Effective and sequential definition by cases on the reals via infinite signed-digit numerals

Abstract The lexicographical and numerical orders on infinite signed-digit numerals are unrelated. However, we show that there is a computable normalization operation on pairs of signed-digit numerals such that for normal pairs the two orderings coincide. In particular, one can always assume without loss of generality that any two numerals that denote the same number are themselves the same. We apply the order-normalization operator to easily obtain an effective and sequential definition-by-cases scheme in which the cases consist of inequalities between real numbers.

[1]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[2]  A. Avizienis.  Binary-Compatible Signed-Digit Arithmetic , 1899 .

[3]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[4]  Klaus Weihrauch.  A Foundation for Computable Analysis , 1997, SOFSEM.

[5]  G.D. Plotkin,et al.  LCF Considered as a Programming Language , 1977, Theor. Comput. Sci..

[6]  Martín Hötzel Escardó,et al.  PCF extended with real numbers : a domain-theoretic approach to higher-order exact real number computation , 1997 .

[7]  Pietro Di Gianantonio.  Real Number Computability and Domain Theory , 1996, Inf. Comput..

[8]  Dana S. Scott,et al.  A Type-Theoretical Alternative to ISWIM, CUCH, OWHY , 1993, Theor. Comput. Sci..

[9]  Claude Brezinski,et al.  Complexity theory of real functions , 1992 .

[10]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[11]  G. Plotkin.  Tω as a Universal Domain , 1978 .

[12]  A. Turing,et al.  On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction , 1938 .

[13]  L. E. J. Brouwer,et al.  Besitzt jede reelle Zahl eine Dezimalbruchentwicklung? , 1921 .

[14]  E. Wiedmer,et al.  Computing with Infinite Objects , 1980, Theor. Comput. Sci..

[15]  Christoph Kreitz,et al.  Representations of the real numbers and of the open subsets of the set of real numbers , 1987, Ann. Pure Appl. Log..

[16]  David Turner,et al.  Research topics in functional programming , 1990 .

[17]  Pietro Di Gianantonio.  A golden ratio notation for the real numbers , 1996 .

[18]  Robert Cartwright,et al.  Exact real arithmetic formulating real numbers as functions , 1990 .

[19]  Hans-Juergen Boehm,et al.  Exact real arithmetic: a case study in higher order programming , 1986, LFP '86.

[20]  Carl A. Gunter.  Semantics of programming languages: structures and techniques , 1993, Choice Reviews Online.

[21]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[22]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[23]  Norbert Th. Müller.  Subpolynomial Complexity Classes of Real Functions and Real Numbers , 1986, ICALP.

[24]  Alex K. Simpson,et al.  Lazy Functional Algorithms for Exact Real Functionals , 1998, MFCS.