equal
deleted
inserted
replaced
1200 This list is then "flattened"--$\ZERO$ will be |
1200 This list is then "flattened"--$\ZERO$ will be |
1201 thrown away by $\textit{flatten}$; $ _0(_0\ONE + _{11}a^*)$ |
1201 thrown away by $\textit{flatten}$; $ _0(_0\ONE + _{11}a^*)$ |
1202 is opened up to make the list consisting of two separate elements |
1202 is opened up to make the list consisting of two separate elements |
1203 $_{00}\ONE$ and $_{011}a^*$, note that $flatten$ |
1203 $_{00}\ONE$ and $_{011}a^*$, note that $flatten$ |
1204 $\fuse$s the bit(s) $_0$ to the front of $_0\ONE $ and $_{11}a^*$. |
1204 $\fuse$s the bit(s) $_0$ to the front of $_0\ONE $ and $_{11}a^*$. |
1205 In a nutshell, the order of simplification causes |
1205 The order of simplification, which impacts the order that alternatives |
|
1206 are opened up, causes |
1206 the bits to be moved differently. |
1207 the bits to be moved differently. |
1207 |
1208 |
1208 \subsubsection{A Failed Attempt To Remedy the Problem Above} |
1209 \subsubsection{A Failed Attempt To Remedy the Problem Above} |
1209 A simple class of regular expression and string |
1210 A simple class of regular expression and string |
1210 pairs $(r, s)$ can be deduced from the above example |
1211 pairs $(r, s)$ can be deduced from the above example |
1370 for the counterexample where |
1371 for the counterexample where |
1371 \begin{center} |
1372 \begin{center} |
1372 $r$ is the regular expression |
1373 $r$ is the regular expression |
1373 $(ab+(a^*+aa))$ and $s$ is the string $aa$ |
1374 $(ab+(a^*+aa))$ and $s$ is the string $aa$ |
1374 \end{center} |
1375 \end{center} |
1375 $\rup\backslash_{simp} \, s \neq \simp(\rup\backslash s)$ |
1376 |
1376 happens again, whereas this does not happen for the old |
1377 \noindent |
|
1378 $\rup\backslash_{simp} \, s$ is equal to |
|
1379 $ _0(_0\ONE + _{11}a^*)$ |
|
1380 $\rup\backslash_{simp} \, s \neq \simp(\rup\backslash s)$, |
|
1381 whereas this does not happen for the old |
1377 version of $\simp$. |
1382 version of $\simp$. |
|
1383 We have changed the algorithm to suppress the old |
|
1384 counterexample, but this gives rise to new counterexamples. |
1378 This dilemma causes this amendment not a successful |
1385 This dilemma causes this amendment not a successful |
1379 attempt to make $\rup\backslash_{simp} \, s = \simp(\rup\backslash s)$ |
1386 attempt to make $\rup\backslash_{simp} \, s = \simp(\rup\backslash s)$ |
1380 under every possible regular expression and string. |
1387 under every possible regular expression and string. |
1381 \subsection{Properties of the Function $\simp$} |
1388 \subsection{Properties of the Function $\simp$} |
1382 |
1389 |