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Top 10 Languages 2015 2 Languages Symbol a, b, . . . Alphabet A finite, nonempty set of symbols usually denoted by String - finite sequence of symbols e.g. abba, b, bb Empty string denoted by *= set of all strings over alphabet e.g. {a, b}* a, b, aa, ab, . . .} language - set of strings defined over 3 Languages Examples = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} L= {0, 1, 2, ..., 12, 13, 14, ...} ={a, b, c} L={ab, ac, cabb} ={1, 2, +, =} L1 = {1+1=2} L2 = {2+2+2+2+2+2=12} What’s about L3={1+1=2} ? And L4 = { 1+1=3 } All strings of even length ={a, b} L= {, aa, ab, ba, bb, aaaa, . . .} All strings of a’s and b’s in equal numbers L= {, ab, ba, aabb, abab, abba, . . .} 4 Languages Let ={a,b,c}. The elemnets of * include Length 0 : Length 1 : a, b, c Length 2 : aa, ab, ac, ba, bb, bc, ca, cb, cc Length 3 : aaa, aab, aac, aba, abb, abc, aca, acb, acc, baa, bab, bac, …, cbc, cca, ccb, ccc … In general, if is an alphabet and L is a subset of *, then L is said to be a language over . Example {0, 11, 001}, {, 10}, and {0, 1}* are subsets of {0, 1}*, and so they are languages over the alphabet {0, 1}. 5 Languages The union of two languages L1 and L2, denoted L1L2, refers to the language that consists of all the strings that are either in L1 or in L2, that is, to { x | x is in L1 or x is in L2 }. The intersection of L1 and L2, denoted L1L2, refers to the language that consists of all the strings that are both in L1 and L2, that is, to { x | x is in L1 and in L2 }. 6 The complementation of a language L over , or just the L complementation of L when is understood, denoted, refers to the language that consists of all the strings over * that are not in L, that is, to { x | x is in * but not in L }. Languages Example Consider the languages L1 = {, 0, 1} and L2 = {, 01, 11}. The union of these languages is L1 L2 = {, 0, 1, 01, 11}, their intersection is L1 L2 = {}, and the complementation of L1 is L1 = {00, 01, 10, 11, 000, 001, . . . }. 7 Languages The difference of L1 and L2, denoted L1 - L2, refers to the language that consists of all the strings that are in L1 but not in L2, that is, { x | x is in L1 but not in L2 }. Example If L1 = {, 1, 01, 11} and L2 = {1, 01, 101} then L1 - L2 = {, 11} and L2 - L1 = {101}. Try this : L1 = Set of all even number and L2 = Set of all number. L2 - L1 = ? and L1 = ? 8 Concatenation Let u,v *. The concatenation of u and v is 1. If length(v) = 0 then v = and uv = u 2. If length(u) and length(v) > 0 , let u = a1a2…an v = b1b2…bn then uv = a1a2…anb1b2…bn The concatenation of X and Y, denoted XY, is the language XY = {uv| u X and v Y} The concatenation of X with itself n times is Denoted Xn, X0 is denoted as {} 9 Concatenation Example Let u = ab, v = ca, w = bb then uv = abca vw = cabb (uv)w = abcabb u(vw) = abcabb So that , (uv)w = u(vw) 10 Concatenation Example Let X = { a, b, c } and Y = {abb, ba} then XY = {aabb, babb, cabb, aba, bba, cba} X0 = {} X1 = X = {a, b, c} X2 = XX = {aa, ab, ac, ba, bb, bc, ca, cb, cc} X3 = X2X = {aaa, aab, aac, aba, abb, abc, aca, acb, acc, …, ccb, ccc} 11 Try this Let X = { 011, 110 } ,Y = { ab, bc } , Z = { d } Find XZY,YZX , ZX2 ,YZ4 Let u = {abcdef} The reversal of u=uR= {fedcba} Let v = {jklmnop}. Find (uv)R and vRuR 12 Kleen star Kleen star of set X, denoted X* X* = X i and X + i0 X i i 1 X* contains all strings that can be built from the elements of X. X+ is the set of nonnull strings built from X. X XX + 13 * Kleen star Example The language L = {a,b}*{bb}{a,b}* consists of the strings over ={a,b} that contain the substring bb. Strings that accepted by L = bb, bba, abb, bbb,… Strings that rejected by L = a,b,aa,ab,ba,aba,... 14 Kleen star Example Let L be the language that consists of all strings over ={a,b} that begin with aa or end with bb. Thus L = {aa}{a,b}* {a,b}*{bb} What’s the different between {a,b}*, {a }*{b }* and {ab}* ? 