CSNB234ARTIFICIAL INTELLIGENCE

Chapter 4Refutation and Resolution Proof (Part I)

Chapter 4Refutation and Resolution Proof (Part I)

Instructor: Alicia Tang Y. C.

(Chapter 13, pp. 547-574, Textbook)

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Proof by Refutation & Resolution

Resolution is a simple iterative process or procedure for deducing conclusions.

In performing resolution to the set of clauses, the negation of the conclusion is also added.

At each step, two clauses - called the parent clauses from this new set of clauses are compared (resolved), yielding a new clause called the Resolvent.

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The two clauses to be resolved are chosen in such a way that they contain a common literal appearing as positive in one and as negative in the other.

The resolvent is obtained by combining the two clauses by an or () connective after removing the common literal from the parent clauses.

Proof by Refutation & Resolution

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Steps:

construct the conflict set (premises + negation of the conclusion).

Convert the conflict set to a set of expressions in clausal form.

Repeatedly applying the resolution rule to try to derive a contradiction.

If a contradiction is found, then the argument is valid; if not, the argument is invalid.

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Let us look at this:

Premises:p qq & s rsp

Conclusion:r

Conflict set:{ p q, q & s r, s, p, r}

Remove implications and we get: p q (from p q) and q s r (from q & s r)

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& we shall get a conflict set that contains all clauses:{p q, q s r, p, s, r}

Apply resolution to derive a contradiction:

1. p q2. q s r3. p4. s5. r6. q s (from 2 and 5)7. q (from 4 and 6)8. p (from 1 and 7)9. Direct contradiction : 3 and 8.

We can conclude that the argument is valid, i.e. the conclusion

r is true.

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A Worked Example

Question:

Given the following facts & rules: all cats are animals lily is a cat all animals will die

Prove that “lily will die”

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all cats are animalslily is a catall animals will die

(1) X. cat(X) animal(X) (2) cat(lily)(3) X. animal(Y) dies(Y)

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Convert (1), (2) & (3) into clausal forms:

1. cat(X) animal(X)2. cat(lily)3. animal(Y) dies(Y)

Don’t forget to negate the conclusion:

4. dies(lily)

5. animal(lily) from 1 & 26. dies(lily) from 3 & 57. Nil from 4 & 6

Direct contradiction

Since we can reach a contradicting situation in the proof steps

Therefore, “lily will die” is true (i.e. the given goal is true)

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CSNB234ARTIFICIAL INTELLIGENCE

Chapter 4Reasoning Methods (part II)

Chapter 4Reasoning Methods (part II)

Instructor: Alicia Tang Y. C.

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Automated Reasoning

Automated Reasoning is arguably the earliest application area of Artificial Intelligence

Throughout the history of AI, automated reasoning has played an important role

Its products include a large number of inferencing techniques and strategies

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What is Reasoning?

Reasoning the set of processes that enables us to go beyond the information provided

Reasoning is the thought process that follows rules of logic.

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What is Reasoning?

We do reasoning in our day-to-day life while drawing conclusions from our knowledge or from information available to us

This is a task that humans are good at

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Automated Reasoning Components

Three components make up an automated reasoning system: an unambiguous representation language,

sound inference rules, and well defined search strategies.

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Reasoning Categories

We are able to make approximate predictions of reasoning so we should be able to build models of reasoning process so that we will be able to solve a problem almost in the same way as a person does. Reasoning is used extensively for problem solving in AI. The reasoning process can be classified into two categories: Monotonic Reasoning and Non-monotonic Reasoning.

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Monotonic Reasoning (I)

In monotonic reasoning if we enlarge at set of axioms we cannot retract any existing assertions or axioms.

Once an assertion is made, that can be considered as an axiom, i.e. during the process of reasoning, if we derive a conclusion, this conclusion can not be disproved throughout the entire process of reasoning.

Even though the newer conclusion may be more valid..Because there is no mechanisms for ‘KB updating’ use…

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Non-Monotonic Reasoning (II)

The traditional logic is monotonic, i.e. if we are able to draw a conclusion from the set of axioms already available, then we will be able to draw the same conclusion after adding some more axioms to this set of axioms. Using the set of newly added axioms we may be able to draw further conclusions.

This monotonicity property is not compatible to our natural ways of thinking.

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Non-Monotonic Reasoning (III)

In non-monotonic reasoning, an already derived conclusion may be removed, if necessary, in case of a newly added assertion forms a contradiction with the set of axioms.

We tend to remove a rule (or axiom) from our memory, whenever we come across a rule that is contradictory to some rule in the memory.

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Other Methods

There are various reasoning methods used for problem solving in AI.

Each method is based on a specific type of logic suitable for that method. Here we discuss different reasoning methods and the logic used for developing these methods.

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Deductive Reasoning (I)

Deductive reasoning allows us to draw conclusions that must hold given a set of premises (facts).

By deductive reasoning, we make inferences about an object based on the information available about a class or category to which the object belongs.

The logic used to do deductive reasoning is deductive logic.

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Deductive Reasoning (II)

Example:

Suppose you are given information that all cats have tails

then you may conclude that ‘sweetie” - your friend’s cat also has a tail even without seeing sweetie.

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Inductive Reasoning (I)

Inductive reasoning is exactly opposite to that of deductive reasoning as far as the way of making inferences is concerned.

Inductive reasoning makes generalisation based on the results available for instances.

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Inductive Reasoning (II)

As a result, conclusions need not be true given premises Category-based induction Mental models

In other words, when we do inductive reasoning, if we have information for a few objects then we conclude that this information is true for any object belonging to a class in which these objects belong.

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Inductive Reasoning (III)

• Suppose you arrived in Bangi town that’s new to you and on the first day you found a new friend who is a vegetarian. This friend introduced you to another friend who is also a vegetarian. Next day when you went to your work place, you were told that your boss is a vegetarian.

• From these, can you conclude that “all people in Bangi are vegetarians”? Surely NOT! But inductive reasoning will say “YES”.

• i.e. all people in Bangi are vegetarian

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Induction & Plausible Reasoning

I have seen 1000 black Perdana.

I have never seen a Perdana that is not black.

Therefore, every Perdana is black.

This conclusion sounds ok... but (still) it is not always right

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The method of drawing conclusions using abductive reasoning is somewhat similar to the inverse process of applying a rule.

It is about when Q is true, P will be true …which is missing from the earlier

inference rules

Abductive Reasoning (I)

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Example: Suppose we have the rule X measles(X) spots(X)

This axiom says that having measles implies having spots. While doing abductive reasoning, if we notice spots, we may conclude measles.

This conclusion is not valid as per the rules of logic. However this can be considered as a good guess when no other information is available. A measure of certainty can be attached to this guess to decide the acceptability of the guess.

Abductive Reasoning (II)

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Default Reasoning [Raymond Reiter 1980]

This is a very common form of non-monotonic reasoning. where We want to draw conclusions based on what is most likely to be true.

Default reasoning is concerned with making inferences in cases where the information at hand is incomplete.

In other words, default reasoning is normally done based on the absence of information.

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Default Reasoning

By default reasoning, we believe any statement or axiom unless it is mentioned that the statement is false.

For example: If Ali is a Professor of Computer Science and there is no proof that Ali does not possess a PhD then you may believe that Ali has a PhD

Conclusions drawn from these type of rules are called defaults.