#### 5.3.4.1 Incorrect Answers

An **incorrect answer** is an
answer that has been proved yet is *false* in the intended interpretation. It is
also called a **false-positive error**. An
incorrect answer can only be produced by a sound proof procedure if an incorrect definite clause
was used in the proof.

Assume that whoever is debugging the knowledge base, such as a domain
expert or a user, knows the intended interpretation of the symbols of
the language and can determine whether a particular proposition is *true* or
*false* in the intended interpretation. The person does not have to know how
the answer was derived. To debug an incorrect answer, a domain
expert needs only to answer yes-or-no questions.

Suppose there is an atom *g* that was proved yet is *false* in the
intended interpretation. Then there must be a rule *g←a _{1}∧...∧a_{k}* in the knowledge base that was used to prove

*g*. Either

- one of the
*a*is_{i}*false*in the intended interpretation, in which case it can be debugged in the same way, or - all of the
*a*are_{i}*true*in the intended interpretation. In this case, the definite clause*g←a*must be incorrect._{1}∧...∧a_{k}

This leads to an algorithm, presented in
Figure 5.6, to debug a knowledge base when an atom that is
*false* in the intended interpretation is derived.

**Procedure**Debug(

*g,KB*)

2:

**Inputs**

3:

*KB*a knowledge base

4:

*g*an atom:

*KB g*and

*g*is

*false*in intended interpretation

5:

**Output**

6: clause in

*KB*that is false

7: Find definite clause

*g←a*used to prove

_{1}∧...∧a_{k}∈KB*g*

8:

**for each**

*a*

_{i}**do**

9: ask user whether

*a*is true

_{i}10:

**if**(user specifies

*a*is

_{i}*false*)

**then**

11:

**return**

*Debug(a*

_{i},KB)12:

13:

14:

**return**

*g←a*

_{1}∧...∧a_{k}This only requires the person debugging the knowledge base to be able to answer yes-or-no questions.

This procedure can also be carried out by the use of the * how*
command. Given a proof for

*g*that is false in the intended interpretation, a user can ask

*that atom was proved. This will return the definite clause that was used in the proof. If the clause was a rule, the user could use*

**how***to ask about an atom in the body that was*

**how***false*in the intended interpretation. This will return the rule that was used to prove that atom. The user can repeat this until a definite clause is found where all of the elements of the body are

*true*(or there are no elements in the body). This is the incorrect definite clause. The method of debugging assumes that the user can determine whether an atom is true or false in the intended interpretation. The user does not have to know the proof procedure used.

**Example 5.14:**Consider Example 5.5, involving the electrical domain, but assume there is a bug in the program. Suppose that the domain expert or user had inadvertently said that whether

*w*is connected to

_{1}*w*depends on the status of

_{3}*s*instead of

_{3}*s*(see Figure 1.8). Thus, the knowledge includes the following incorrect rule:

_{1}*live_w*

_{1}←live_w_{3}∧up_s_{3}.All of the other axioms are the same as in Example 5.5.
Given this axiom set, the atom *lit_l _{1}* can be
derived, which is

*false*in the intended interpretation. Consider how a user would go about finding this incorrect definite clause when they detected this incorrect answer.

Given that *lit_l _{1}* is

*false*in the intended interpretation, they ask how it was derived, which will give the following rule:

*lit_l*

_{1}←light_l_{1}∧live_l_{1}∧ok_l_{1}.They check the atoms in the body of this rule.
*light_l _{1}* and

*ok_l*are

_{1}*true*in the intended interpretation, but

*live_l*is

_{1}*false*in the intended interpretation. So they ask

**how**2.

The system presents the rule

*live_l*

_{1}←live_w_{0}.*live_w _{0}* is

*false*in the intended interpretation, so they ask

**how**1.

The system presents the rule

*live_w*

_{0}←live_w_{1}∧up_s_{2}.live_w_{1}* is *false* in the intended interpretation, so they ask
*

*
*

**how**1.

The system presents the rule

*live_w*

_{1}←live_w

_{3}∧up_s

_{3}.

Both elements of the body are true in the intended interpretation, so this is the buggy rule.

The user or domain expert can find the buggy definite clause without
having to know the internal workings of the system or how the proof
was found. They only require knowledge about the intended interpretation
and the disciplined use of * how*.