Consider the two$-$bit machine presented in class. The machine has two bits of state information, HIGH and LOW $($for $$high$$ and $$low$$ classifications, respectively$).$ The machine has two commands, xor$0$ and xor$1,$ which exclusive$-$or both bits with $0$ and $1,$ respectively. There are two users: Holly $($who can read high and low information$)$ and Lucy $($who can read only low information$).$ The system keeps two bits of state $($HIGH; LOW$).$

Let the set of states $=\{(0$; $0)$; $(0$; $1)$; $(1$; $0)$; $(1$; $1)\},$ the set of subjects S $=\{$Holly; Lucy$\},$

and the set of possible commands C $=\{$xor$0$; xor$1\}.$ Let the initial system state $0=(0$; $1).$ Suppose Lucy applies the command xor$0,$ then Holly the command xor$0,$ then Holly the command xor$0,$ and finally Lucy the command xor$1.$

Assuming a command affects both the HIGH and LOW bits, give the values of $($For parts $($a$)($h$),$ xor$0$ performs an XOR of $0$ across both bits, and xor$1$ performs an XOR of $1$ across both bits.$)$:

$$ proj$($Holly; cs; $0)$

$$ proj$($Lucy; cs; $0)$

$$ Lucy$($cs$)$

$$ Holly$($cs$)$

$$ Lucy, xor$0($cs$)$

$$ Holly, xor$0($cs$)$

$$ Holly, xor$1($cs$)$

$$ xor$1($cs$)$

Now, assuming a command affects only the bit at the level of the user, give the values of $($For parts $($i$)($p$),$ xor$0$ performs an XOR of $0$ on the corresponding bit of the user$$s level only, and xor$1$ performs an XOR of $1$ on the corresponding bit of the user$$s level only.$)$:

$$ proj$($Holly; cs; $0)$

$$ proj$($Lucy; cs; $0)$

$$ Lucy$($cs$)$

$$ Holly$($cs$)$

$$ Lucy, xor$0($cs$)$

$$ Holly, xor$0($cs$)$

$$ Holly, xor$1($cs$)$

$$ xor$1($cs$)$