want to know (1) which elements of {x1 , . . . , xn} are in A, (2) how many elements of {x1 , . . . , xn} are in A, or (3) is {x1 , . . . , xn}#A  even. All of these can be determined with n queries to A. For which A,n can we get by with fewer queries? Other questions involving `how many queries do you need to . . .' have been posed and (some) answered. This article is a survey of the gems in the fieldthe results that both answer an interesting question and have a nice proof. Keywords: Queries, Computability
Extending and Interpreting Post’s Programme
, 2008
"... Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic in ..."
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Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computabilitytheoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Posts programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Posts programme, and look at two interesting developments of Posts approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computabilitytheoretic infrastructure. We will conclude by viewing Posts programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1