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2(5 points)Myhill-Nerode Versus The Pumping Lemma.Consider the languageF={aibjck|i, j, k≥0and ifi= 1,thenj=k}over the alphabet{a, b, c}.1. (2 points) Show thatFis not regular.2. (2 points) Show thatFacts like a regular language in the standard pumping lemma (slide 3, lecture 4). In otherwords, give a pumping lengthpand demonstrate thatFhas the three properties guaranteed by the pumpinglemma for this value ofp.3. (1 point) Explain why parts (a) and (b) do not contradict the standard pumping lemma.3(1 point)Streaming AlgorithmsRecall the Distinct Elements problem from lecture: we have ann-element data streamx:= (x0, . . . , xn-1)in which each elementxiis not a bit, but rather an element of the universeU={1, . . . ,2k}.When the streamxends, the goal is to output the number of distinct elements inx. We noted that Distinct Elementscan be computed using a trivial streaming algorithm which usesn×kbits of memory (storing the entire stream).Show that Distinct Elements can also be computed using a streaming algorithm which usesO(2k+ logn)bits ofmemory. You do not have to give formal proofs – just give the algorithm and explain informally how they work.4(3 points)Communication ComplexityIn this problem, letnbe a power of two for simplicity.Recall from lecture thedistributed version of the majority problem: for any integern≥1andx, y∈ {0,1}n,MAJORITY(x,y)= 1⇐⇒the number of1’s inxyexceeds the number of0’s.Prove that for alln= 2k, the communication complexity of MAJORITY onn-bit strings is at leastk= log2n.That is, the simple communication protocol for MAJORITY given in class is essentially optimal! (Again, you don’thave to be super-rigorous here; your answer will be perfectly fine if its formality is at the same level as our proof inclass of the communication lower bound for EQUALS.)2

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- Fall '19