Kleene's algorithm

In theoretical computer science, in particular in formal language theory, Kleene's algorithm transforms a given deterministic finite automaton (DFA) into a regular expression. Together with other conversion algorithms, it establishes the equivalence of several description formats for regular languages.

Algorithm description

According to Gross and Yellen (2004),[1] the algorithm can be traced back to Kleene (1956).[2]

This description follows Hopcroft and Ullman (1979).[3] Given a deterministic finite automaton M = (Q, Σ, δ, q0, F), with Q = { q0,...,qn } its set of states, the algorithm computes

the sets Rk
ij of all strings that take M from state qi to qj without going through any state numbered higher than k.

Here, "going through a state" means entering and leaving it, so both i and j may be higher than k, but no intermediate state may. Each set Rk
ij is represented by a regular expression; the algorithm computes them step by step for k = -1, 0, ..., n. Since there is no state numbered higher than n, the regular expression Rn
0j represents the set of all strings that take M from its start state q0 to qj. If F = { q1,...,qf } is the set of accept states, the regular expression Rn
01 | ... | Rn
0f represents the language accepted by M.

The initial regular expressions, for k = -1, are computed as

R−1
ij = a1 | ... | am       if i≠j, where δ(qi,a1) = ... = δ(qi,am) = qj
R−1
ij = a1 | ... | am | ε, if i=j, where δ(qi,a1) = ... = δ(qi,am) = qj

After that, in each step the expressions Rk
ij are computed from the previous ones by

Rk
ij = Rk-1
ik (Rk-1
kk)* Rk-1
kj | Rk-1
ij

By induction on k, it can be shown that the length[4] of each expression Rk
ij is at most 4k+1(6s+7) - 4/3 symbols, where s denotes the number of characters in Σ. Therefore, the length of the regular expression representing the language accepted by M is at most 4n+1(6s+7)f - f - 3/3 symbols, where f denotes the number of final states.

Example

Example DFA given to Kleene's algorithm

The automaton shown in the picture can be described as M = (Q, Σ, δ, q0, F) with

  • the set of states Q = { q0, q1, q2 },
  • the input alphabet Σ = { a, b },
  • the transition function δ with δ(q0,a)=q0,   δ(q0,b)=q1,   δ(q1,a)=q2,   δ(q1,b)=q1,   δ(q2,a)=q1, and δ(q2,b)=q1,
  • the start state q0, and
  • set of accept states F = { q1 }.

Kleene's algorithm computes the initial regular expressions as

R−1
00    		 = a | ε
R−1
01		 = b
R−1
02		 = ∅
R−1
10		 = ∅
R−1
11		 = b | ε
R−1
12		 = a
R−1
20		 = ∅
R−1
21		 = a | b
R−1
22		 = ε

After that, the Rk
ij are computed from the Rk-1
ij step by step for k = 0, 1, 2. Kleene algebra equalities are used to simplify the regular expressions as much as possible.

Step 0:

R0
00    		 = R−1
00 (R−1
00)* R−1
00 | R−1
00    		 = (a | ε) (a | ε)* (a | ε) | a | ε     = a*
R0
01		 = R−1
00 (R−1
00)* R−1
01 | R−1
01		 = (a | ε) (a | ε)* b | b = a* b
R0
02		 = R−1
00 (R−1
00)* R−1
02 | R−1
02		 = (a | ε) (a | ε)* ∅ | ∅ = ∅
R0
10		 = R−1
10 (R−1
00)* R−1
00 | R−1
10		 = ∅ (a | ε)* (a | ε) | ∅ = ∅
R0
11		 = R−1
10 (R−1
00)* R−1
01 | R−1
11		 = ∅ (a | ε)* b | b | ε = b | ε
R0
12		 = R−1
10 (R−1
00)* R−1
02 | R−1
12		 = ∅ (a | ε)* ∅ | a = a
R0
20		 = R−1
20 (R−1
00)* R−1
00 | R−1
20		 = ∅ (a | ε)* (a | ε) | ∅ = ∅
R0
21		 = R−1
20 (R−1
00)* R−1
01 | R−1
21		 = ∅ (a | ε)* b | a | b = a | b
R0
22		 = R−1
20 (R−1
00)* R−1
02 | R−1
22		 = ∅ (a | ε)* ∅ | ε = ε

