Past exam of the computer science course of the University of Oxford / 2015 / Computer aided formal verification / 2 / a / Solution

Use the two incoming initial arrows in the source figure: $I=\{s_0,s_3\}$. An atomic proposition is true precisely at the states whose displayed labels contain it. In particular, $c$ is false at every state. Hence $GF\neg c$ is true on every infinite path in a transition system, and the first implication in the fairness assumption imposes no restriction.

Write $p=a\wedge b$. The remaining implication is $FGp\to GF\neg b$. By the linear temporal logic duality $\neg FGp=GF\neg p$, it is equivalent tosince $\neg b$ implies $\neg p$. Thus a path is fair exactly when it visits states not labelled with both $a$ and $b$ infinitely often. This condition also follows directly: if $p$ eventually holds forever, then $b$ eventually holds forever and the required $GF\neg b$ is impossible; if $p$ does not eventually hold forever, the implication has a false antecedent.

Every initial path in a transition system in the upper component either alternates forever between $s_0$ and $s_1$, or makes finitely many alternations and then stays at $s_2$. These are all fair: the first visits $s_1$ infinitely often, and the second visits $s_2$ infinitely often; neither state satisfies $a\wedge b$.

In the lower component, a path in a transition system can stay at $s_3$ forever, or leave it after finitely many visits and pass through $s_4$ to $s_5$. The constant path in a transition system $s_3^{\omega }$ is unfair because $a\wedge b$ holds forever while $\neg b$ never holds. Every departing path in a transition system is fair because $s_5$ satisfies neither $a$ nor $b$. Here $v^{\omega }$ means infinite repetition of the finite word $v$. ThereforeThis notation lists initial paths in a transition system. If one instead allows arbitrary starting states, their suffixes starting at $s_1,s_2,s_4,s_5$ are fair as well; the fair paths in a transition system from each initial state, and hence all verdicts below, are unchanged.