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CS 461 – Aug. 31

CS 461 – Aug. 31. Section 1.2 – Nondeterministic FAs How to trace input √ NFA design makes “union” operation easier Equivalence of NFAs and DFAs. NFA’s using “or”. Can you draw NFA for: { begin with 0 or end with 1 } ?. Old start 1. ε. New start. Old start 2. ε. Amazing fact.

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CS 461 – Aug. 31

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  1. CS 461 – Aug. 31 Section 1.2 – Nondeterministic FAs • How to trace input √ • NFA design makes “union” operation easier • Equivalence of NFAs and DFAs

  2. NFA’s using “or” • Can you draw NFA for: { begin with 0 or end with 1 } ? Old start 1 ε New start Old start 2 ε

  3. Amazing fact • NFA = DFA • In other words, the two kinds of machines have the same power. • Proof idea: we can always convert a DFA into an NFA, or vice versa. Which do you think is easier to do?

  4. Formal NFA def’n • The essential difference with DFA is in the transition function: DFA δ: Q x Σ Q NFA δ: Q x Σε P(Q) • Thus, converting DFA  NFA is easy. We already satisfy the definition!

  5. NFA  DFA construction • When creating DFA, states will be all possible subsets of states from NFA. • This takes care of “all possible destinations.” • In practice we won’t need whole subset: only create states as you need them. • “empty set” can be our dead state. • DFA start state = NFA’s start state or anywhere you can begin for free. Happy state will be any subset containing NFA’s happy state. • Transitions: Please write as a table. Drawing would be too cluttered. When finished, can eliminate useless states.

  6. Example #1 • NFA transition table given to the right. • DFA start state is {1, 3}, or more simply 13. • DFA accept state would be anything containing 1. Could be 1, 12, 13, 123, but we may not need all these states.

  7. continued • The resulting DFA could require 2n states, but we should only create states as we need them. Let’s begin: If we’re in state 1 or 3, where do we go if we read an ‘a’ or a ‘b’? δ(13, a) = 1, but we can get to 3 for free. δ(13, b) = 2. We need to create a new state “2”. Continue the construction by considering transitions from state 2.

  8. NFA DFA answer Notice that the DFA is in fact deterministic: it has exactly one destination per transition. Also there is no column for ε.

  9. Example #2 • NFA transition table given to the right. • DFA start state is A. • DFA accept state would be anything containing D.

  10. continued Let’s begin. δ(A, 0) = A δ(A, 1) = AC We need new state AC. δ(AC, 0) = A δ(AC, 1) = ABC Continue from ABC…

  11. NFA DFA answer

  12. final thoughts • NFAs and DFAs have same computational power. • NFAs often have fewer states than corresponding DFA. • Typically, we want to design a DFA, but NFAs are good for combining 2+ DFAs. • After doing NFA  DFA construction, we may see that some states can be combined. • Later in chapter, we’ll see how to simplify FAs.

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