2005 Fiscal Year Final Research Report Summary
Gray-code representation of real number and the induced computability structure
| Project/Area Number |
15500010
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Fundamental theory of informatics
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| Research Institution | Kyoto University |
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Principal Investigator |
TSUIKI Hideki Kyoto University, Graduate School of Human and Environmental Studies, associate professor, 大学院人間・環境学研究科, 助教授 (10211377)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Gray-code / Topology / Real number computation / subbase / domain theory / bottom / ボトム |
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| Research Abstract |
We studied computational, topological, and domain-theoretic problems related to the computable structure of real numbers derived form representation with {0,1,bot}-infinite sequences (Gray-code representation) and indeterministic multi-head machine (IM2-machine) operating on them.
1.IM2-machines can be realized with logic programming languages with guards like GHC (Guarded Horn Clauses). The set of functions expressible in GHC is larger than the set of IM2-computable functions when we consider functions over {0,1,bot}-infinite sequences but they are equal when only real number representation is considered.
2.IM2-machines cannot be realized with functional languages, but by extending a functional language with a restricted amb operator which is realized with sequential graph reduction rules instead of term reduction rules, one can implement the ability of IM2-machines. We have implemented it as an extension of the Haskell language.
3.The number of bottoms required to represent a topological space and the topological dimension of the space is equal. We have a more general result on an algebraic domain, which is proved domain-theoretically.
4.Codings and subbases of a topological space are closely related. Codings which induce natural computational structure will induce a dyadic subbase, which is based on pairs of regular open sets. The efficiency of the Gray-code representation can be characterized through properties of the corresponding subbase, which are studied topologically.
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