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Reducing DFA: Finding the Smallest Equivalent DFA through Distinguishable States

This article explores the process of minimizing deterministic finite automata (DFA) to find the smallest equivalent representation of a given language. It introduces an algorithm that identifies indistinguishable states using a two-step approach: marking state pairs as distinguishable or indistinguishable and then merging indistinguishable states. The Mark and Reduce algorithms are detailed, highlighting the significance of removing inaccessible states and rewriting transitions. Illustrated with examples, this guide underscores the importance of efficient DFA representation in computational theory.

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Reducing DFA: Finding the Smallest Equivalent DFA through Distinguishable States

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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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