News
1 June, 2022: The slide file was uploaded.
Title
Generality of computable measures
Type
Leeds Computability Days 2022
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LCD
宮部賢志(ミヤベケンシ)
News
1 June, 2022: The slide file was uploaded.
Title
Generality of computable measures
Type
Leeds Computability Days 2022
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LCD
News
18 Dec, 2021: The slide file was uploaded.
Title
The rate of convergence of computable predictions
Type
RIMS workshop: Theory and application of Proof and computation
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rims
News
16 Sep, 2020 The slide file was uploaded.
Title
The rate of convergence of computable predictions
Type
The 2020 annual meeting of the Deutsche Mathematiker-Vereinigung (German Mathematical Society)
Minisymposia: The impact of randomness on computation
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DMV
News
4 Nov, 2019 The slide file was uploaded.
Title
Reverse randomness
Type
Colloquium talk at Kyoto University on 8 Nov, 2019.
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kyoto
News
19 June 2019, Online
Title
Uniform relativization
Type
Conference survey paper
Publication
In: Manea F., Martin B., Paulusma D., Primiero G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science, vol 11558. Springer, Cham
Abstract
This paper is a tutorial on uniform relativization. The usual relativization considers computation using an oracle, and the computation may not work for other oracles, which is similar to Turing reduction. The uniform relativization also considers computation using oracles, however, the computation should work for all oracles, which is similar to truth-table reduction. The distinction between these relativizations is important when we relativize randomness notions in algorithmic randomness, especially Schnorr randomness. For Martin-Löf randomness, its usual relativization and uniform relativization are the same so we do not need to care about this uniform relativization.
We focus on two specific examples of uniform relativization: van Lambalgen’s theorem and lowness. Van Lambalgen’s theorem holds for Schnorr randomness with the uniform relativization, but not with the usual relativization. Schnorr triviality is equivalent to lowness for Schnorr randomness with the uniform relativization, but not with the usual relativization. We also discuss some related known results.