+and+construct+the+following+sets:+R(0)11+%3D+%EF%81%A5+%2B+1%3B+R(0)12+%3D+0%3B+R(0)21+%3D+%EF%83%86%3B+and+R(0)22+%3D+%EF%81%A5+%3B+each+corresponding+the+single-arc+transitions+from+one+state+to+another+state.+Using+the+recursive+rule%2C+we+can+now+construct+the+R(k)ij+%E2%80%98s+with+k+%3D+1:+Recursive+rule+Simplified.jpg)
ij+are+all+regular+expression+because+their+base+step+(when+k+%3D+0)+start+with+regular+expression+and%2C+in+each+step+of+the+recursive+rule%2C+only+regular+expression+operations+(i.e.%2C+%2B%2C+*%2C+and+concatenation)+are+used.+To+complete+the+proof+that+a+DFA+M+can+be+converted+to+an+equivalent+regular+expression%2C+we+construct+R(n)1j+for+each+of+the+accepting+states+j.+(Recall+the+states+are+labeled+1+through+n%2C+and+state+1is+the+start+state.).+Then+L(M)+%3D+the+sum+(i.e.+the+union)+of+these+R(n)1j+%E2%80%99s%2C+where+state+j+ranges+over+all+accepting+states+of+M..jpg)
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