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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
Revision as of 19:38, 15 March 2009
, 19:38, 15 March 2009→Note 27: markup
==Note 27==
==Note 27==
===Interpretation of the Propositional Program (cont.)===
Interpretation of the Propositional Program (cont.)
'''State Partition'''
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{l}
\texttt{((~p0\_q0~),(~p0\_q1~),(~p0\_q\#~),(~p0\_q*~))}
\\
\texttt{((~p1\_q0~),(~p1\_q1~),(~p1\_q\#~),(~p1\_q*~))}
\\
\texttt{((~p2\_q0~),(~p2\_q1~),(~p2\_q\#~),(~p2\_q*~))}
\end{array}</math>
|}
In cactus syntax, an expression of the form <math>\texttt{((} e_1 \texttt{),(} e_2 \texttt{),(} \ldots \texttt{),(} e_k \texttt{))}</math> expresses a statement to the effect that ''exactly one of the expressions <math>e_j\!</math> is true, for <math>j = 1 ~\mathit{to}~ k.</math>'' Expressions of this form are called ''universal partition'' expressions, and the corresponding ''painted and rooted cactus'' (PARC) has the following shape:
<br>
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| ((e_1),(e_2),(...),(e_k)) |
| ((e_1),(e_2),(...),(e_k)) |
o---------------------------------------o
o---------------------------------------o
</pre>
<br>
The State Partition segment of the propositional program consists of three universal partition expressions, taken in conjunction expressing the condition that <math>M\!</math> has to be in one and only one of its states at each point in time under consideration. In short, we have the constraint:
</pre>
{| align="center" cellpadding="8" width="90%"
|
<p>At each of the points in time <math>p_i,\!</math> for <math>i\!</math> in the set <math>\{ 0, 1, 2 \},\!</math></p>
<p><math>M\!</math> can be in exactly one state <math>q_j,\!</math> for <math>j\!</math> in the set <math>\{ 0, 1, \#, * \}.</math></p>
|}
==Note 28==
==Note 28==