$($XC$)$ Parsing a Star Language. Let $=\{0,1\}$ and let $C$ be a finite set of strings in ${}^{**}.$ We say

that $C$ forms a unique code if there is no string in $C^{**}$ that can be formed in two different ways as a

concatenation of strings in $C.$

$($a$)$ A prefix code is a finite set $x$ of nonempty strings such that no string in $x$ is a proper prefix

of any other. Prove that any prefix code is a unique code. A suffix code is defined similarly $-$

prove that any suffix code is a unique code.

$($b$)$ Find an example of a finite language that is not a prefix code, suffix code or unique code.

$($c$)$ Let $S_1=C^{-1}\frac{C}{\frac{?}{?}}\{\},$ and for all positive $i,$ let $S_{i+1}$ be the language $C^{-1}{S}_i{S}_i{C}^{-1},$ where $x^{-1}Y$

is defined as in Problem $7.$ Prove that if $C$ is a unique code, then there must exist $S_i=S_{j}Ci{S}_{i}C\frac{O}{?}in{S}_{i}Ci$ with

$S_i=S_j.$ Prove that $ifCis$ not a unique code, then there must $be$ some i with either $S_{i}C\frac{O}{?}$

$orin{S}_i.$

$(d)$ Explain how $we$ can use this fact $to$ test whether $an$ arbitrary $Cis$ a unique code.