Step 1:

R1
00    		 = R0
01 (R0
11)* R0
10 | R0
00    		 = a*b (b | ε)* ∅ | a*         = a*
R1
01		 = R0
01 (R0
11)* R0
11 | R0
01		 = a*b (b | ε)* (b | ε) | a* b = a* b* b
R1
02		 = R0
01 (R0
11)* R0
12 | R0
02		 = a*b (b | ε)* a | ∅ = a* b* ba
R1
10		 = R0
11 (R0
11)* R0
10 | R0
10		 = (b | ε) (b | ε)* ∅ | ∅ = ∅
R1
11		 = R0
11 (R0
11)* R0
11 | R0
11		 = (b | ε) (b | ε)* (b | ε) | b | ε = b*
R1
12		 = R0
11 (R0
11)* R0
12 | R0
12		 = (b | ε) (b | ε)* a | a = b* a
R1
20		 = R0
21 (R0
11)* R0
10 | R0
20		 = (a | b) (b | ε)* ∅ | ∅ = ∅
R1
21		 = R0
21 (R0
11)* R0
11 | R0
21		 = (a | b) (b | ε)* (b | ε) | a | b = (a | b) b*
R1
22		 = R0
21 (R0
11)* R0
12 | R0
22		 = (a | b) (b | ε)* a | ε = (a | b) b* a | ε

Step 2:

R2
00    		 = R1
02 (R1
22)* R1
20 | R1
00    		 = a*b*ba ((a|b)b*a | ε)* ∅ | a* = a*
R2
01		 = R1
02 (R1
22)* R1
21 | R1
01		 = a*b*ba ((a|b)b*a | ε)* (a|b)b* | a* b* b = a* b (a (a | b) | b)*
R2
02		 = R1
02 (R1
22)* R1
22 | R1
02		 = a*b*ba ((a|b)b*a | ε)* ((a|b)b*a | ε) | a* b* ba = a* b* b (a (a | b) b*)* a
R2
10		 = R1
12 (R1
22)* R1
20 | R1
10		 = b* a ((a|b)b*a | ε)* ∅ | ∅ = ∅
R2
11		 = R1
12 (R1
22)* R1
21 | R1
11		 = b* a ((a|b)b*a | ε)* (a|b)b* | b* = (a (a | b) | b)*
R2
12		 = R1
12 (R1
22)* R1
22 | R1
12		 = b* a ((a|b)b*a | ε)* ((a|b)b*a | ε) | b* a = (a (a | b) | b)* a
R2
20		 = R1
22 (R1
22)* R1
20 | R1
20		 = ((a|b)b*a | ε) ((a|b)b*a | ε)* ∅ | ∅ = ∅
R2
21		 = R1
22 (R1
22)* R1
21 | R1
21		 = ((a|b)b*a | ε) ((a|b)b*a | ε)* (a|b)b* | (a | b) b* = (a | b) (a (a | b) | b)*
R2
22		 = R1
22 (R1
22)* R1
22 | R1
22		 = ((a|b)b*a | ε) ((a|b)b*a | ε)* ((a|b)b*a | ε) | (a | b) b* a | ε     = ((a | b) b* a)*

Since q0 is the start state and q1 is the only accept state, the regular expression R2
01 denotes the set of all strings accepted by the automaton.

See also

References

  1. ↑ Jonathan L. Gross and Jay Yellen, ed. (2004). Handbook of Graph Theory. Discrete Mathematics and it Applications. CRC Press. ISBN 1-58488-090-2. Here: sect.2.1, remark R13 on p.65
  2. ↑ Kleene, Stephen C. (1956). "Representation of Events in Nerve Nets and Finite Automate" (PDF). Automata Studies, Annals of Math. Studies. Princeton Univ. Press. 34. Here: sect.9, p.37-40
  3. ↑ John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here: Theorem 2.4, p.33-34
  4. ↑ More precisely, the number of regular-expression symbols, "ai", "ε", "|", "*", "·"; not counting parantheses.
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