Let A$=\{$w$$w is a binary string containing at least $2$ consecutive $1$s and an even number of $1$ s$\}.$ Let B$=\{$a$^$i

b$^$jc$^$k$0<$ i $<$ j $<$ k$\}.$

Show the YES$$NO table for A and B$.$

Construct a mapping reduction function to show that A $$m B$.$ Write the Grafstate code for a Turing machine that computes your mapping reduction function.

GRAFSTATE TURING MACHINE CODE FORMAT:

:$+$ tm Problem$3$

Q$=\{$q$0,$q$1,$qa$,$qr$,$q$2,$q$3\}$

S$=\{$a$,$b$\}$

T$=\{$a$,$b$,\backslash \_\}$

q$0=$q$0$

tapes$=2$

q$0->$qr:$\backslash \_,$S; $\backslash \_,$S

q$0->$q$1$:a$,$S; $\backslash \_,$R

q$0->$qa:b$,$S; $\backslash \_,$S

q$1$:a$,$R; $\backslash \_->$a$,$R

q$1->$q$2$:b$,$S; $\backslash \_,$L

q$1->$qa:$\backslash \_,$S;$\backslash \_,$S

q$2$:b$,$S; a$,$L

q$2->$q$3$:b$,$S; $\backslash \_,$R

q$3$:b$,$R; a$,$R

q$3->$qr:$\backslash \_,$S; $\backslash \_,$S

q$3->$qa:$\backslash \_,$S; a$,$S

q$3->$qr:a$,$S; a$,$S

q$3->$qr:a$,$S; $\backslash \_,$S

q$3->$qa:b$,$S; $\backslash \_,$S

done.

Choose a string wY that represents YES for A and a string wN that represents NO for A$.$ Show the simulation output for your TM on wY and wN$.$

Prove that your function form has the mapping reduction property.

Prove that ECFG $$m EPDA.

a Find a mapping reduction function.

b Prove that your function has the mapping reduction property.