1
Here are the first few even natural numbers:
$$
\begin{array}{r}
0 \\ 10 \\ 100 \\ 110 \\ 1000
\end{array}
$$
Here are the first few odd natural numbers:
$$
\begin{array}{r}
1 \\ 11 \\ 101 \\ 111 \\ 1001
\end{array}
$$
Notice any pattern? You take it from here.
1
This proof looks correct. You can shorten it a bit (it doesn't need to be so descriptive), and you would get the following proof:
$h(L)=h(L(r_1\cup r_2))=h(L(r_1)\cup L(r_2)) = h(L(r_1))\cup h(L(r_2)) = L(r'_1)\cup L(r'_2)=L(r'_1\cup r'_2)$
Where the transition $h(L(r_1)\cup L(r_2)) = h(L(r_1)) \cup h(L(r_2))$ can be shown for any function $h$ (not ...
1
The $+$ sign stands for union, but you got the order of operations wrong. Your regular expression is the sum of $(b+c)^*$ and
$$ (b+c)^*((a+aa)(b+c)^+)^*(a+aa)(b+c)^*. $$
1
Note that $abbbb+aba^*bbb = aba^*bbb$. Also, note that you can "factorize" common expressions in the parentheses.
Your regular expression is equivalent to $b(a(\varepsilon + ba^*b)bb)^*$ (I don't know if it can be simplified again).
1
They're not regular, but finite state transducers can be used to
generate and recognize complicated morphology, such as Finnish
numerals. – phipsgabler May 7 '20 at 7:31
Similarly, a long time ago I worked using FST to identify multiword verbal expressions (such as 'tomar el pelo' [to pull somebody's leg]) in Spanish texts; words were previously tagged. ...
1
Any of the following should work:
$(1+01)^*(0+\epsilon)$
$(0+\epsilon)(1+10)^*$
$1^*(011^*)^*(0+\epsilon)$
$(0+\epsilon)(11^*0)^*1^*$
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