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19 Pages

### sectregpropS

Course: CPS 140, Fall 2009
School: Duke
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Word Count: 689

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Properties Section: of Regular Languages Example L = {anban | n &gt; 0} Closure Properties A set is closed over an operation if L1, L2 class L1 op L2 = L3 L3 class 1 L1={x | x is a positive even integer} L is closed under addition? multiplication? subtraction? division? Closure of Regular Languages Theorem 4.1 If L1 and L2 are regular languages, then L1 L2 L1 L2 L1 L2 L1 L 1 are regular languages. 2...

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Properties Section: of Regular Languages Example L = {anban | n > 0} Closure Properties A set is closed over an operation if L1, L2 class L1 op L2 = L3 L3 class 1 L1={x | x is a positive even integer} L is closed under addition? multiplication? subtraction? division? Closure of Regular Languages Theorem 4.1 If L1 and L2 are regular languages, then L1 L2 L1 L2 L1 L2 L1 L 1 are regular languages. 2 Proof(sketch) L1 and L2 are regular languages reg. expr. r1 and r2 s.t. L1 = L(r1) and L2=L(r2) r1 + r2 is r.e. denoting L1 L2 closed under union r1r2 is r.e. denoting L1L2 closed under concatenation r1 is r.e. denoting L 1 closed under star-closure 3 complementation: L1 is reg. lang. DFA M s.t. L1 = L(M) Construct M s.t. nal states in M are nonnal states in M nonnal states in M are nal states in M closed under complementation 4 intersection: L1 and L2 are reg. lang. DFA M1 and M2 s.t. L1 = L(M1) and L2 = L(M2) M1=(Q,,1, q0, F1) M2=(P,,2, p0, F2) Construct M=(Q,,, (q0, p0), F) Q = (QP) : ((qi, pj ), a) = (qk , pl ) if w L(M) w L1L2 closed under intersection 5 Example: a b a 2 A B a a,b 1 C Regular languages are closed under reversal dierence right quotient homomorphism Right quotient Def: L1/L2 = {x|xy L1 for some y L2} Example: L1={ab ba} L2={bn|n is even, n > 0} L1/L2 = 6 LR L1-L2 L1/L2 h(L) Theorem If L1 and L2 are regular, then L1/L2 is regular. Proof (sketch) DFA M=(Q,,,q0,F) s.t. L1 = L(M). Construct DFA M=(Q,,,q0,F) For each state i do Make i the start state (representing Li) if Li L2 = then put qi in F in M QED. 7 Homomorphism Def. Let , be alphabets. A homomorphism is a function h: Example: = {a, b, c}, = {0, 1} h(a)=11 h(b)=00 h(c)=0 h(bc) = h(ab) = 8 Questions about regular languages : L is a regular language. Given L, , w , is wL? Is L empty? Is L innite? Does = L1 L2? 9 Identifying Nonregular Languages If a language L is nite, is L regular? If L is innite, is L regular? L1 = {anbm|n > 0, m > 0} = aabb L2 = {anbn|n > 0} Prove that L2 = {anbn|n > 0} is ? Proof: 10 Pumping Lemma: Let L be an innite regular language. a constant m > 0 such that any w L with |w| m can be decomposed into three parts as w = xyz with |xy| m |y| 1 xy iz L for all i 0 11 To Use the Pumping Lemma to prove L is not regular: Proof by Contradiction. Assume L is regular. L satises the pumping lemma. Choose a long string w in L, |w| m. Show that there is NO division of w into xyz (must consider all possible divisions) such that |xy| m, |y| 1 and xy iz L i 0. The pumping lemma does not hold. Contradiction! L is not regular. QED. 12 Example L={ancbn|n > 0} L is not regular. Proof: Assume L is regular. the pumping lemma holds. Choose w = 13 Example L={anbn+scs|n, s > 0} L is not regular. Proof: Assume L is regular. the pumping lemma holds. Choose w= So the partition is: 14 Example = {a, b}, L={w | na(w) > nb(w)} L is not regu...

