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

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====Output Conditions for Tape Input "1"====

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Let <math>p_1\!</math> be the proposition that we get by conjoining the proposition that describes the initial conditions for tape input "1" with the proposition that describes the truncated turing machine <math>\operatorname{Stunt}(2).</math> As it turns out, <math>p_1\!</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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<pre>

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~~Let P_1 be the proposition that we get by conjoining~~

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~~the proposition that describes the initial conditions~~

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~~for tape input "1" with the proposition that describes~~

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~~the truncated turing machine Stunt(2). As it turns out,~~

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~~P_1 has a single satisfying interpretation, and this is~~

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~~represented as a singular proposition in terms of its~~

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~~positive logical features in the following display:~~

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o-------------------------------------------------o

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Line 2,658: Line 2,654:

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o-------------------------------------------------o

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</pre>

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The Output Conditions for Tape Input "1" 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 "1", 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{1},</math> and

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At the time p_1, M is in the state q_1, 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 "1", and

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At the time <math>p_1,\!</math> machine <math>\operatorname{M}</math> is in the state <math>q_1,\!</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{1},</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 "1", 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{1},</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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|}

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The output of Stunt(2) being the symbol that rests under

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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> when and if the machine <math>\operatorname{M}</math> reaches one of its resting states, we get the result that <math>\operatorname{Parity}(1) = 1.</math>

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the tape head H when and if the machine M reaches one of

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its resting states, we get the result that Parity(1) = 1.

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</~~pre~~>

==Work Area==

Jon Awbrey

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