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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)

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→‎Output Conditions for Tape Input "0": markup

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====Output Conditions for Tape Input "0"====

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Let <math>p_0\!</math> be the proposition that we get by conjoining the proposition that describes the initial conditions for tape input "0" with the proposition that describes the truncated turing machine <math>\operatorname{Stunt}(2).</math> As it turns out, <math>p_0\!</math> has a single satisfying interpretation~~, and this~~ is ~~represented as~~ a singular proposition in ~~terms of~~ its positive logical features in the following display:

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Let <math>p_0\!</math> be the proposition that we get by conjoining the proposition that describes the initial conditions for tape input "0" with the proposition that describes the truncated turing machine <math>\operatorname{Stunt}(2).</math> As it turns out, <math>p_0\!</math> has a single satisfying interpretation. This interpretation is expressible in the form of a singular proposition, which can in turn be indicated by its positive logical features, as shown in the following display:

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The Output Conditions for Tape Input "0" can be read as follows:

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At the time p_0, M is in the state q_0, and

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{| align="center" cellpadding=8" width="90%"

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At the time p_0, H is reading cell r_1, and

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|

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At the time p_0, cell r_0 contains "#", and

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At the time <math>p_0,\!</math> machine <math>\operatorname{M}</math> is in the state <math>q_0,\!</math> and

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At the time p_0, cell r_1 contains "0", and

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At the time <math>p_0,\!</math> scanner <math>\operatorname{H}</math> is reading cell <math>r_1,\!</math> and

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At the time p_0, cell r_2 contains "#", and

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At the time <math>p_0,\!</math> cell <math>r_0\!</math> contains the symbol <math>\texttt{\#},</math> and

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At the time <math>p_0,\!</math> cell <math>r_1\!</math> contains the symbol <math>\texttt{0},</math> and

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At the time p_1, M is in the state q_0, and

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At the time <math>p_0,\!</math> cell <math>r_2\!</math> contains the symbol <math>\texttt{\#},</math> and

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At the time p_1, H is reading cell r_2, and

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|-

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At the time p_1, cell r_0 contains "#", and

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|

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At the time p_1, cell r_1 contains "0", and

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At the time <math>p_1,\!</math> machine <math>\operatorname{M}</math> is in the state <math>q_0,\!</math> and

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At the time p_1, cell r_2 contains "#", and

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At the time <math>p_1,\!</math> scanner <math>\operatorname{H}</math> is reading cell <math>r_2,\!</math> and

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At the time <math>p_1,\!</math> cell <math>r_0\!</math> contains the symbol <math>\texttt{\#},</math> and

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At the time p_2, M is in the state q_#, and

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At the time <math>p_1,\!</math> cell <math>r_1\!</math> contains the symbol <math>\texttt{0},</math> and

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At the time p_2, H is reading cell r_1, and

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At the time <math>p_1,\!</math> cell <math>r_2\!</math> contains the symbol <math>\texttt{\#},</math> and

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At the time p_2, cell r_0 contains "#", and

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|-

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At the time p_2, cell r_1 contains "0", and

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|

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At the time p_2, cell r_2 contains "#".

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At the time <math>p_2,\!</math> machine <math>\operatorname{M}</math> is in the state <math>q_\#,\!</math> and

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At the time <math>p_2,\!</math> scanner <math>\operatorname{H}</math> is reading cell <math>r_1,\!</math> and

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At the time <math>p_2,\!</math> cell <math>r_0\!</math> contains the symbol <math>\texttt{\#},</math> and

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At the time <math>p_2,\!</math> cell <math>r_1\!</math> contains the symbol <math>\texttt{0},</math> and

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At the time <math>p_2,\!</math> cell <math>r_2\!</math> contains the symbol <math>\texttt{\#}.</math>

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|}

The output of <math>\operatorname{Stunt}(2)</math> being the symbol that rests under the tape head <math>\operatorname{H}</math> if and when the machine <math>\operatorname{M}</math> reaches one of its resting states, we get the result that <math>\operatorname{Parity}(0) = 0.</math>

Jon Awbrey

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