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

To apply bounded model checking, seek an initial path in a transition system satisfyingA satisfying assignment for its bounded encoding is enough to disprove the specification. Use a lasso path: positions $q_0,\dots ,q_k$ follow the transition relation, and a final arrow returns from $q_k$ to some $q_{\ell }$, $0\le \ell \le k$. The repeated part is $q_{\ell }\cdots q_k$. On such a lasso path, $GFr$ is equivalent to having $r$ true at some position of the repeated part.

For this example an explicit Boolean formula for the bounded search isHere $I(q)$ means $q\in \{s_0,s_3\}$ and $T(q,r)$ means that the source diagram has the arrow $q\to r$; each can be encoded by finite Boolean variables for the states. The loop requirements are the fairness assumption and the negated specification respectively. Unrolling the transition relation and adding a loop closure is the usual bounded model checking construction.^[ref]

At bound $k=0$, an initial self-loop would be needed. The only such choice is $q_0=s_3$, whose repeated label satisfies $a\wedge b$ and $a$. It violates both loop requirements. The other initial state, $s_0$, has no self-loop. Thus $E_0$ has no satisfying assignment.

At bound $k=1$, chooseBoth required arrows $s_0\to s_1$ and $s_1\to s_0$ exist. At $q_1$, $a$ is false and hence $a\wedge b$ is false, so both repeated-part requirements are true. Equivalently, the label values are $a(q_0)=b(q_0)=b(q_1)=1$ and $a(q_1)=c(q_0)=c(q_1)=0$. These values give a satisfying assignment of $E_1$, representing the fair counterexample $(s_0{s}_1{)}^{\omega }$. ConsequentlyHere the bound counts forward arrows before the returning arrow, so $k=1$ stores two positions and describes a two-state repeated part. There is no need for a completeness bound: the exhibited fair counterexample settles the negative answer. These satisfying assignments are checked directly; no claim of executing an external SAT solver is needed.