In backward chaining, if it is desired to prove the conclusion C of a
clause, the system tries to do so by proving the premises
* P _{1} ... P_{n}*.

*&forall x CAR(x) &and RED(x) &rarr EXPENSIVE(x)*

Given this axiom, an attempt to prove that *BMW _{1}* is expensive would be
reduced to the subproblems of proving that it is a car and that it is red.

** Problems:**

- Infinite loops. For example, consider transitivity:
*&forall x &forall y &forall z GREATER(x,y) &and GREATER(y,z) &rarr GREATER(x,z)* - The system has to keep reproving (and failing to prove) the same mundane facts.