More Applications of The Pumping Lemma

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More Applicationsof The Pumping Lemma

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The Pumping Lemma
For infinite context-free language

there exists an integer such that
for any string
we can write
with lengths
and it must be
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Non-context free languages
Context-free languages
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Theorem
The language
is not context free
Proof
Use the Pumping Lemma for context-free languages
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Assume for contradiction that
is context-free
Since is context-free and infinite we can
apply the pumping lemma
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Pumping Lemma gives a magic number such that
Pick any string of with length at least
we pick
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We can write
with lengths and
Pumping Lemma says
for all
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We examine all the possible locations of string
in
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Case 1
is within the first
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Case 1
is within the first
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Case 1
is within the first
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Case 1
is within the first
However, from Pumping Lemma
Contradiction!!!
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Case 2
is in the first
is in the first
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Case 2
is in the first
is in the first
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Case 2
is in the first
is in the first
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Case 2
is in the first
is in the first
However, from Pumping Lemma
Contradiction!!!
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Case 3
overlaps the first
is in the first
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Case 3
overlaps the first
is in the first
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Case 3
overlaps the first
is in the first
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Case 3
overlaps the first
is in the first
However, from Pumping Lemma
Contradiction!!!
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Case 4
in the first
Overlaps the first
Analysis is similar to case 3
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Other cases
is within
or
or
Analysis is similar to case 1
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More cases
overlaps
or
Analysis is similar to cases 2,3,4
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There are no other cases to consider
Since , it is impossible
to overlap
nor
nor
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In all cases we obtained a contradiction
Therefore
The original assumption that
is context-free must be wrong
Conclusion
is not context-free
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Non-context free languages
Context-free languages
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Theorem
The language
is not context free
Proof
Use the Pumping Lemma for context-free languages
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Assume for contradiction that
is context-free
Since is context-free and infinite we can
apply the pumping lemma
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Pumping Lemma gives a magic number such that
Pick any string of with length at least
we pick
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We can write
with lengths and
Pumping Lemma says
for all
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We examine all the possible locations of string
in
There is only one case to consider
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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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Since , for we have
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(No Transcript)
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However, from Pumping Lemma
Contradiction!!!
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We obtained a contradiction
Therefore
The original assumption that
is context-free must be wrong
Conclusion
is not context-free
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Non-context free languages
Context-free languages
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Theorem
The language
is not context free
Proof
Use the Pumping Lemma for context-free languages
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Assume for contradiction that
is context-free
Since is context-free and infinite we can
apply the pumping lemma
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Pumping Lemma gives a magic number such that
Pick any string of with length at least
we pick
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We can write
with lengths and
Pumping Lemma says
for all
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We examine all the possible locations of string
in
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Most complicated case
is in
is in
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(No Transcript)
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Most complicated sub-case
and
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Most complicated sub-case
and
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Most complicated sub-case
and
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and
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(No Transcript)
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However, from Pumping Lemma
Contradiction!!!
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When we examine the rest of the cases we also
obtain a contradiction
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In all cases we obtained a contradiction
Therefore
The original assumption that
is context-free must be wrong
Conclusion
is not context-free
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YACC

• Yet Another Compiler Compiler

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Yacc is a parser generator
Input A Grammar Output A parser for the
grammar
Reminder a parser finds derivations
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Example grammar
expr -gt ( expr ) expr '' expr
expr '-' expr expr '' expr
expr '/' expr - expr INT

The yacc code
expr '(' expr ')' expr '' expr
expr '-' expr expr '' expr
expr '/' expr - expr INT

