Exercise 9.9

This question considers Horn KBs, such as the following: $\begin{array}{l} P(F(x)) {\:\;{\Rightarrow}\:\;}P(x)\\ Q(x) {\:\;{\Rightarrow}\:\;}P(F(x))\\ P(A)\\ Q(B) \end{array}$ Let FC be a breadth-first forward-chaining algorithm that repeatedly adds all consequences of currently satisfied rules; let BC be a depth-first left-to-right backward-chaining algorithm that tries clauses in the order given in the KB. Which of the following are true?

1. FC will infer the literal $Q(A)$.

2. FC will infer the literal $P(B)$.

3. If FC has failed to infer a given literal, then it is not entailed by the KB.

4. BC will return ${true}$ given the query $P(B)$.

5. If BC does not return ${true}$ given a query literal, then it is not entailed by the KB.