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Regular Expression to determine if a base 10 number is divisible by 3

Original posted on Erik Vold's Blog

The Problem:

For the regular language L = { w | w mod 3 = 0 }, where the alphabet is {0,1,2,3,4,5,6,7,8,9}; give the deterministic finite automaton (DFA) for L, and convert this to a regular expression.

The Solution:

The DFA ( S, Σ, T, s, A ):

S = {q0,q1,q2}

Σ = {0,1,2,3,4,5,6,7,8,9}

T = (doing the state diagram below)

s = {q0}

A = {q0}

For shorthand I will divide the alphabet, Σ, into:

  • A={0,3,6,9}
  • B={1,4,7}
  • C={2,5,8}

The state diagram:

DFA State Diagram

Now to convert the DFA state diagram into a regular expression. This is done by converting the DFA into generalized non deterministic finite automaton (GNFA), and then converting the GNFA into a regular expression.

Conversion of DFA into GNFA, and removal of q0 state


Notice in the above that I did two steps in one; I first converted the DFA into a GNFA (which is the easy part), then I removed the q0 state.

Removing the q1 state:

Removing the q1 state

Finally, removing the q2 state:

Removing the q2 state

Therefore the regular expression that defines the regular language L is:
(A+)∪((B∪A*B)(A∪CA*B)*CA*)∪((C∪A*C∪(B∪A*B)(A∪CA*B)*(B∪CA*C))(A∪BA*C∪(C∪BA*B)(A∪CA*B)*(B∪CA*C))*(BA*∪(C∪BA*B)(A∪CA*B)*CA*))

For further reading please see "Introduction to the Theory of Computation" by Michael Sipser


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