Suppose that we have formulas:[Backward chaining only works for Horn clauses, which have at most one positive literal.]
A
B
D
A &and B &rarr C
C &and D &rarr E

A conclusion E can be proved recursively:

1. First check whether the desired conclusion is in the database of facts. If so, return True.

2. Otherwise, for each rule that has the desired conclusion (right-hand side), call the algorithm recursively for each item in the premise (left-hand side). If all of the premises are true, return True.

3. Otherwise, return False.

In this example, we would know that E is true if we knew that C and D were true; we would know that C is true if we knew A and B ; A and B are in the database, so C must be true; and D is in the database, so E is true.

With careful implementation, backchaining can run in linear time.

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