1. Show that the axioms predict that nothing will happen when an action is executed in a state where its preconditions are not satisfied.

2. Consider a plan $p$ that contains the actions required to achieve a goal but also includes illegal actions. Is it the case that $$ initial state \land successor-state axioms \land p {\models} goal ? $$ 3. With first-order successor-state axioms in situation calculus, is it possible to prove that a plan containing illegal actions will achieve the goal?

Up to now we have assumed that the
plans we create always make sure that an action’s preconditions are
satisfied. Let us now investigate what propositional successor-state
axioms such as ${HaveArrow}^{t+1} {\;\;{\Leftrightarrow}\;\;}{}$
$({HaveArrow}^t
\land \lnot {Shoot}^t)$ have to say about actions whose preconditions
are not satisfied.

1. Show that the axioms predict that nothing will happen when an action
is executed in a state where its preconditions are not satisfied.

2. Consider a plan $p$ that contains the actions required to achieve a
goal but also includes illegal actions. Is it the case that
$$
initial state \land successor-state axioms \land
p {\models} goal ?
$$
3. With first-order successor-state axioms in situation calculus, is it
possible to prove that a plan containing illegal actions will
achieve the goal?