Let ?= (a, b). The Language L = {w E ?. : na(w) < na(w)) is not regular. (Note: na(w) and nu(w) are the number of a's and 's in tw, respectively.) To show this language is not regular, suppose you are given p. You now have complete choice of w. So choose wa+1, Of course you see how this satisfies the requirements of words in the language. Now, answer the following: (a) What is the largest value of lryl?...
(d) Let L be any regular language. Use the Pumping Lemma to show that In > 1 such that for all w E L such that|> n, there is another string ve L such that lvl <n. (4 marks) (e) Let L be a regular language over {0,1}. Show how we can use the previous result to show that in order to determine whether or not L is empty, we need only test at most 2" – 1 strings. (2...
1. Let L = {ambm cn | m <n}. Use the pumping lemma to show that L is not a CFL.
Question 8 10 pts Let S = {a,b,c}. Write a grammar that generates the language: L = {(ac)"6n+1w: n > 0, W € 2*, W contains the substring acb}
5. Prove that the following languages are not regular: (a) L = {a"bak-k < n+1). (b) L-(angla": kメn + 1). (c) L = {anglak : n = l or l k} . (d) L = {anb : n2 1} L = {w : na (w)关nb (w)). "(f) L = {ww : w E {a, b)'). (g) L = {w"www" : w E {a,b}*}
3. Let T = {< M > | m accepts w" when it accepts w. }. Show T is undecidable.
In the binomial replacement branching model with , let . (a) Show that P[T=n] for n≥1 is . (b) Find P[T=n] for . P(S) = q + ps T = inf{n: Zn=0 We were unable to transcribe this image0 < ? = 7
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.
Let An = {j EN:j<n} for n = 1,2,.... What are D. and A.
Let REPEATTM = { | M is a TM, and for all s L(M), s = uv where u = v }. Show that REPEATTM is undecidable. Do not use Rice’s Theorem. Let REPEATTM = { <M>M is a TM, and for all s E L(M), s = uv where u = v}. Show that REPEATM is undecidable. Do not use Rice's Theorem.