Hello$-$World programs are impossible to grade. A Java program P is called a Hello$-$World$-$program, if the following is true: When given the empty string $\backslash$epsi as input, P outputs the string Hello World and then terminates. $($We do not care what P does when the input string is non$-$empty.$)$

Consider the language

HW $=\{|$ P is a Hello$-$World$-$program$\}.$

You are asked to prove that HW is undecidable.

$($a$)$ Use a Turing reduction from HALT to HW where

HALT $=\{|$ P is a Java program that halts on input w$\}$

is undecidable. Your reduction may bear similarities to the reduction in class that preceded the proof of Rice$$s Theorem

Just for this problem, we switch our computation model from Turing machines to Java programs $($as these models are equivalent, all results from class also apply here$).$

$($b$)$ Show that your reduction is a mapping reduction by defining the corresponding function f : $\backslash$Sigma $*->\backslash$Sigma $*$ and explaining why it satisfies the definition of a mapping reduction.