Coinductive Proofs for Basic Real Computation
We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.
KeywordsReal computation Coinductive proof Signed digit streams Computability Minlog
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- 1.Bertot, Y.: Coinduction in Coq. In: Lecture Notes of TYPES Summer School 2005, Sweden, August 15-26, 2005, vol. II (2005), http://www.cs.chalmers.se/Cs/Research/Logic/TypesSS05/Extra/bertot.pdf
- 5.Ciaffaglione, A.: Certified reasoning on real numbers and objects in co-inductive type theory. Ph.D Thesis. Department of Mathematics and Computer Science, University of Udine, and INPL-ENSMNS, Nancy, France (2003)Google Scholar
- 7.Escardo, M.H., Simpson, A.: A universal characterization of the closed Euclidean interval (extended abstract). In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, pp. 115–125 (2001)Google Scholar
- 9.Jones, C.: Completing the rationals and metric spaces in LEGO. In: Huet, G., Plotkin, G. (eds.) Proceedings of the Second Annual Workshop on Logical Frameworks (1992)Google Scholar
- 10.Ko, K.-I.: Complexity theory of real functions. Birkhauser, Boston (1991)Google Scholar
- 11.Lenisa, M.: From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems. In: Jacobs, B., Rutten, J. (eds.) Coalgebraic Methods in Computer Science CMCS 1999 Conference Proceedings. ENTCS, vol. 19 (1999)Google Scholar
- 13.Plume, D.: A Calculator for Exact Real Number Computation. 4th year project. Departments of Computer Science and Artificial Intelligence, University of Edinburgh (1998)Google Scholar