15 Try this Let L be a language over ={a,b} consists of all even-length strings Let L be a language over ={a} consists of even number of a. 16 Regular sets Let be an alphabet. The regular sets over Arer defined recursively as follows: i) Basis: , {} and {a}, for every a, are regular sets over ii) Recursive step: Let X and Y be regular sets over . The sets X Y XY X* are regular sets over . III. Closure: X is a regular set over only if it can be obtained from the basis elements by a finite number of applications of the recusive step. 17 Regular sets Example Language L is the set of strings over ={a,b} that containing the substring bb is a regular set L = {a,b}*{bb}{a,b}* From basis : {a} and {b} are regular sets Apply union and kleen star operation produce {a,b}*, the set of all strings over {a,b}. By concatenation, {b}{b} ={bb} is regular. By concatenation {a,b}*{bb} and then concatenation {a,b}*{bb}{a,b}* is regular sets. 18 Regular sets Example Language L is the set of strings over ={a,b} that Begin and end with a and contain at least one b. L = {a}{a,b}*{b}{a,b}*{a} From basis : {a} and {b} are regular sets Apply union and kleen star operation produce {a,b}*, then concat {a} and {a,b}* = {a}{a,b}* then concat {a}{a,b}* and {b} = {a}{a,b}*{b} then concat {a}{a,b}*{b} and {a,b}* = {a}{a,b}*{b} {a,b}* then concat {a}{a,b}*{b} {a,b}* and {a} = {a}{a,b}*{b} {a,b}* {a} is a regular set. 19 Regular expression Let be an alphabet. The regular sets over Arer defined recursively as follows: i) Basis: , and a, for every a, are regular expressions over ii) Recursive step: Let u and v be regular expressions over . The expressions (u + v) (uv) (u*) are regular expressions over . III. Closure: u is a regular expression over only if it can be obtained from the basis elements by a finite number of applications of the recursive step. 20 Regular expression Example A regular expression for the set of strings over ={a,b} that contain exactly two b’s. Any number of a’s may occur before, between, and after b’s. r = a*ba*ba* Example The regular expression for the set of strings over ={a,b,c} containing the substring bc. r = (a+b+c)*bc(a+b+c)* 21 Regular expression Example The regular expression for the set of strings over ={a,b} with an even number of a’s and an even number of b’s Strings in this language will have clumps of 3 kinds: Type 1 — aa Type 2 — bb Type 3 — starting with ab (or with ba ) and eventually balancing with another ab or ba. So: ( aa+bb+ ((ab+ba)(aa+bb)*(ab+ba)) )* 22 Regular Expression Example The regular expression R is the string over ={a,b} with an odd number of a’s. String in R : a,ab,ba,bab,abbbb,bbbba,bbabbb,…, aaa, ababa, aabbbba,ababab,abababbb, abbbabbabbb,bbbbabbabbabbb,… String not in R : ,aa,baa,aab,bbaa,baba,abab,aaaa,… We can do in 2 cases : a aa or aa a Case 1 : bbbb…b a bb..abb..abbb.. / bb…b a bbb…b / bbbb…b a bb..babb..ba / bb…b a abb..ba / bb..b a aa R = b*a(b + ab*a)* Case 2 : bb..baa a b..b / b..bab..bab..b a b..b / bb..b a bb..b R = (b + ab*a)*ab* So that : R = (b*a(b + ab*a)*) + ((b + ab*a)*ab*) 23 Regular Expression Example The regular expression R is the string over ={a,b} ending with b and not containing aa. String in R : b,ab,bab,abb..b,abab,bb..b, ababb..b, bb..babb..bab String not in R : ,aa,ba,baa,bbaa,aabb,aba,aaaab,… So, string must be the combination of b and ab. The two smallest string are b, ab. R= ? (b + ab) , R = (b + ab)*(b+ab) 24 Try this The regular expression of the string over ={a,b,c} in which all the a’s precede the b’s which in turn precede the c’s. The regular expression of the string over ={0,1} in which the string end with 1 and does not contain the substring 00. The regular expression of the string over ={a,b} consists of all odd-length strings 25 Try this The regular expression of the string over ={a,b,c} in which all the a’s precede the b’s which in turn precede the c’s, but without the empty string . The regular expression of the string over ={a,b,c} that begin with a, contain exactly two b's, and end with cc. The regular expression of the string over ={a,b,c} that do not contain substring aa. 26 Try this The regular expression of the string over ={a,b} with an even number of a's or an odd number of b's. The regular expression of the string over ={a,b,c} with an even length and contain exactly one a. The regular expression of the string over ={a,b} with an odd length and contain exactly two b’s. 27