flac-hw4

flac-hw4 - FLAC Assignment 4 Exercise 1. Consider a class...

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FLAC Assignment 4 Exercise 1. Consider a class of automata in which the set of states is S = { 0 ,..., 7 } , the alphabet is Σ = { a , b } , the starting state is s 0 = 0, and the transition relation δ is given by: δ a b 0 5 4 1 4 4 2 1 0 3 0 1 4 7 7 5 6 7 6 3 2 7 3 3 a. Assign a set of ﬁnal states F so that M = ( S, Σ ,s 0 ,δ,F ) is already a reduced automa- ton. b. Assign a set of ﬁnal states F , where ∅ 6 = F 6 = S , so that M reduces to the smallest possible automaton. Exercise 2. We say that an equivalence relation E on Σ * is invariant iﬀ for all u,v Σ * , wE w 0 ( uwv ) E ( uw 0 v ). Prove that L Σ * is a regular language iﬀ there is an invariant equivalence relation E of ﬁnite index 1 such that w L ⇒ { w 0 | w E w 0 } ⊆ L . Exercise 3. Prove that the following language is regular: The set of sequences of 1000 × 1000 binary matrices that, when multiplied together modulo 2, yield the 1000 × 1000 identity matrix I = 1 0 ··· 0 0 1 ··· 0 . . . . . . . . . . . . 0 0

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flac-hw4 - FLAC Assignment 4 Exercise 1. Consider a class...

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