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,...,q_n } its set of states, the algorithm computes

the sets Rk
ij of all strings that take M from state q_i to q_j 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 q_j. If F = { q₁,...,q_f } 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 | ... | a_m       if i≠j, where δ(q_i,a₁) = ... = δ(q_i,a_m) = q_j

R-1
ij = a₁ | ... | a_m | ε, if i=j, where δ(q_i,a₁) = ... = δ(q_i,a_m) = q_j

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

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	=
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	=
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^*	=
R2 12	= R1 12 (R1 22)^* R1 22 | R1 12	= b^* a	((a|b)b^*a | ε)^*	((a|b)b^*a | ε)	| 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^*	=
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	=

((step 2 simplification to be completed))

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

• Floyd-Warshall algorithm — an algorithm on weighted graphs that can be implemented by Kleene's algorithm using a particular Kleene algebra
• Star height problem — what is the minimum stars' nesting depth of all regular expressions corresponding to a given DFA?
• Generalized star height problem — if a complement operator is allowed additionally in regular expressions, can the stars' nesting depth of Kleene's algorithm's output be limited to a fixed bound?
• Thompson's construction algorithm — transforms a regular expression to a finite automaton

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.
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