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1 Summary Showing regular construct DFA, NFA construct regular expression show L is the union, concatenation, intersection (regular operations) of regular languages. Showing non-regular pumping lemma assume regular, apply closure properties of regular languages and obtain a known non-regular language.

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2 Equivalence & Minimization of Automata

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3 Equivalence of a state in a DFA regular languages Minimization of DFA smallest possible DFA for a given regular language Outline

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4 Consider the accept states c and g. They are both sinks meaning that any string which ever reaches them is guaranteed to be accepted later. Q: Do we need both states? Equivalent States. Example a b 1 d 0,1 e 1 c g f

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5 A: No, they can be unified as illustrated below. Q: Can any other states be unified because any subsequent string suffixes produce identical results? Example (cont) a b 1 d 0,1 e 1 cg f

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6 A: Yes, b and f. Notice that if you’re in b or f then: if string ends, reject in both cases if next character is 0, forever accept in both cases if next character is 1, forever reject in both cases So unify b with f. Example (cont) a b 1 d 0,1 e 1 cg f

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7 Intuitively two states are equivalent if all subsequent behavior from those states is the same. Come up with a formal characterization of state equivalence. Q: Any other ways to simplify the automaton? Example (cont) a 0,1 d e 1 cg bf 0 0 1

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8 A: Get rid of d. i.e. Getting rid of unreachable useless states doesn’t affect the accepted language. Example (cont) a 0,1 e cg bf 0 1

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9 Equivalent States. Definition DEF: Two states q and q’ in a DFA M = (Q, Σ, δ, q 0, F ) are said to be equivalent (or indistinguishable) if for all strings u Σ *, the states on which u ends on when read from q and q’ are both accept, or both non-accept. Equivalent states may be glued together without affecting M’ s behavior.

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10 DEF: An automaton is irreducible if it contains no useless states, and no two distinct states are equivalent. The goal of minimization algorithm is to create irreducible automata from arbitrary ones. Later: remarkably, the algorithm actually produces smallest possible DFA for the given language, hence the name “minimization”. Minimization Algorithm. Goals

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11 Minimization of DFA’s Example: The DFA has equivalence classes {{A,E}, {B,H}, {C}, {D,F}, {G}}

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12 Equivalence Class Transitivity: If p ≡ q and q ≡ r, then p ≡ r Proof: Suppose to the contrary that p ≢ r Then or vice versa. Now is either accepting or not If Otherwise The vice versa case is proved symmetrically

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13 Minimization of DFA’s (cont)

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14 Example Can be minimized to

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15 Cannot apply the TF-algo to NFA’s For Example, to minimize Simply remove state C However, A ≢ C

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16 Let B be the minimized DFA obtained by applying the TF-Algo to DFA A. We know that L(A) = L(B) What if there existed a DFA C with L(C) = L(B) and fewer states than B? If we run the TF-Algo on B “union” C Also If we could distinguish successors then Minimized DFA Cannot be Beaten

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17 Claim: For each state p in B there is at least one state q in C, s.t. Proof: There are no inaccessible states, so Now and Since C has fewer states than B, there must be two states r & s of B s.t. For some state t of C. But CONTRADICTION!! Minimized DFA Cannot be Beaten (cont)

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