Abstract Geometrical Computation and Computable Analysis

Jérôme Durand-Lose (1)

1 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans

Extended Signal machines are proven able to compute any computable function in the understanding of recursive/computable analysis (CA), here type-2 Turing machines (T2-TM) with signed binary encoding. This relies on an intermediate representation of any real number as an integer (in signed binary) plus an exact value in $(-1,1)$ which allows to have only finitely many signals present outside of the computation. Extracting a (signed) bit, improving the precision by one bit and iterating the 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 constructions emulating the black hole model of computation on an \emph{extended} signal machine. Extracting/encoding an infinite sequence of bits is achieved as the limit of the approximation process with a careful handling of accumulations and singulari ties.

Keywords: Type-2 Turing machine; Analog computation; Abstract geometrical computation; Computable analysis; Signal machine; Type-2 Turing machine.

Domains: Logic in Computer Science [cs.LO]

Jérôme Durand-Lose. Abstract Geometrical Computation and Computable Analysis. International Conference on Unconventional Computation 2009 (UC~'09), Aug 2009, Portugal. pp.158-167, ⟨10.1007/978-3-642-03745-0_20⟩. ⟨hal-00447965⟩