$3$ LTL Semantics $2$

Consider the labeled transition system with the following transition graph, where each state is

characterized by the values of the Boolean variables a and $b.$

Using $-$regular expressions, for each formula $F$ below,

describe a path from the initial state $s_3$ that satisfies $F,$ if any;

describe a path from the initial state $s_3$ that falsifies $F,$ if any.

When no satisfying or falsifying path exists, write "None".

$Oa$

$not{b}^{{?}^{?}}(notavvnotb)$

$((not{a}^{{?}^{?}}$notb$)b)$

$((not{a}^{{?}^{?}}$notb$)b)$

diamond$b$

$diamondb$

$diamondbdiamondnotb$

$(not{a}^{{?}^{?}}b)$

$(avvb)U(not{a}^{{?}^{?}}$notb$)$

bRa