COMS W3261
Computer Science Theory
Lecture 6: September 28, 2009
Constructing a Regular Expression from a DFA

1. Outline

• Review
• From DFA's to regular expressions by Kleene's algorithm
• From DFA's to regular expressions by eliminating states

2. Review

• In past lectures we have shown DFA's, NFA's, and ε-NFA's are equivalent in definitional capability.
• Today we show regular expressions are equivalent in definitional capability to finite automata.

3. From DFA's to Regular Expressions by Kleene's algorithm

• Given a DFA A, Kleene's algorithm constructs a regular expression R from A such that L(R) = L(A).
• Suppose the states of A are numbered 1, 2, ..., n.
• Kleene's algorithm constructs a set of regular expressions R[i,j,k] representing all paths from state i to state j with no intermediate node in the path numbered higher than k:

for (i = 1; i <= n; i++)
  for (j = 1; j <= n; j++)
    if (i != j)
      if (there are transitions from state i to state j labeled a1, a2, ..., ak)
        R[i,j,0] = a1 + a2 + ... + ak;
      else
        R[i,j,0] = ∅;
    else if (i == j)
      if (there are transitions from state i to state i labeled a1, a2, ..., ak)
        R[i,i,0] = &epsilon + a1 + a2 + ... + ak;
      else
        R[i,i,0] = ε;
for (i = 1; i <= n; i++)
  for (j = 1; j <= n; j++)
    for (k = 1; k <= n; k++)
      R[i,j,k] = R[i,j,k-1] + R[i,k,k-1](R[k,k,k-1])*R[k,j,k-1];
  

• Assuming the start state is 1, the regular expression for the DFA A is then the sum (union) of all expressions R[1,j,n] such that j is a final state.
• Note that Kleene's algorithm for constructing a regular expression from a DFA reduces to the transitive closure algorithm (Warshall's algorithm) and to the all-pairs shortest paths algorithm (Floyd's algorithm) for directed graphs.
• Example 3.5, HMU, pp. 95-97.

4. From DFA's to RE's by Eliminating States

• Kleene's algorithm gives us a mechanical way to construct a regular expression from a DFA, or for that matter, from any NFA or ε-NFA.
• Another approach that avoids duplicating work is to eliminate states, one at a time, from the DFA using the procedure outlined in Section 3.2.2 of HMU.
• Example 3.6, HMU, pp. 101-102.
• You can see an expanded treatment of state elimination with more examples in pp. 583-588 of Chapter 10 of Aho and Ullman, Foundations of Computer Science.

5. Practice Problems

1. Consider the DFA D with:
1. Q = {1, 2, 3}
2. Σ = {a, b}
3. δ:



State	Input Symbol
	a	b
1	2	1
2	3	1
3	3	2



4. Start state: 1
5. F = {3}


a) Use Kleene's algorithm to construct a regular expression for L(D). Simplify your expressions as much as possible at each stage.
b) Construct a regular expression for L(D) by eliminating state 2.


2. HMU, Exercise 3.2.3, p. 107.

6. Reading Assignment

• HMU: Sects. 3.1-3.3