Formal regular expression for a language over a,b,c such that a is never adjacent to b

I am trying to write a regex query for a language with letters a,b,c such that a is never adjacent to b.

Can it be done by using only the alternation (plus), concatenation and repetition (multiplication) operators?

L = w belongs to {a,b,c}* such that a is never adjacent to b

Solution

(Lets see if I recall enough formal language theory.)

Such a regular expression could be built with help of a DFA like this:

A = aA + cC + F      // only a or c can follow a
B = bB + cC + F      // only b or c can follow b
C = cC + aA + bB + F // any char can follow c

Where A, B and C are states representing the state when a, b and c respectively was the previous character. Since any character can follow c we can make C our start state. F being the final end state (end of string).

This DFA can be converted to a regular expression like this:

A = a*(cC+F) // eliminate recursion
B = b*(cC+F) // eliminate recursion

C = cC + aA + bB + F
  = cC + aa*(cC+F) + bb*(cC+F) + F       // substitute A and B
  = (c + aa*c + bb*c)C + aa*F + bb*F + F // regroup
  = (c + aa*c + bb*c)*(aa*F + bb*F + F)  // eliminate recursion
  = (c + aa*c + bb*c)*(aa* + bb* + e)F   // regroup

So the expression would be:

(c + aa*c + bb*c)*(aa* + bb* + e) // e being the empty/null string

Or in informal regex format:

(c|a+c|b+c)*(a+|b+)?

Which can be shortened to:

(a+c|b*c)*(a*|b*)