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Exampe Input
10 3 4
Yacc Derivation
expr gt expr expr gt expr expr expr
gt 103 4
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Resolving Ambiguities
left '', '-' left '', '/' left UMINUS
expr '(' expr ')' expr '' expr
expr '-' expr expr '' expr expr
'/' expr '-' expr prec UMINUS
INT
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Actions
left '', '-' left '', '/' left UMINUS
expr '(' expr ')' 2 expr
'' expr 1 3 expr '-'
expr 1 - 3 expr '' expr
1 3 expr '/' expr
1 / 3 '-' expr prec
UMINUS -2 INT
1
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A Complete Yacc program
union int int_val left '', '-' left
'', '/' left UMINUS token ltint_valgt INT type
ltint_valgt expr start program
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program expr printf("Expr value d
\n", 1) error printf("YACC
syntax error near line d \n", linenum)
abort() expr '('
expr ')' 2 expr '' expr
1 3 expr '-' expr 1
- 3 expr '' expr 1 3
expr '/' expr 1 / 3 '-'
expr prec UMINUS -2
INT 1 include
"lex.yy.c"
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Execution Example
Input
10 20(3 - 4 25)
Output
Expr value 490
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The Lex Code
int linenum1 int temp_int \n
linenum \t / skip spaces
/ \/\/\n / ignore comments / ""
return '' "-" return '-' ""
return '' "/" return '/' ")"
return ')' "(" return '('
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0-9 sscanf(yytext, "d", temp_int)
yylval.int_val temp_int return
INT . printf("LEX unknown input string
found in line d \n", linenum) abort()
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Compiling
yacc YaccFile lex LexFile cc y.tab.c -ly -ll -o
myparser
Executable myparser
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Another Yacc Program
union int int_val left '', '-' left
'', '/' left UMINUS token ltint_valgt INT type
ltint_valgt expr start program
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program stmt_list error
printf("YACC syntax error near line d \n",
linenum) abort()
stmt_list stmt_list stmt
stmt stmt expr ''
printf("Expr value d \n", 1)
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expr '(' expr ')' 2 expr
'' expr 1 3 expr '-' expr
1 - 3 expr '' expr
1 3 expr '/' expr 1 /
3 '-' expr prec UMINUS
-2 INT 1
include "lex.yy.c"
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Execution Example
Input
10 20(30 -67) / 4 34 35 - 123
-001 178/6
Output
Expr value -175 Expr value 1066 Expr value
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Lex Code
int linenum1 int temp_int \n
linenum \t / skip spaces
/ \/\/\n / ignore comments /
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"" return '' "-" return '-' ""
return '' "/" return '/' ")"
return ')' "(" return '(' "" return
'' 0-9 sscanf(yytext, "d",
temp_int) yylval.int_val temp_int
return INT . printf("LEX unknown
input string found in line d \n", linenum)
abort()
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Another Yacc Program
union int int_val char str_val left
'', '-' left '', '/' left UMINUS token
PRINT token NEWLINE token ltstr_valgt
STRING token ltint_valgt INT type ltint_valgt
expr start program
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program stmt_list error
printf("YACC syntax error near line d \n",
linenum) abort()
stmt_list stmt_list stmt stmt
stmt expr ''
printf("expression found\n") PRINT
expr '' printf("d", 2)
PRINT STRING '' printf("s", 2)
PRINT NEWLINE '' printf("\n")
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expr '(' expr ')' 2 expr
'' expr 1 3 expr '-' expr
1 - 3 expr '' expr
1 3 expr '/' expr 1 /
3 '-' expr prec UMINUS
-2 INT 1
include "lex.yy.c"
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Execution Example
Input
print "The value of expression 123 25 is
" print 123 25 print newline 10 5
8 print "end of program" print newline
Output
The value of expression 123 25 is
3075 expression found end of program
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Lex Code
int linenum1 int temp_int char
temp_str200 \n linenum \t
/ skip spaces / \/\/\n /
ignore comments /
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"" return '' "-" return '-'
"" return '' "/" return
'/' ")" return ')' "("
return '(' "" return '' "print"
return PRINT "newline" return NEWLINE
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0-9 sscanf(yytext, "d", temp_int)
yylval.int_val temp_int
return INT \""\n\" strncpy(temp_str,
(yytext1), strlen(yytext)-2)
temp_strstrlen(yytext)-2 (char) 0
yylval.str_val temp_str
return STRING . printf("LEX unknown
input string found in line d \n", linenum)
abort()