Especificando el rendimiento

The derivation of negative conclusions from the lack of positive information about a predicate is justified by a belief or assumption that we have all the positive information that there is to be had about the predicate. This applies both to the conclusions we derive by actively observing the world, and to the conclusions we derive by consulting our beliefs. For example:

You look for your keys in their usual place, and you cannotfind them. On the assumption that you have done a thorough investigation, you conclude that they are not in their usual place.

If you believe that Christopher Columbus discovered America in 1492, and you believe that a person can discover something only once, then it follows that Christopher Columbus did not discover America in 2010 or in any year other than 1492.

If you believe that the last train is at 22:52, and you believe that the only trains on a given day are between thefirst and last train, then there is no train scheduled to leave at 22:53 or at any other time after 22:52 on the same day.

If you believe that you know how to add two numbers, that every pair of numbers has only one sum, and that when you add 2 + 2 you get 4, then you can conclude that 2 + 2≠ a pot of gold.

Deriving a negative conclusion from the failure to solve a positive goal is called negation as failure in logic programming:

to show that the negation of a positive sentence holds, show that the positive sentence does not hold.

Negation as failure extends the much simpler if–then–else statement of more conventional programming languages. Analogues of the if–then–else statement are also familiar in natural languages like English. For example, the second and third sentences of the London Underground Emergency Notice are expressed in a variant of the if–then–else form:

if any part of the train is in a station, then the driver will stop the train,

else the driver will stop the train at the next station.

The use of negation as failure to derive a negative conclusion is justified by the closed-world assumption that you have complete knowledge about all the condi- tions under which the positive conclusion holds. It might better be called the closed- mind assumption, since an agent’s beliefs are not held externally in the world, but internally in its mind. The assumption can be represented as a meta-belief:

the negation of a sentence holds if the sentence does not hold.

This meta-belief is a meta-sentence, because it talks about sentences. It can also be understood as an epistemic or autoepistemic sentence,2because it can be phrased in terms of what an agent knows or believes:

the negation of a sentence holds

if I do not know (or believe) that the sentence itself holds. The term epistemic comes from the same root as epistemology, the study of knowledge.

As we will see inChapter 17, the language of Computational Logic can be extended to include goals and beliefs that are meta-logical or epistemic. Because

2 _{Epistemic logic and meta-logic are very similar when understood informally, but they are very}

different when they are formalised. The relationship between them is touched upon in later chapters, but to some extent is still an open research issue.

the closed-world assumption has conditional form, it can be used to reason backwards or forwards, like any other conditional. Backward reasoning with the closed world assumption is equivalent to negation as failure. Therefore, neg- ation as failure is a natural complement to the use of backward reasoning in general. Given a conditional with negative conditions of the form:

positive conclusion if positive conditions and negative conditions backward reasoning uses the conditional as a goal-reduction procedure:

to show or make the positive conclusion hold, show or make the positive conditions hold and show or make the negative conditions fail to hold.

To illustrate the negation as failure rule (abbreviated naf), suppose that we are trying to decide on whether or not to go to a party and suppose:

mary will go if john will go. john will go if bob will not go.

Suppose we are interested in whether mary will go. Then we can reason back- wards as follows:

Initial goal: mary will go. Subgoal: john will go. Subgoal: bob will not go.

Naf: bob will go. Failure: no!

Success: yes!

In accordance with the closed-world assumption, because we have no way of showing that bob will go, it follows that bob will not go, and therefore that mary will go.

The same conclusion that mary will go can also be derived by reasoning forward, once we get off the ground by starting with the assumption that bob will not go:

Assume: bob will not go.

Forward reasoning: john will go. Forward reasoning: mary will go.

Now suppose Bob decides to be difficult. Believing that mary will go, he decides to go as well. Let’s see what Mary thinks about that:

Initial goal: mary will go. Subgoal: john will go. Subgoal: bob will not go.

Success: yes!

Failure: no!

So it seems that Bob will be going to the party on his own. The addition of the new information that bob will go defeats the previous argument that mary will go. It similarly defeats any attempt to show that john will go.

This property of negation as failure and the closed-world assumption is called defeasibility or non-monotonicity.3It is a form of default reasoning, in which an agent jumps to a conclusion, but then withdraws the conclusion given new information that leads to the contrary of the conclusion.

Looked at in this way, the closed-world assumption is not so close-minded after all, because any conclusion obtained with its aid is always subject to revision. It is as though the conclusion had an extra, hidden auto-epistemic qualification, as far as I know. For example:

Conclusion: Mary and John will not go the party, as far as I know.

The development of logics for default reasoning has been one of the most important achievements of Artificial Intelligence. Most of the research has been concerned with exploring alternative “semantics” of default reasoning and with developing efficient proof procedures. The closed-world assumption is an informal semantics, but it needs to be refined to deal with more difficult cases, as the following example shows.

Suppose that Bob is now out of the picture, but Mary and John are still having trouble deciding what to do:

mary will go if john will go. john will go if mary will go. Initial goal: mary will go.

Subgoal: john will go. Subgoal: mary will go. ad infinitum . . .

Since it cannot be shown that mary will go, it follows from the closed-world assumption that mary will not go. Similarly john will not go. As far as we know. The example shows that default reasoning can involve the need to reason with an infinite amount of resources. For this reason, the semantics is said to be non-constructive. However, in this as in many other cases, the infinite chain of

3 _{Monotonicity in mathematics means that the more you put into a system, the more you get out.}

Classical logic is monotonic in this sense. Default reasoning is non-monotonic, because putting in more information can result in previously derived conclusions being withdrawn.

reasoning needed to show that a negative conclusion holds can be detected finitely by noticing that the same subgoal reoccurs as a subgoal of itself. But in the general case, infinite failure cannot be detected by finite means.

This is an example of the same phenomenon underlying Kurt Gödel’s (1931,

1951) proof of the incompleteness theorem, which states that there exist true, but unprovable sentences of arithmetic. We will return to this issue inChapters 15,17

andA2. Moreover inChapter 15, we will investigate afinite, constructive version of negation as failure and discuss its relationship with proof in arithmetic.