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A.: How much incomputable is the separable HahnBanach Theorem
 Conference on Computability and Complexity in Analysis. Number 348 in Informatik Berichte, FernUniversität Hagen (2008) 101 – 117
"... Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable ..."
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Cited by 7 (2 self)
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Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sepcomputable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the HahnBanach Extension Theorem is Sepcomplete. 1.
The effective theory of Borel equivalence relations
 Annals of Pure and Applied Logic
"... The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and HarringtonKechrisLouveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality ..."
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Cited by 2 (2 self)
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The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and HarringtonKechrisLouveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [6]) establishing for any recursive ordinal α the existence of Π0 1 singletons whose αjumps are Turing incomparable. 1
Effective cardinals of boldface pointclasses
 Journal of Mathematical Logic
"... Abstract. Assuming AD + DC(R), we characterize the selfdual boldface pointclasses which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank ≤ ω ξ 1, and those of Wadge rank < ωξ 1 when ξ ..."
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Cited by 2 (1 self)
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Abstract. Assuming AD + DC(R), we characterize the selfdual boldface pointclasses which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank ≤ ω ξ 1, and those of Wadge rank < ωξ 1 when ξ is limit. 1.
Pipes and Filters: Modelling a Software Architecture Through Relations
, 2002
"... A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream proces ..."
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A pipeline is a popular architecture which connects computational components/filers) through connectors (pipes) so that computations are performed in a stream like fashion. The data are transported through the pipes between filers, gradually transforming inputs to outputs. This kind of stream processing has been made popular through UNIX pipes that serially connect independent components for performing a sequence of tasks. We show in this paper how to formalize this architecture in terms of monads, hereby including relational specifications as special cases. The system is given through a directed acyclic graph the nodes of which carry the computational structure by being labelled with morphisms from the monad, and the edges provide the data for these operations. It is shown how fundamental compositional operations like combining pipes and filers, and refining a system by replacing simple parts through more elaborate ones, are supported through this construction.
BORELAMENABLE REDUCIBILITIES FOR SETS OF REALS
, 804
"... Abstract. We show that if F is any “wellbehaved ” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on P ( ω ω) induced by F turns out to look like the Wadge hierarchy (which is the special case where F is the set of continuous functions). 1. ..."
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Abstract. We show that if F is any “wellbehaved ” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on P ( ω ω) induced by F turns out to look like the Wadge hierarchy (which is the special case where F is the set of continuous functions). 1.
HOW MUCH INCOMPUTABLE IS THE SEPARABLE HAHNBANACH THEOREM?
, 808
"... Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable ..."
Abstract
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Abstract. We determine the computational complexity of the HahnBanach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sepcomputable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the HahnBanach Extension Theorem is Sepcomplete. 1.