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**Summary Showing regular Showing non-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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**Equivalence & Minimization of Automata**

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**Outline Equivalence of a state in a DFA regular languages**

Minimization of DFA smallest possible DFA for a given regular language

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**Equivalent States. Example**

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? 0,1 b c 1 0,1 1 a d e 0,1 1 1 f g

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**Example (cont) A: No, they can be unified as illustrated below.**

Q: Can any other states be unified because any subsequent string suffixes produce identical results? b 0,1 1 1 a d 0,1 e cg 1 1 f

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**Example (cont) 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. b 0,1 1 1 a d 0,1 e cg 1 1 f

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Example (cont) 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? 0,1 1 a d 0,1 e cg 0,1 1 bf

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**Example (cont) A: Get rid of d.**

i.e. Getting rid of unreachable useless states doesn’t affect the accepted language. 0,1 a 0,1 e cg 0,1 1 bf

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**Equivalent States. Definition**

Two states q and q’ in a DFA M = (Q, Σ, δ, q0, 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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**Minimization Algorithm. Goals**

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”.

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**Minimization of DFA’s Example: The DFA has equivalence classes**

{{A,E}, {B,H}, {C}, {D,F}, {G}}

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**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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**Minimization of DFA’s (cont)**

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Example Can be minimized to

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**Cannot apply the TF-algo to NFA’s**

For Example, to minimize Simply remove state C However, A ≢ C

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**Minimized DFA Cannot be Beaten**

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

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**Minimized DFA Cannot be Beaten (cont)**

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!!

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