Show that the grammars

S$->$aSb$|$bSa$|$SS$|$a

and

S$->$aSb$|$bSa$|$a

are not equivalent.

$1\mathrm{.}3$ #$2$

Suppose that in some programming language, numbers are restricted as follows:

a$.$ A number may be signed or unsigned.

b$.$ The value field consists of two nonempty parts, separated by a decimal point.

c$.$ There is an optional exponent field. If present, this field must contain the letter $\backslash$theta $,$ followed by a signed two$-$digit integer.

Design a grammar set of such numbers.

$2\mathrm{.}1$#$5$ a$,$ c

Give dfa's for the languages

a$.$ L$=$ab$^(4)$wb$^(2)$:win$\{$a$,$b$\}^(*).$

b$.$ L$=\{$ab$^($n$)$a$^($m$)$:n$>=3,$m$>=2\}.$

c$.$ L$=\{$w$\_(1)$abbw$\_(2)$:w$\_(1)$in$\{$a$,$b$\}^(*),$w$\_(2)$in$\{$a$,$b$\}^(*)\}.$

d$.$ L$=\{$ba$^($n$)$:n$>=1,$n$!=4\}.$

#$7$ d$,$e

Find dfa's for the following languages on $\backslash$Sigma $=\{$a$,$b$\}.$

a$.$ L$=\{$w$.|$w$|$mod$3!=0\}.$

b$.$ L$=\{$w:$|$w$|$mod$5=0\}.$

c$.$ L$=\{$w$.$n$\_($a$)($w$)$mod

d$.$ L$=\{$w$.($n$\_($a$)($w$)-$n$\_($b$)($w$))$mod$3=0\}$L$=\{$w:$($n$\_($a$)($w$)+2$n$\_($b$)($w$))$mod$3<mn>1</mn><mi>\}</mi>$L$=\{$w$.|$w$|$mod$3=0,|$w$|!=5\}$