Real number computation with committed choice logic programming languages

Abstract

As shown in [H. Tsuiki, Real number computation through gray code embedding, Theoretical Computer Science 284 (2002) 467], the real line can be embedded topologically in the set Σ ⊥ , 1 ω of infinite sequences of {0, 1, ⊥} containing at most one ⊥. Moreover, there is a nondeterministic multi-headed machine, called an IM2-machine, which operates on Σ ⊥ , 1 ω and which induces the standard notion of computation over the reals via this embedding. In this paper, we study how the behavior of an IM2-machine can be expressed in “real” programming languages. When we use a lazy functional language like Haskell and represent a sequence as an infinite list, we cannot express the behavior of an IM2-machine. However, when we use a logic programming language with guarded clauses and committed choice, such as Concurrent Prolog, PARLOG, and GHC (Guarded Horn Clauses), we can express the behavior of IM2-machines naturally and execute them on an ordinary computer. We show that GHC-computability implies IM2-computability but not vice versa when we consider functions defined on Σ ⊥ , 1 ω in general, but they coincide when we only consider functions defined on the reals. We give some GHC program examples, such as the conversions between Gray-code and the signed-digit representations, and the addition function on reals.