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- W3170611460 abstract "Infinite Gray code has been introduced by Tsuiki~cite{ts} as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also cite{tsug}). Berger and Tsuiki~cite{btifp,bt} reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates $bS$ and $bG$ are defined and the inclusion $bS subseteq bG$ is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version $§_{2}$ of $S$ and $bG subseteq bS_{2}$ is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version $bG^{*}$ of $bG$ and a modification $bS^{*}$ of $bS_{2}$ are presented such that $bS^{*} = bG^{*}$. A crucial tool in cite{btifp} is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that $bS^{*}$ and $bG^{*}$ coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals." @default.
- W3170611460 created "2021-06-22" @default.
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- W3170611460 date "2021-05-29" @default.
- W3170611460 modified "2023-09-27" @default.
- W3170611460 title "Computing with Infinite Objects: the Gray Code Case." @default.
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