Abstract geometrical computation 5: embedding computable analysis
Extended Signal machines are proven capable to compute any computable function in the understanding of recursive/computable analysis (CA), represented here with type-2 Turing machines (T2-TM) and signed binary. This relies on a mixed representation of any real number as an integer (in signed binary) plus an exact value in (?1, 1). This permits to have only finitely many signals present simultaneously. Extracting a (signed) bit, improving the precision by one bit and iterating a T2-TM only involve standard signal machines. For exact CA-computations, T2-TM have to deal with an infinite entry and to run through infinitely many iterations to produce an infinite output. This infinite duration can be provided by an infinite acceleration construction. Extracting/encoding an infinite sequence of bits is achieved as the limit of the approximation process with a careful handling of accumulations.
abstract geometrical computation computable analysis signal machine type-2 turing machine analog computation