Decomposition of 2,3 Turing Machine Evolutions

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Turing machines are simple models of a computing machine invented by Alan Turing. They consist of a head that switches between states while reading from and writing on a tape. The symbols read or written are called colors. Generalizations of such machines can have multiple heads or tapes.

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Evolutions of 2-state, 3-color Turing machines can be decomposed into 2-color images which represent individual colors. Two of these 2-color images include enough information to reconstruct the image of the original 3-color evolution.

The row of labels shows an example tape featuring three colors and then gives the results of replacing one of the digits in the list {0,1,2} by a 1-digit and all others by 0-digits.. This map defines the decomposed representations of the original 3-color evolution shown in the four main plots.

The four main plots show evolutions of 2-state, 3-color Turing machines and their decompositions. Each row of squares gives one step in the evolution of the tape. Squares correspond to bits on the tape. In the left graph: white squares represent 0-digits, yellow squares stand for 1-digits and orange squares for 2-digits. The small tears represent states: a falling corresponds to a 1-state, a rising disk indicates a 2-state. In the three graphs on the right, 1-digits are represented as gray squares.

The initial position of the head is given in relation to the specified initial condition of the tape, which can be cyclic, padded by an infinite number of 0-, 1- or 2-digits, or padded by repetitions of the selected 9-digit initial tape specification.

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Contributed by: Michael Schreiber (March 2011)
Open content licensed under CC BY-NC-SA


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