|
I wouldn't want to actually compute it, but here's one way you can do it: Let R_i be an unknown regexp that matches base 10 expansion of numbers with residue i mod 7 (for simplicity, include the empty string --representing 0-- in the strings matched for R_0). We want R_0 and will get it from a system of equations for R_0, ..., R_6. The key is that any string matched by, say (R_j)k is a number congruent to 10j+k mod 7, so we get the equation R_i = sum( R_j k ) + delta_i0 where * the sum means union, i.e., operator | for regexes, and is taken over all 0<=j<=6 and 0<=k<=9 such that 10j+k = i mod 7, and * delta_i0 is a term which is not there unless i=0 in which case it is equal to epsilon, (the language consisting solely of) the empty string. You solve such a system by solving for one unknown at a time and substituting. An equation of the form R = RD | A, where R is one of the unknowns, and does not appear in either D or A, has solution R = AD*. You take that solution and plug it into all other equations, and repeat until you just get an equation for R_0. It's solution is the regex you want (except maybe you didn't want it to also match the empty string). Obviously this should not be done by hand. |