2015 Fiscal Year Research-status Report
Research on complete quasi-metric spaces with algebraic structure
| Project/Area Number |
15K15940
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| Research Institution | National Institute of Information and Communications Technology |
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Principal Investigator |
ディブレクト マシュー 国立研究開発法人情報通信研究機構, 脳情報通信融合研究センター・脳情報通信融合研究室, 研究員 (20623599)
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Keywords | quasi-Polish space / topological algebra / semilattices / domain theory / powerspace |
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| Outline of Annual Research Achievements |
The purpose of this research is to investigate topological algebra for quasi-Polish (countably based completely quasi-metrizable) spaces. For the past year I have focused on quasi-Polish semilattices, and through collaborative work we have succeeded in extending many results from domain theory to the class of quasi-Polish spaces. It is known from domain theory that certain topological dcpo semilattices can be described as Eilenberg-Moore algebras of powerspace monads. A major research achievement was proving that these powerspace monads are well defined on the category of quasi-Polish spaces, and verifying that most of the desired properties and connections with topological semilattices are maintained.
The significance of the results are as follows. First, they provide a uniform approach to topological semilattices which includes both domains and Polish spaces. This provides useful hints for developing the theory of other semigroups and semirings. Second, the powerspace monads can model non-determinstic dynamical systems on quasi-Polish spaces. This has been applied in domain theory to provide semantics of non-determinsitic programs, where the semilattice operations interpret the syntax for non-deterministic branching operations. Third, via Stone duality, the powerspace monads provide coalgebraic models of (geometric) modal logic. Quasi-Polish spaces are closely related to geometric propositional logic, and we are now extending this to modal logic.
These and related results have been presented at several international conferences, and journal papers are under preparation.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We have developed a large fragment of the theory of quasi-Polish semilattices via the use of powerspace monads, and we have made much progress on the relevant duality theory. It is therefore fair to say the research is progressing rather smoothly.
It is known that the powerspace monads provide free topological semilattices within the category of domains, but the situation within the class of quasi-Polish spaces is not completely clear yet. In particular, characterizing the algebras of the upper powerspace monad is known to be a difficult problem, and we have not yet completely solved this for the class of quasi-Polish spaces. Also, we have only made slight progress on other kinds of topological algebras (such as non-idempotent semigroups). Some of our initial results developing probability theory for quasi-Polish spaces using the probabilistic powerspace monad suggests that the approach we have taken using certain monads will generalize to other classes of topological semirings. However, in order to understand how general this approach is, we will need to further compare our results with those of R. Heckmann, which relates powerspaces and semiring modules, and with work by A. Simpson and others on observationally induced algebras.
Since it is yet unclear how well the approach we have adopted will generalize to other topological algebras, I hesitate to give a status of “progressing more smoothly than initially planned.”
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| Strategy for Future Research Activity |
Our results open up several research directions, including applications to non-deterministic dynamical systems and co-algebraic models of geometric modal logic. We expect the methods used in domain theory to model non-deterministic programs with the powerspace monads will have generalizations to the category of quasi-Polish spaces. However, a crucial aspect of these applications is that the powerspace monads create free topological semilattices in the category of domains. Investigating the “freeness” of the powerspace constructions in the category of quasi-Polish spaces is an important research problem.
We are also interested in classifying the modal logics which can be modeled using quasi-Polish co-algebras. Via Stone duality, this problem is related to understanding certain algebras on the frame of open subsets of a quasi-Polish space. We can view these algebras as topological algebras if we go to a slightly larger subcategory of QCB spaces, and understanding the completeness properties of these more general spaces will be useful in understanding their role in modal logic. However, it is known from previous research that working with these more general spaces implicitly involves co-analytic complete sets, and is thus rather challenging.
The other main project is to extend our results to other quasi-Polish algebras, including semigroups and semirings. As mentioned in the previous section, this involves better understanding the relationship between powerspaces and semiring modules, and also observationally induced algebras, within the category of quasi-Polish spaces.
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| Causes of Carryover |
Only a modest amount of 4万 was left over, and it was deemed to be more useful to use it in the following year along with the additional funds than try to spend it all this year.
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| Expenditure Plan for Carryover Budget |
The remaining funds will be combined with this years funds for conference expenses, inviting researchers, and other supplies.
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| Remarks |
I have recently changed institutions, and the new webpage is still under construction.
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