Coinduction for Exact Real Number Computation

Berger, Ulrich; Hou, Tie

doi:10.1007/s00224-007-9017-6

“…Whilst such algorithms have been verified before (see eg. [CDG06,MRE07,GNSW07,BH08]), in the present paper we show by means of an example how to extract them. We extract a program which for every rational number a ∈ [−1, 1] computes a signed binary digit representation, that is, a (finite or infinite) stream of digits…”

Section: Coalgebras For Exact Real Numbersmentioning

confidence: 96%

Minlog - A Tool for Program Extraction Supporting Algebras and Coalgebras

Berger

¹

,

Miyamoto

²

,

Schwichtenberg

³

et al. 2011

Algebra and Coalgebra in Computer Science

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Abstract. Minlog is an interactive system which implements prooftheoretic methods and applies them to verification and program extraction. We give an overview of Minlog and demonstrate how it can be used to exploit the computational content in (co)algebraic proofs and to develop correct and efficient programs. We illustrate this by means of two examples: one about parsing, the other about exact real numbers in signed digit representation.

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“…A similar type of trees has been studied independently in [GHP06], however, not in the context of analysis and realisability. The definition of C 1 is motivated by earlier works on the development and verification of exact real number algorithms based on the signed digit representation of real numbers [MRE07,GNSW07,EH02] some of which make use of coinductive methods [CDG06,Ber07,BH08,Niq08].…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Proofs, Programs, Processes

Berger

¹

,

Seisenberger

²

2011

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Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation.

show abstract

“…A similar type of trees has been studied independently in [GHP06], however, not in the context of analysis and realisability. The definition of C 1 is motivated by earlier works on the development and verification of exact real number algorithms based on the signed digit representation of real numbers [MRE07,GNSW07,EH02] some of which make use of coinductive methods [CDG06,Ber07,BH09,Niq08].…”

Section: Conclusion and Further Workmentioning

confidence: 99%