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218 these kind of not-regular-expressions is. Try to write down the regular expression which can match any |
218 these kind of not-regular-expressions is. Try to write down the regular expression which can match any |
219 string except {\it "hello"} and {\it "world"}. It is possible in principle, but often it is easier to just include |
219 string except {\it "hello"} and {\it "world"}. It is possible in principle, but often it is easier to just include |
220 $\sim{}r$ in the definition or regular expressions. Whenever we do so, we will state it explicitly. |
220 $\sim{}r$ in the definition or regular expressions. Whenever we do so, we will state it explicitly. |
221 |
221 |
222 So far we have only considered informally what the \emph{meaning} of a regular expression is. |
222 So far we have only considered informally what the \emph{meaning} of a regular expression is. |
223 Formally, we associate with every regular expression a set of strings which are matched by this |
223 To do so more formally we will associate with every regular expression a set of strings |
224 regular expression. This can be formally defined as |
224 that is supposed to be are matched by this |
|
225 regular expression. This can be defined recursively as follows |
225 |
226 |
226 \begin{center} |
227 \begin{center} |
227 \begin{tabular}{rcl} |
228 \begin{tabular}{rcl} |
228 $L(\varnothing)$ & $\dn$ & $\{\,\}$\\ |
229 $L(\varnothing)$ & $\dn$ & $\{\,\}$\\ |
229 $L(\epsilon)$ & $\dn$ & $\{""\}$\\ |
230 $L(\epsilon)$ & $\dn$ & $\{""\}$\\ |