Minimization of DFA
- 1. Minimization of DFA • Name: Kunj Desai • Enrollment No.:140950107022 • Branch: CSE-A • Year : 2017
- 2. DFA • Deterministic Finite Automata (DFSA) • (Q, Σ, δ, q0, F) • Q – (finite) set of states • Σ – alphabet – (finite) set of input symbols • δ – transition function • q0 – start state • F – set of final / accepting states
- 3. DFA • Often representing as a diagram:
- 4. DFA Minimization • Some states can be redundant: • The following DFA accepts (a|b)+ • State s1 is not necessary
- 5. DFA Minimization • So these two DFAs are equivalent:
- 6. DFA Minimization • This is a state-minimized (or just minimized) DFA • Every remaining state is necessary
- 7. DFA Minimization • The task of DFA minimization, then, is to automatically transform a given DFA into a state-minimized DFA • Several algorithms and variants are known • Note that this also in effect can minimize an NFA (since we know algorithm to convert NFA to DFA)
- 8. DFA Minimization Algorithm • Recall that a DFA M=(Q, Σ, δ, q0, F) • Two states p and q are distinct if • p in F and q not in F or vice versa, or • for some α in Σ, δ(p, α) and δ(q, α) are distinct • Using this inductive definition, we can calculate which states are distinct
- 9. DFA Minimization Algorithm • Create lower-triangular table DISTINCT, initially blank • For every pair of states (p,q): • If p is final and q is not, or vice versa • DISTINCT(p,q) = ε • Loop until no change for an iteration: • For every pair of states (p,q) and each symbol α • If DISTINCT(p,q) is blank and DISTINCT( δ(p,α), δ(q,α) ) is not blank • DISTINCT(p,q) = α • Combine all states that are not distinct
- 10. Very Simple Example s0 s1 s2 s0 s1 s2
- 11. Very Simple Example s0 s1 ε s2 ε s0 s1 s2 Label pairs with ε where one is a final state and the other is not
- 12. Very Simple Example s0 s1 ε s2 ε s0 s1 s2 Main loop (no changes occur)
- 13. Very Simple Example s0 s1 ε s2 ε s0 s1 s2 DISTINCT(s1, s2) is empty, so s1 and s2 are equivalent states
- 14. Very Simple Example Merge s1 and s2
- 16. More Complex Example •Check for pairs with one state final and one not:
- 17. More Complex Example • First iteration of main loop:
- 18. More Complex Example • Second iteration of main loop:
- 19. More Complex Example • Third iteration makes no changes • Blank cells are equivalent pairs of states
- 20. More Complex Example •Combine equivalent states for minimized DFA:
- 21. Conclusion • DFA Minimization is a fairly understandable process, and is useful in several areas • Regular expression matching implementation • Very similar algorithm is used for compiler optimization to eliminate duplicate computations • The algorithm described is O(kn2 ) • John Hopcraft describes another more complex algorithm that is O(k (n log n) )