“Viral” Turing Machines, computation from noise and combinatorial hierarchies

Abstract

Abstract:The interactive computation paradigm is reviewed and a particular example is extended to form the stochastic analog of a computational process via a transcription of a minimal Turing Machine into an equivalent asynchronous Cellular Automaton with an exponential waiting times distribution of effective transitions. Furthermore, a special toolbox for analytic derivation of recursive relations of important statistical and other quantities is introduced in the form of an Inductive Combinatorial Hierarchy.

Persistent order in Schramm-Loewner Evolution driven by the primes last digit sequence

“…In previous work [20], it was pointed out by one of the authors that many sequences computed by simple automata acting on a lexicographically ordered powerset of all strings of constant legnth, also termed a "Hamming Space", may inherit an internal self-similarity out of the periodicities involved which can be seen as a type of semi-direct product between a set of dilations and a set of automorphisms of the Z 2 group. This then allows an arithmetized form of any such based on an iterated list concatenation system utilising much simpler "reproducing maps" of lower complexity in the form…”

Encoding discrete quantum algebras in a hierarchy of binary words

“…Recursive relations like (11) can often be found using successive sequence folding (reshaping into matrix form) until the simplest possible reproducing maps can be located as explained in [44]. Notably, the class of polynomials defined in (9) can be extended into arbitrary higher radices b > 2 using complex coefficients as…”

“…In the next sections, we attempt to first set up a unique framework allowing the examination of several different themes under the common theme of their arithmetized encoding, a kind of Gödelianization method for bounded string tuples. In section 2, we prove that a direct transfer of the axiomatic framework of abstract C*-algebras is possible for a hierarchy of lexicographically ordered (LEX) dictionaries of binary words using the construct of an Inductive Combinatorial Hierarchy (ICH) that was introduced in [44] as a generic toolbox for the study of global maps of automata. In section 3, we observe that the result of the necessary conjugation introduced in the previous section, sends directly to a special subset of words of a Dyck language [45], [46] which is isomorphic with the diagrammatic interpretation of the elements of any TL n algebra.…”