Polynomial differential equations compute all real computable functions on computable compact intervals

Detalhes bibliográficos
Autor(a) principal: Bournez, Olivier
Data de Publicação: 2007
Outros Autores: Campagnolo, Manuel, Graça, Daniel, Hainry, Emmanuel
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10400.1/1011
Resumo: In the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models.
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spelling Polynomial differential equations compute all real computable functions on computable compact intervalsIn the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models.SapientiaBournez, OlivierCampagnolo, ManuelGraça, DanielHainry, Emmanuel2012-04-13T08:20:29Z20072007-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.1/1011engAUT: DGR01772;info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-24T10:11:56Zoai:sapientia.ualg.pt:10400.1/1011Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:55:16.570911Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Polynomial differential equations compute all real computable functions on computable compact intervals
title Polynomial differential equations compute all real computable functions on computable compact intervals
spellingShingle Polynomial differential equations compute all real computable functions on computable compact intervals
Bournez, Olivier
title_short Polynomial differential equations compute all real computable functions on computable compact intervals
title_full Polynomial differential equations compute all real computable functions on computable compact intervals
title_fullStr Polynomial differential equations compute all real computable functions on computable compact intervals
title_full_unstemmed Polynomial differential equations compute all real computable functions on computable compact intervals
title_sort Polynomial differential equations compute all real computable functions on computable compact intervals
author Bournez, Olivier
author_facet Bournez, Olivier
Campagnolo, Manuel
Graça, Daniel
Hainry, Emmanuel
author_role author
author2 Campagnolo, Manuel
Graça, Daniel
Hainry, Emmanuel
author2_role author
author
author
dc.contributor.none.fl_str_mv Sapientia
dc.contributor.author.fl_str_mv Bournez, Olivier
Campagnolo, Manuel
Graça, Daniel
Hainry, Emmanuel
description In the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models.
publishDate 2007
dc.date.none.fl_str_mv 2007
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