An Abstract Data Type for Real Numbers

We present a calculus having real numbers as a basic data type. The calculus is defined by its denotational semantics. We prove the universality of the calculus. We show how the definition of an operational semantics is problematic. We discuss this problem and present a possible solution.

[1]  Jon L. White Proceedings of the 1992 ACM conference on LISP and functional programming , 1992 .

[2]  Oliver Aberth,et al.  Computable analysis , 1980 .

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

[4]  Stephen A. Cook,et al.  Computability and Complexity of Higher Type Functions , 1992 .

[5]  Abbas Edalat,et al.  Integration in Real PCF , 2000, Inf. Comput..

[6]  A. Grzegorczyk On the definitions of computable real continuous functions , 1957 .

[7]  Abbas Edalat,et al.  A new representation for exact real numbers , 1997, MFPS.

[8]  Gérard Berry,et al.  Stable Models of Typed lambda-Calculi , 1978, ICALP.

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

[10]  Thomas Streicher,et al.  A universality theorem for PCF with recursive types, parallel-or and ∃ , 1994, Mathematical Structures in Computer Science.

[11]  Jean Vuillemin,et al.  Exact real computer arithmetic with continued fractions , 1988, IEEE Trans. Computers.

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

[13]  Valerie Menissier-Morain Menissier Arithmetique exacte : conception, algorithmique et performances d'une implementation informatique en precision arbitraire , 1994 .

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

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

[16]  Martín Hötzel Escardó,et al.  PCF Extended with Real Numbers , 1996, Theor. Comput. Sci..

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

[18]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[19]  erie M enissier-Morainy Arbitrary Precision Real Arithmetic: Design and Algorithms , 1996 .

[20]  P. Martin-Löf Notes on constructive mathematics , 1970 .

[21]  H. G. Rice,et al.  Recursive real numbers , 1954 .

[22]  A. Troelstra Constructivism in mathematics , 1988 .

[23]  Kenneth W. Regan,et al.  Computability , 2022, Algorithms and Theory of Computation Handbook.

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