chapter 4 pumping lemma properties of regular languages decidable questions on regular languages

Download Chapter 4 Pumping Lemma Properties of Regular Languages Decidable questions on Regular Languages

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  • Chapter 4Pumping LemmaProperties of Regular LanguagesDecidable questions on Regular Languages

  • Theorem 4.1: Pumping lemma for regular languagesLet L be a regular language. Then there is a constant n (which depends on L) such that for every string w in L such that |w| n, we can break w into three strings, w = xyz, such that y , |xy| n, and for all i 0, xyiz is also in L.

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  • Proof: Let m be the number of states in the smallest FA accepting L, and let w = a1a2an where n m. If w is in L, then we have (q, a1a2aj) = ql, (ql, aj+1..ak) = ql, and (ql, ak+1an) = f. choose x = a1a2aj y = aj+1..ak z = ak+1an It is obvious that (ql, vi) = ql for i 0. So, if the FA accepts w = xyz, it also accepts zyiz.

  • Applications of the pumping lemma: Useful to prove a language L is not a regular set

    Method select an arbitrary n choose a string w in L where |w| n for any partition of w = xyz such that |xy| n and |y| 1, show a contradiction; i.e. show that there is a string xykz not in L; k will depend on n, x, y, and z

  • Example L = {0i1i | k > 0} given arbitrary n, choose w = 0n1nfor any partition of w as xyz, y is one of 0j, 1j, or 0j1l in all cases xy2z is not in L So, L is not a regular set

  • Example: L = { w in (a+b)* | # as = # bs}

  • Example: L = {w | w = ap where p is a prime}

  • Definition of concatenationLet u, v *. The concatenation of u and v, written as uv, is a binary operation on the strings of * defined as follows: 1. Basis: If lenghth(v) = 0, then v = and uv = u 2. Recursive step: Let v be a string of length n > 0 and v = wa where a and lenghth(w) = n-1. Then uv = (uw)a.

    Definition of uR (reversal of u)Basis: length(u) = 0, then u = and = RRecursive step: If length(u) = n > 0, then u = wa for some string w such that length(w) = n-1 and some a , and uR = awR.

  • u is a substring of v if v = xuyu is a prefix of v if v = uxu is a suffix of v if v = xu

    Theorem 1: (uv)w = u(vw). Concatenation is associative.Theorem 2: (uv)R = vRuR

  • A language L such that no string in L is a proper prefix (suffix) of any other string L is said to have the prefix(suffix) property.Example:{a}* does not have prefix property{aib| i 0} does have the prefix property

    Definition of homomorphism:Let 1 and 2 be alphabets. A homomorphism is a mapping h: 1 2*. It is extended to 1* as follows: h: 1* 2* is defined as h() = and h(xa) = h(x)h(a) for all x1* and a1.

  • Homomorphism applied to a languageLet L be a language and h be a homomorphism.h(L) = {h(w) | w is in L}

    Definition of inverse homomorphism:If h: 1 2* is a homomorphism, then the relation h-1: 2* (1*), called an inverse homomorphism and is defined as follows:If y 2* then h-1(y) = {x | h(x) = y} h-1(L) = h-1(y) = {x | y = h(x) L}

  • Examples:Let 1 = {0, 1}, 2 = {a, b}, L = {0n1n | n > 0}.Let h(0) = a, h(1) = bb Then, h(L) = {anb2n | n > 0} h-1(abb) = {01}

    Let h(0) = ab and h(1) = Then, h(L) = (ab)n h-1(ab) = {1n01n}

  • Closure Properties of Regular Languages

    The class of regular sets is closed underUnionIntersectionComplementDifferenceReversalClosure (*)ConcatenationHomomorphismInverse homomorphism

  • Theorem (4.5): If L is a regular language over an alphabet , then L = * L is also regularProof: Let L = L(A) for some DFAA = (Q, , , q0, F). Let B be the DFAB = (Q, , , q0, Q F). Then L(B) = L.Therefore L is regular.

    Corollary: If L and M are regular languages, then L M is regular.

  • Theorem (4.11): If L is a regular language, so is LR.Proof: (Use structural induction on the size of the regular expression E representing L.)Claim: ER represents LRBasis: If L is , {} or {a} then claim is trueInduction step: If E = E1 + E2, then ER = E1R + E2RIf E = E1 + E2, then ER = E2RE1RIf E = E1*, then ER = (E1R)*

  • Theorem (4.14): If L is a regular language over an alphabet , and h is a homomorphism on , then h(L) is also regular.Proof: Let r be a regular expression, representing L, and s be a regular expression obtained from r by replacing every symbol a of by h(a), then s represents h(L) (use structural induction to prove this). s is a regular expression. So, h(L) is regular language.

  • Theorem (4.16): if h is a homomorphism from an alphabet to * and L *, then h-1(L) is also a regular set.Proof: Let M=(Q, , , q0, F) be such that L = L(M). Let M1 = (Q, , 1, q0, F) where (q, a) in M1 = (q, h(a)) in M.Then L(M1) = h-1(L).

  • Example: Show that {anban | n 1} is not regular.

  • Example: Show that {0,1}* - {0n1n | n > 0} is not regular

  • Note:The Class of Regular sets included in * is a Boolean algebra of sets for anyalphabet .



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