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Reducing DFA’s

Reducing DFA’s. Section 2.4. Reduction of DFA. For any language, there are many DFA’s that accept the language Why would we want to find the smallest? Algorithm: Finds smallest equivalent DFA. Distinguishable States.

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Reducing DFA’s

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  1. Reducing DFA’s Section 2.4

  2. Reduction of DFA • For any language, there are many DFA’s that accept the language • Why would we want to find the smallest? • Algorithm: Finds smallest equivalent DFA

  3. Distinguishable States • A state p is indistinguishable from another q if, for all walks w, δ*(p,w)  F implies δ*(q,w)  F and δ*(p,w)  F implies δ*(q,w)  F • Otherwise, they are distinguishable

  4. Two Step Algorithm • First, mark all pairs of states as distinguishable or indistinguishable • Then, merge indistinguishable states into one state for the smaller graph

  5. Mark Algorithm • Remove inaccessible states • Mark all states in F as distinguishable from those not in F. • Repeat until all pairs are marked: For all pairs (p,q) and all symbols (a), if δ(p,a) is distinguishable from δ(q,a), then p is distinguishable from q.

  6. Reduce Algorithm • Create a state for each set of indistinguishable states from the Mark algorithm. • Rewrite transitions between states. If δ(p,a) = q, then make a transition from the node containing the original p to the node containing the original q and label it a.

  7. Example q1 1 0 0 0 1 q4 q0 q2 0,1 0 1 1 q3

  8. Example Distinguishable Pairs Final –Nonfinal states (q0,q4) (q1,q4) (q2,q4) (q3,q4)

  9. Example Distinguishable Pairs: Chart Compare (q0,q4) (q1,q4) (q2,q4) (q3,q4) (q0,q1) (q0,q2) (q0,q3)

  10. Example Distinguishable Pairs: (q0,q4) (q1,q4) (q2,q4) (q3,q4) (q0,q1) (q0,q2) (q0,q3) Indistinguishable Pairs: {q0} {q1, q2,q3} {q4}

  11. q1 1 0 0 0 1 q4 q0 q2 0,1 0 1 1 q3 Example 0 0,1 0,1 1 0 1,2,3 4

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