# Artificial Intelligence. Clause Form and The Resolution Rule. Prof. Deepak Khemani. Department of Computer Science and Engineering

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1 Artificial Intelligence Clause Form and The Resolution Rule Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Module 07 Lecture 03 Okay so we are looking at resolution method. And we had said that we came to the resolution method because of the fact that forward chaining and backward chaining were not complete. Now a resolution rule if you keep applying repeatedly is not complete in general. To give you a simple example so what is the notion of completeness that if something is entailed then they must be derived. So to give you an example if you have some random database in which there is only one statement P it could be first order logic it could be in propositional logic it doesn t matter and this entails the statement S OR NOT S which is not surprising because S OR NOT S is a tautology in any language so it must be true. Infact you don t even need that P to be true. But as you can see that there is no way of deriving S OR NOT S from P in the sense that you cannot derive. Or in other words from this statement P you cannot derive S OR NOT S. in that sense it is not complete but resolution refutation is complete. And so what do we mean by this that if the input is unsatisfiable then one can drive the empty clause. Which in our language we can say if a set of formulas S entails an empty clause it means it is unsatisfiable. Anticlause stands for false if you can derive false that means your original set must be unsatisfiable or contradiction. So if this then you can and this was shown by Robinson when he actually proposed the method the refutation is unsatisfiable that if you want to show that something is the refutation is complete that if you want to show that a set of formulas is unsatisfiable then there would always exist a proof that you can find. So you can derive the null clause.

2 (Refer Slide Time 4:16) Now ofcourse this doesn t create any particular difficulty for us because in general if a set of clauses S entails a formula alpha and which is what you want to show that alpha is a consequence of a set of clauses S then if you take the union of S and the negation of alpha then this is empty. So remember that entailment basically stand for a tautology of some kind that S implies alpha is true so a simple way to make it unsatisfiable is to take the negation of what you want to show which is ofcourse we know it as proof by contradiction add it to the set of clauses and then entailment follows that the empty clause is listed. And because we have said that this is true this is the definition of compleness. We have already spoken about soundness so infact you can replace this if then statement with an if and only if statement that you can derive the null clause if and only if the original set is a contradiction. So its not a problem to work with refutation method you can easily adapt that. So we are talking of FOL in general so resolution refutation so we already see that the choice of your proof methods determines if it is complete or not. so with resolution refutation it is sound and complete. There is also a third property we are interested in that will our algorithms terminate that s called decidability and it turns out that for all kinds of methods FOL is it is semi decidable. What do we mean by this. That if the input is unsatisfiable we can derive the null clause, so for example one could use something like breadth first search but if the input is satisfiable could loop forever. Let me give you an example.

3 (Refer Slide Time: 8:32) And we have seen in backward chaining for example that you could write rules which could go into infinite loops. So just look at this rule this says that if let me first write it in logic. That if the successor of s remember s is a successor function. If the successor of s is less than y then s is less than y and we can accept that this is a true statement. So which if we write in clause form we will write it as less than let me use question mark here. I convert P implies Q into not P or Q and write it as a clause. Now supposing that our goal is that is zero less than zero. So first of all when we say that s the goal the basic question we are asking is is this entailed by the knowledge base and our knowledge base has only one sentence. And you can see that this will not entail. So we negate it this is our strategy for refutation and add the negation of the clause which is. You can see that if I resolve these two clauses I will cancel this not less 0 0 by lessthan x y by saying x is equal to 0 and y is equal 0 and I will get a new clause successor of 0 which I will write as 1 or lets just write it as 0 to avoid confusion. Now we can match this with the same clauses we have only two clauses to start with and we can produce a new formula. And this process can continue indefinitely. So with just these two statements we can get into an infinite loop we can keep resolving this whole process to generate new clauses which can be resolved with the first clause again and again. So this goes into a loop.

5 So we will go through our process again to resolve this with this. And what do we get not plus this time I will just use natural numbers 3 R1 or. Remember that the unifier for this will have R is equal to successor of R1 and ofcourse x is equal to 2 and y is equal to 3. Sorry x is equal to 1 and y is equal to 3. From this and again from this we will get not plus 0 3 R1 OR. Now this will resolve with the second statement given in our knowledge base which says that plus 0 z z. okay so we wont get the null clause this time but what we will get is only this answer successor successor I will just write 3 here. (Refer Slide Time: 19:48) so we will not terminate here. So at termination the answer predicate tells you what is the answer. What was the question we asked we asked what is 2 plus 3 and this answer predicate says that the answer is 5. So successor of successor of 3 is 5. Remember the example Answer we had considered in which our KB was the following ontable a ontable b green a or green b. so we have these four clauses. And if our query is does there exist something which is on the table and green and we had shown that resolution method will allow you to do this. Now what will we do we will convert this into clauses so ontable We will negate and convert it into clause form. so what will we get now. So what are the clauses this is one clause. Then ontable a is another clause ontable b is third clause and green a or green b Is fourth clause. Andwe have shown that from these four clauses this is the negation of the

7 So this is given to us so without using real names we will use the names a and b. so this statements says that either a is a culprit or b is a culprit that is known to us that is given to us. You could have written this statement as follows. For all x you could have written this as well and basically two of them are equivalent. And we have seen that one of the problems in first order logic is how to express things. So this is one question that one should look at. And then we have some more clauses there is a clause that says that I will just use uppercase in prolog style to stand for the variable here. So if you read it in first order logic you should really read it as culprits wear red shirts. Or for all x if x is a culprit then the color of x s shirt is red. So remember we had talked about properties and there are different ways of writing color and so on so I will chosen this particular set of predicate. As an exercise you should convert to triple. But I will leave that as an exercise for you. There are different ways of expressing the same fact, so anyway we are told that culprits wear red shirts and we are also told that one of the two culprits that we have a was actually wearing a blue shirt. And then we have some general knowledge about colors that colors are different which we can express as follows. So if you go and sort this a little bit carefully what I am saying is that either color c1 is equal to color c2 or they are different you cannot have u for some object shirt lets say shirt wearing both red shirt and blue shirt at same time. One of them must be atleast false. So this is basically saying that colors are distinct and I have in my knowledge base certain statements like red is not equal to green and blue is not equal to red and all kinds of such statements. I just need blue is not equal to red so I am not writing the others here. And this is the knowledge base given to us this is one clause this is another clause this is the third clause which says that a was wearing a blue color. This is the clause which says that colors are different that you cannot wear the color of your whatever you are wearing is unique. And this is the clause which says red is not equal to green and blue is not equal to red. (Refer Slide Time: 29:25)

9 Then we can resolve the fact that there is a culprit with this one to give us the fact that b is the culprit. Infact by now we have arrived at that answer but since we are looking at the resolution refutation method you see that between this clause and the goal clause you will terminate with answer. (Refer Slide Time: 33:59) So as we saw that there were two derivations one which is in green did not pinpoint the culprit for us but we have an alternate derivation which can exploit the information we have in the knowledge base that culprits wear red shirts and since we know that a is not wearing red shirt we could figure out that a is not the culprit and b is the culprit and that shows up in answer. So if the resolution method had taken this as the derivation then the answer would have been answer b. notice that both answers are logically correct. If you say b is the culprit it is correct if you say a is the culprit or b is the culprit even that is correct so there is nothing wrong which is going on. Its Just that we have not been able to identify what is. so in the next class we will take a closer look at equality and we will see that equality is a special predicate its not like any other predicate like father mother loves and so on because it carries with it a certain amount of mathematical knowledge which we will need to make explicit if we are to exploit the fact that equality has certain properties. So we will do that in the next class.

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