More Applications of the Pumping Lemma 1 The

More Applications of the Pumping Lemma 1

The Pumping Lemma: • Given a infinite regular language • there exists an integer • for any string with length • we can write • with • such that: and 2

Non-regular languages Regular languages 3

Theorem: The language is not regular Proof: Use the Pumping Lemma 4

Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma 5

Let be the integer in the Pumping Lemma Pick a string such that: length pick 6

Write From the Pumping Lemma it must be that length 7

We have: From the Pumping Lemma: Thus: 8

Therefore: BUT: CONTRADICTION!!! 9

Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language 10

Non-regular languages Regular languages 11

Theorem: The language is not regular Proof: Use the Pumping Lemma 12

Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma 13

Let be the integer in the Pumping Lemma Pick a string such that: length pick 14

Write From the Pumping Lemma it must be that length 15

We have: From the Pumping Lemma: Thus: 16

Therefore: BUT: CONTRADICTION!!! 17

Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language 18

Non-regular languages Regular languages 19

Theorem: The language is not regular Proof: Use the Pumping Lemma 20

Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma 21

Let be the integer in the Pumping Lemma Pick a string such that: length pick 22

Write From the Pumping Lemma it must be that length 23

We have: From the Pumping Lemma: Thus: 24

Therefore: And since: There is : 25

However: for any 26

Therefore: BUT: and CONTRADICTION!!! 27

Therefore: Our assumption that is a regular language is not true Conclusion: is not a regular language 28

Lex 29

Lex: a lexical analyzer • A Lex program recognizes strings • For each kind of string found the lex program takes an action 30

Output Input Var = 12 + 9; if (test > 20) temp = 0; else while (a < 20) temp++; Lex program Identifier: Var Operand: = Integer: 12 Operand: + Integer: 9 Semicolumn: ; Keyword: if Parenthesis: ( Identifier: test. . 31

In Lex strings are described with regular expressions Lex program Regular expressions “+” “-” “=“ /* operators */ “if” “then” /* keywords */ 32

Lex program Regular expressions (0|1|2|3|4|5|6|7|8|9)+ (a|b|. . |z|A|B|. . . |Z)+ /* integers */ /* identifiers */ 33

integers (0|1|2|3|4|5|6|7|8|9)+ [0 -9]+ 34

identifiers (a|b|. . |z|A|B|. . . |Z)+ [a-z. A-Z]+ 35

Each regular expression has an associated action (in C code) Examples: Regular expression Action n linenum++; [0 -9]+ prinf(“integer”); [a-z. A-Z]+ printf(“identifier”); 36

Default action: ECHO; Prints the string identified to the output 37

A small program %% [ tn] ; /*skip spaces*/ [0 -9]+ printf(“Integern”); [a-z. A-Z]+ printf(“Identifiern”); 38

Output Input 1234 test var 566 9800 78 Integer Identifier Integer 39

%{ int linenum = 1; %} Another program %% [ t] n ; /*skip spaces*/ linenum++; [0 -9]+ prinf(“Integern”); [a-z. A-Z]+ printf(“Identifiern”); . printf(“Error in line: %dn”, 40 linenum);

Output Input 1234 test var 566 9800 + temp 78 Integer Identifier Integer Error in line: 3 Identifier 41

Lex matches the longest input string Example: Regular Expressions Input: Matches: ifend if “ifend” “if” “ifend” ifn nomatch 42

Internal Structure of Lex Regular expressions NFA DFA Minimal DFA The final states of the DFA are associated with actions 43