Logic for Gray-code Computation

Gray-code is a well-known binary number system where neighboring values differ in one digit only. Tsuiki (2002) has introduced Gray code to the field of real number computation. He assigns to each number a unique 1 perpendicular to-sequence, i.e., an infinite sequence of {-1, 1, perpendicular to} containing at most one copy of perpendicular to (meaning undefinedness). In this paper we take a logical and constructive approach to study real number computation based on Gray-code. Instead of Tsuiki's indeterministic multihead Type-2 machine, we use pre-Gray code, which is a representation of Gray-code as a sequence of constructors, to avoid the difficulty due to perpendicular to which prevents sequential access to a stream. We extract real number algorithms from proofs in an appropriate formal theory involving inductive and coinductive definitions. Examples are algorithms transforming pre-Gray code into signed digit code of real numbers, and conversely, the average for pre-Gray code and a translation of finite segments of pre-Gray code into its normal form. These examples are formalized in the proof assistant Minlog.