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Costas Busch - RPI1 The Pumping Lemma for Context-Free Languages
Costas Busch - RPI2 Take an infinite context-free language Example: Generates an infinite number of different strings
Costas Busch - RPI3 A derivation: In a derivation of a long string, variables are repeated
Costas Busch - RPI4 Derivation treestring
Costas Busch - RPI5 repeated Derivation treestring
Costas Busch - RPI6
7 Repeated Part
Costas Busch - RPI8 Another possible derivation from
Costas Busch - RPI9
10 A Derivation from
Costas Busch - RPI11
Costas Busch - RPI12
Costas Busch - RPI13 A Derivation from
Costas Busch - RPI14
Costas Busch - RPI15
Costas Busch - RPI16
Costas Busch - RPI17
Costas Busch - RPI18 A Derivation from
Costas Busch - RPI19
Costas Busch - RPI20
Costas Busch - RPI21
Costas Busch - RPI22 In General:
Costas Busch - RPI23 Consider now an infinite context-free language Take so that I has no unit-productions no -productions Let be the grammar of
Costas Busch - RPI24 (Number of productions) x (Largest right side of a production) = Let Example : Let
Costas Busch - RPI25 Take a string with length We will show: in the derivation of a variable of is repeated
Costas Busch - RPI26
Costas Busch - RPI27 maximum right hand side of any production
Costas Busch - RPI28 Number of productions in grammar
Costas Busch - RPI29 Number of productions in grammar Some production must be repeated Repeated variable
Costas Busch - RPI30 Some variable is repeated Derivation of string
Costas Busch - RPI31 Last repeated variable repeated Strings of terminals Derivation tree of string
Costas Busch - RPI32 Possible derivations:
Costas Busch - RPI33 We know: This string is also generated:
Costas Busch - RPI34 This string is also generated: The original We know:
Costas Busch - RPI35 This string is also generated: We know:
Costas Busch - RPI36 This string is also generated: We know:
Costas Busch - RPI37 This string is also generated: We know:
Costas Busch - RPI38 Therefore, any string of the form is generated by the grammar
Costas Busch - RPI39 knowing that we also know that Therefore,
Costas Busch - RPI40 Observation: Since is the last repeated variable
Costas Busch - RPI41 Observation: Since there are no unit or -productions
Costas Busch - RPI42 The Pumping Lemma: there exists an integer such that for any string we can write For infinite context-free language with lengths and it must be:
Costas Busch - RPI43 Applications of The Pumping Lemma
Costas Busch - RPI44 Context-free languages Non-context free languages
Costas Busch - RPI45 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages
Costas Busch - RPI46 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma
Costas Busch - RPI47 Pumping Lemma gives a magic number such that: Pick any string with length We pick:
Costas Busch - RPI48 We can write: with lengths and
Costas Busch - RPI49 Pumping Lemma says: for all
Costas Busch - RPI50 We examine all the possible locations of string in
Costas Busch - RPI51 Case 1: is within
Costas Busch - RPI52 Case 1: and consist from only
Costas Busch - RPI53 Case 1: Repeating and
Costas Busch - RPI54 Case 1: From Pumping Lemma:
Costas Busch - RPI55 Case 1: From Pumping Lemma: However: Contradiction!!!
Costas Busch - RPI56 Case 2: is within
Costas Busch - RPI57 Case 2: Similar analysis with case 1
Costas Busch - RPI58 Case 3: is within
Costas Busch - RPI59 Case 3: Similar analysis with case 1
Costas Busch - RPI60 Case 4: overlaps and
Costas Busch - RPI61 Case 4: Possibility 1:contains only
Costas Busch - RPI62 Case 4: Possibility 1:contains only
Costas Busch - RPI63 Case 4: From Pumping Lemma:
Costas Busch - RPI64 Case 4: From Pumping Lemma: However: Contradiction!!!
Costas Busch - RPI65 Case 4: Possibility 2:contains and contains only
Costas Busch - RPI66 Case 4: Possibility 2:contains and contains only
Costas Busch - RPI67 Case 4: From Pumping Lemma:
Costas Busch - RPI68 Case 4: From Pumping Lemma: However: Contradiction!!!
Costas Busch - RPI69 Case 4: Possibility 3:contains only contains and
Costas Busch - RPI70 Case 4: Possibility 3:contains only contains and Similar analysis with Possibility 2
Costas Busch - RPI71 Case 5: overlaps and
Costas Busch - RPI72 Case 5: Similar analysis with case 4
Costas Busch - RPI73 There are no other cases to consider (since, string cannot overlap, and at the same time)
Costas Busch - RPI74 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free
1 Let’s Recapitulate. 2 Regular Languages DFAs NFAs Regular Expressions Regular Grammars.
4.5 Inherently Ambiguous Context-free Language For some context-free languages, such as arithmetic expressions, may have many different CFG’s to generate.
Fall 2006Costas Busch - RPI1 Non-regular languages (Pumping Lemma)
Prof. Busch - LSU1 Simplifications of Context-Free Grammars.
Lecture 15UofH - COSC Dr. Verma 1 COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 15.
Courtesy Costas Busch - RPI1 A Universal Turing Machine.
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1 More Properties of Regular Languages. 2 We have proven Regular languages are closed under: Union Concatenation Star operation Reverse.
Costas Busch - RPI1 NPDAs Accept Context-Free Languages.
Recursively Enumerable and Recursive Languages
Costas Busch - RPI1 Standard Representations of Regular Languages Regular Languages DFAs NFAs Regular Expressions Regular Grammars.
Courtesy Costas Busch - RPI1 NPDAs Accept Context-Free Languages.
Costas Busch - RPI1 Grammars. Costas Busch - RPI2 Grammars Grammars express languages Example: the English language.
Courtesy Costas Busch - RPI1 The Pumping Lemma for Context-Free Languages.
Costas Buch - RPI1 Simplifications of Context-Free Grammars.
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Courtesy Costas Busch - RPI
1 The Pumping Lemma for Context-Free Languages. 2 Take an infinite context-free language Example: Generates an infinite number of different strings.
Courtesy Costas Busch - RPI1 Context-Free Languages.
1 More Applications of the Pumping Lemma. 2 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we.
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