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| * [http://arxiv.org/pdf/cs/0703128.pdf Implementation of a Kolmogorov–Uspensky machine on a biological substrate. (Adamatzky 2007)] | | * [http://arxiv.org/pdf/cs/0703128.pdf Implementation of a Kolmogorov–Uspensky machine on a biological substrate. (Adamatzky 2007)] |
| * [http://arxiv.org/pdf/0901.4556v1.pdf Programmable reconfiguration of Physarum machines (Adamatzky 2009)] | | * [http://arxiv.org/pdf/0901.4556v1.pdf Programmable reconfiguration of Physarum machines (Adamatzky 2009)] |
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| === Computable Discrete Elements in the Turing Machine ===
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| In a 1936 paper by Turing, the concept of the machine is proposed as the simple idea of an apparatus which is able to compute discrete values – zeros and ones. In the same paper, Turing introduces a computing machine with an infinite length of tape and a tape head acting upon seven commands: a) read the tape, b) move the tape left, c) move tape right, d) write “zero” on the tape, e) write “one” on the tape, f) jump to another command, and g) halt. The idea of these commands is to show that output B could be processed having an initial state and some input A. The position of the tape head on the proposed apparatus processing the information is dependent on the information stored on the tape: If the input information is defined, so is the output. The problem in such a computational model is any numerically undefined variable which would cause the machine to stop processing information, or to "halt." The halting state or, according to Turing, the “decision problem" (Enscheidungsproblem) is the problem of digital computation being defined by numerical variables. Thus, the Turing machine is limited to computing all input information and to solving all given problems (Turing 1936).
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| Turing Machines: https://www.youtube.com/watch?v=gJQTFhkhwPA
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| === Markov chain ===
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| "A Markov chain (discrete-time Markov chain or DTMC[1]), named after Andrey Markov, is a random process that undergoes transitions from one state to another on a state space. It must possess a property that is usually characterized as "memorylessness": the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes."(wikipedia)
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| === Kolmogorov Machine ===
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| "Kolmogorov, or Kolmogorov-Uspensky, machines [Ko1, KU, US] are similar to Turing machines except that the tape can change its topology."(Gurevich) Also, as far as I understand, Kolmogorov Machine isn't described by discrete 0 and 1 values. Also its functions could be updated in real time over the recursive method. On the other hand both Turing Machine and Kolmogorov machine, could emulate each other, so at the end the difference is just in the way how the machines compute their functions.
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| "Мы остановимся на следующих вариантах математического опреде ления вычислимой функции или алгоритма:
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| A) Определение вычислимой функции как функции, значения которой выводимы в некотором логическом исчислении (Гёдель [4], Чёрч [5]1)). Б) Определение вычислимой функции как функции, значения кото
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| рой получаются при помощи исчисления Х-коиверсии Чёрча [5], [7].
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| B) Определение вычислимой функции как функции частично-рекур сивной (см. работу Клини [8])2) или —для случая всюду определенной функции —как общерекурсивной (Клини [10]). (Термины «частично-рекур сивная» и «общерекурсивная» понимаются здесь в смысле приложения I).
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| Г) Вычислительная машина Тьюринга [ И ] 3 ) .
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| Д) Финитный комбинаторный процесс Поста [13].
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| Е) Нормальный алгорифм А. А. Маркова [1], [2]." (Колмогоров & Успенский 1958)
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| "Kolmogorov machines tape similarly to Schönhage’s tape is a finite connected graph with a distinguished (active) node. They work upon partly recursive function, changing instructions in real time." (Gurevich)
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| "Instructions:
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| *1. add a new node together with a pair of edges of some colors between the active node and the new one,
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| *2. remove a node and the edges incident to it,
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| *3. add a pair of edges of some colors between two existing nodes,
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| *4. remove the two edges between two existing nodes,
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| *5. halt. "(Gurevich)
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| "Grigoriev [Gr] exhibited a function real-time computable by some KU machine but not real-time computable by any Turing machine."(Gurevich)
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| == References == | | == References == |
