TY - GEN

T1 - Highlights of the rice-shapiro theorem in computable topology

AU - Korovina, Margarita

AU - Kudinov, Oleg

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Computable topological spaces naturally arise in computer science for continuous data type representations that have tools for effective reasoning about quite complex objects such as real numbers and functions, solutions of differential equations, functionals and operators. Algebraic and continuous domains, computable metric spaces, computable Polish spaces have been successfully used in the theoretical foundation of computer science. In this paper we consider generalisations of the famous Rice-Shapiro theorem in the framework of effectively enumerable topological spaces that contain the weakly-effective ω –continuous domains and computable metric spaces as proper subclasses. We start with the classical case when the spaces admit principal computable numberings of computable elements and one can investigate arithmetical complexity of index sets. We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. It turns out that if we relax these requirements then the Rice-Shapiro theorem does not hold. Then we discuss the perspective of extensions of the Rice-Shapiro theorem to spaces that do not have computable numberings of computable elements, in particular to computable Polish spaces.

AB - Computable topological spaces naturally arise in computer science for continuous data type representations that have tools for effective reasoning about quite complex objects such as real numbers and functions, solutions of differential equations, functionals and operators. Algebraic and continuous domains, computable metric spaces, computable Polish spaces have been successfully used in the theoretical foundation of computer science. In this paper we consider generalisations of the famous Rice-Shapiro theorem in the framework of effectively enumerable topological spaces that contain the weakly-effective ω –continuous domains and computable metric spaces as proper subclasses. We start with the classical case when the spaces admit principal computable numberings of computable elements and one can investigate arithmetical complexity of index sets. We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. It turns out that if we relax these requirements then the Rice-Shapiro theorem does not hold. Then we discuss the perspective of extensions of the Rice-Shapiro theorem to spaces that do not have computable numberings of computable elements, in particular to computable Polish spaces.

KW - Arithmetical complexity

KW - Continuous data type

KW - Program semantics

KW - The Rise-Shapiro theorem

KW - INDEX SETS

KW - INSEPARABILITY

UR - http://www.scopus.com/inward/record.url?scp=85041749979&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-74313-4_18

DO - 10.1007/978-3-319-74313-4_18

M3 - Conference contribution

AN - SCOPUS:85041749979

SN - 9783319743127

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 241

EP - 255

BT - Perspectives of System Informatics - 11th International Andrei P. Ershov Informatics Conference, PSI 2017, Revised Selected Papers

A2 - Petrenko, AK

A2 - Voronkov, A

PB - Springer-Verlag GmbH and Co. KG

T2 - 11th International Andrei Ershov Memorial Conference on Perspectives of System Informatics, PSI 2017

Y2 - 27 June 2017 through 29 June 2017

ER -