Abstract geometrical computation 4: Small Turing universal signal machines

Abstract

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Abstract geometrical computation 4: small Turing universal signal machinesJérôme Durand-Lose To cite this version:Jérôme Durand-Lose. Abstract geometrical computation 4: small Turing universal signal machines. Theoretical Computer Science, Elsevier, 2011, 412, pp. AbstractThis article provides several very small signal machines able to perform any computation -in the classical understanding-generated from Turing machines, cellular automata and cyclic tag systems. A halting universal signal machine with 13 metasignals and 21 collision rules is presented (resp. 15 and 24 for a robust version). If infinitely many signals are allowed to be present in the initial configuration, 5 metasignals and 7 collision rules are enough to achieve non-halting weak universality (resp. 6 and 9 for a robust version).

Abstract geometrical computation 8: Small machines, accumulations & rationality

“…The computing power of 2-speed signal machines is very limited since the length of a computation is at most quadratic in the number of signals in the initial configuration. The last constructions in [Durand-Lose, 2011], provides a Turing-universal signal machine with four speeds. The case of 3 speeds has been studied in [Durand-Lose, 2013]: the same dichotomy arises.…”

“…Signal machines are able to compute in the classical Turing understanding. This paper is somehow a companion to [Durand-Lose, 2011] where a Turing-universal signal machine is presented with only 13 meta-signals and 4 speeds. This research takes place in the quest for minimality in order to compute, and thus get unpredictable behavior.…”

“…Being in a continuous setting, signal machines can carry out analog computations in the sense of both the BSS's model [Blum et al, 1989, Durand-Lose, 2008a and Computable analysis [Weihrauch, 2000, Durand-Lose, 2009b.…”

Exact Discretization of 3-Speed Rational Signal Machines into Cellular Automata

“…It is possible to simulate a CA with a SM [11]. However, the other way round is done only on particular ad hoc cases, often leaving the technical details out.…”

Abstract geometrical computation 5: embedding computable analysis

“…The presentation of the implementation of (classical) Turing machines (TM) is only illustrated; it is quite straightforward and already done in [Durand-Lose, 2009a, 2010. A Type-2 TM needs infinitely many iterations to complete a computation.…”