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a V o dA V r V }t r \$ o trc &amp;V sH V V b b 5b s( b b b b b b 5 V V tH d b 2 b b b A( V c V trfV V
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6.63 15.04 14.26 15.56 16.83 16.98 14.88 14.37 14.23 15.15 15.74 26.15 26.52 25.84 25.69 26.08 25.32 25.95 26.95 25.57 29.57 36.99 38.77 39.59 37.07 37.7 37.22 37.45 37.89 38.99 38.79 49.35 48.21 48.63 410.33 48.51
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W X b i C2f6&quot;egvCvrdvh6iCvY2yvtrr2vCvyr0egvYCgEl2lv ie b u b u b u x w q i x x e b u b n q i ie e e e X X be u b s i u s u b b u v V Cgvvd&amp;v2!add6Y{ V|Xf CYElevCbhCvuYxfwvrCgr)vCy2v02)V|V 2egv26\$vEvd b b b
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( {~ m { zyG}~ 0 G ~ t 0y k m| { 0~ p~m { 0y k { Y ym d dy ~ k { ym d k 2X B T4 RXaa22 D h 0c`C0bbSSIpap4h D2 B q { T4 2 B4 h 2 8 T4 2 22 D 2R Ta H F h h H 2 B4 2 42 1 rW) }0y YbICwD#Q`CSSYSPpHpD`AICQ8 k 5@8 ( Y}pm Y0Y
Duke - STA - 244
APMAM APSAB APSLAKE OPBPC OPRC OPSLAKE Y Year 9.13 3.58 3.91 4.1 7.43 6.47 54235 1948 5.28 4.82 5.2 7.55 11.11 10.26 67567 1949 4.2 3.77 3.67 9.52 12.2 11.35 66161 1950 4.6 4.46 3.93 11.14 15.15 11.13 68094 1951 7.15 4.99 4.88 16.34 20.05 22.81
Duke - STA - 244
BigMac Bread BusFare EngSal EngTax Service TeachSal TeachTax VacDays WorkHrs 31 9 1.27 44.3 44.1 280 21.8 28.2 31.9 1714 33 9 0.27 19.4 23.7 170 9.4 14.8 23.5 1792 98 23 0.09 15.4 20.3 100 2.2 4.3 17.4 2152 131 27 0.09 4.7 37.6 70 1.1 11.7 30.6 2
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Exotic Sire Total Trt 9 1 9 1 5 1 8 2 5 1 8 3 6 1 8 4 3 2 9 1 0 2 9 2 5 2 9 3 5 2 8 4 5 3 8 1 5 3 8 2 6 3 9 3 5 3 6 4 7 4 7 1 7 4 8 2 3 4 6 3 4 4 8 4 8 5 9 1 4 5 8 2 4 5 7 3 6 5 9 4 5 6 9 1 5 6 9 2 4 6 9 3 2 6 7 4 8 7 9 1 4
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Height Length Type 75 502 0 80 522 0 68 425 0 64 344 0 83 407 0 80 451 0 70 551 0 76 530 0 74 547 0 100 519 1 75 225 1 52 300 1 62 418 1 68 409 1 86 425 1 57 370 1 82 506 1 82 506 1 88 295 1 55 273 1 67 415 1 45 182 1 103 530 1
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D F S W 7.2 0 0 10.404 8.2 0 0 18.161 10.3 0 0 25.778 10.1 0 0 20.511 10.7 0 0 21.87 13.3 0 0 47.186 5.1 1 0 4.447 7.2 1 0 8.682 10.2 1 0 19.511 11.3 1 0 37.682 12.6 1 0 25.775 17.1 1 0 67.363 5.1 0 1 4.02 6.5 0 1 7.504 8.4 0 1 13.391
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smoke disease sex 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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Duke - STA - 113
0 49 376 726 736 990 2008 2574 2718 2857 2920 3423 3678 3739 4465 4879 5056 5217 6027
Duke - STA - 205
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This is from the same paper as the etchratedata.txt file.The 490 measurements in etchratedata.txt were used tocompute a measure of nonuniformity for each of the ten wafers. The nonuniformity is actually the standard deviation of the 49etch ra
Duke - CH - 113
&quot;x1&quot;&quot;x2&quot;&quot;x3&quot;&quot;x4&quot;&quot;y&quot;8410011.42418072.27418014.610712054.97418054.67718014.771314014.65416074.54714034.85110071.481014034.72410031.641018034.56712074.7101318034.84101605
Duke - CH - 113
&quot;C1&quot;&quot;C2&quot;&quot;C3&quot;1.2&quot;pH 3&quot;&quot;Diseased&quot;1.4&quot;pH 3&quot;&quot;Diseased&quot;1&quot;pH 3&quot;&quot;Diseased&quot;1.2&quot;pH 3&quot;&quot;Diseased&quot;1.4&quot;pH 3&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;.6&quot;pH 5.5&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;1&quot;pH 5.5&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;1&quot;pH 7&quot;&quot;Dis
Duke - CH - 113
&quot;response&quot;&quot;type&quot;&quot;subject&quot;12111022733744815926837748919152114321443111411251136124711181329123113421313102483511461217828103910411212923934745101611
Duke - CH - 113
&quot;Obs:&quot;&quot;x:&quot;&quot;y:&quot;1.41.022.421.213.48.884.51.985.571.526.61.837.71.58.751.89.751.7410.781.6311.84212.952.813.992.48141.032.47151.123.05161.153.18171.23.76181.253.68191.253.82201.283.21211.3
Duke - CH - 113
&quot;Linoleic&quot;&quot;Kerosene&quot;&quot;Antiox&quot;&quot;Betacaro&quot;303010.7303010.63303018.41.01340405.049303010.713.183010.120405.04204015.006540205.202303010.6330301.59.04402015.132404015.15303010.73046.8210.34630
Duke - CH - 113
Duke - CH - 113
&quot;stiffness&quot;&quot;plate lengths&quot;309.24409.543114326.54316.84349.84309.74402.16347.263616404.563316348.96381.76392.48366.283518357.18409.98367.383828346.710452.910461.410433.110410.610384.210362.6104
Duke - CH - 113
&quot;temp&quot;&quot;removal%&quot;7.6898.096.5198.256.4397.825.4897.826.5797.8210.2297.9315.6998.3816.7798.8917.1398.9617.6398.916.7298.6815.4598.6912.0698.5111.4498.0910.1798.259.6498.368.5598.277.57986.9498.098.3298.2510.59
Duke - CH - 113
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Duke - CH - 113
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Duke - STA - 113
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Duke - STA - 113
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Duke - STA - 244
STA2441/08/2003Homework 1Due 1/15/2003.Please provide concise, neatly written or typed solutions. All work should be your own and not copied from other texts or sources. Do feel free to discuss questions with me, the TA, others in class, or po
Duke - STA - 113
Students0510152025Range: 69.38% - 96.08%, 84 Students Median = 82.07, Quantiles = [76.36, 86.09] Mean = 81.4, Std Dev = 6.255060708090100Course Averages for STA113