# nerode - Myhill-Nerode Handout Definition An equivalence...

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Unformatted text preview: Myhill-Nerode Handout Definition. An equivalence relation E on strings is right invariant iff concatenating a string w onto two equivalent strings u and v produces two strings ( uw and vw ) that are also equivalent; i.e., for all strings u , v , and w , we have uE v ⇒ uw E vw . Theorem 1. A language L is accepted by a DFA iff L is the union of some equivalence classes of a right-invariant equivalence relation of finite index. Proof, Part A. Suppose that a lan- guage L is accepted by a DFA M = h S, Σ ,δ,s ,F i . Define an equivalence relation E so that two strings u and w are equivalent iff the DFA (starting in state s ) would transition to the same state by reading either u or w , i.e., uE v iff ˆ δ ( s ,u ) = ˆ δ ( s ,v ) Let “ EC ( s i )” denote the equivalence class { w | ˆ δ ( s ,w ) = s i } (i.e., the set of strings that transition the DFA to state s i ). It is easy to verify that all members of EC ( s i ) are indeed equivalent to each other. E is of finite index, because the set of equivalence classes is...
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nerode - Myhill-Nerode Handout Definition An equivalence...

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