I have the problem of needing to construct a regular expression corresponding to the set of strings of 0's and 1's whose number of 0's is divisible by five and whose number of 1's is even.
Construction of a regular expression that satisfies either of these conditions is not difficult, but I'm having difficulty constructing one that satisfies both conditions. Any suggestions?
Hmm. Only for fun, but not as something remotely practical.
Consider the 64 expressions of the form $\alpha_1 0 \alpha_2 0 \alpha_3 0 \alpha_4 0 \alpha_5 0 \alpha_6$, where $\alpha_i$ is either $(11)^*$ or $(11)^*1$. Together they will represent all strings that have five $0$'s and any number of $1$'s. Group 32 of them into an expression $E$ for an even number of $1$'s and 32 of them into expression $O$ for an odd number of $1$'s. Then your expression is $(E+OE^*O)^*$.
You can halve the size by first considering only strings that end in a $0$ (and omitting $\alpha_6$) and dealing with trailing $1$'s separately.