A Stream Calculus of Bottomed Sequences for Real Number Computation

https://doi.org/10.1016/j.entcs.2013.09.023Get rights and content
Under a Creative Commons license
open access

Abstract

A calculus XPCF of 1⊥-sequences, which are infinite sequences of {0,1,} with at most one copy of bottom, is proposed and investigated. It has applications in real number computation in that the unit interval I is topologically embedded in the set Σ,1ω of 1⊥-sequences and a real function on I can be written as a program which inputs and outputs 1⊥-sequences. In XPCF, one defines a function on Σ,1ω only by specifying its behaviors for the cases that the first digit is 0 and 1. Then, its value for a sequence starting with a bottom is calculated by taking the meet of the values for the sequences obtained by filling the bottom with 0 and 1. The validity of the reduction rule of this calculus is justified by the adequacy theorem to a domain-theoretic semantics. Some example programs including addition and multiplication are shown. Expressive powers of XPCF and related languages are also investigated.

Keywords

Bottom
stream
real number computation
domain model
PCF
adequacy
parallel or

Cited by (0)