 Welcome back in the last few lectures we discussed various decision procedure methods in the context of propositional logic for instance we first began with the most simplistic kind of method so that is constituted with the truth table method and then we saw that when the number of variables increases from 2 to 3 maybe 5 etc and all the number of entries will also increase the number of rows will also increase then it will be very difficult for us to manage and in that context we discussed about another method which is directly relevant to the truth table method so that is the indirect truth table method and instead of checking all the rows for the validity of a given argument we checked only few rows where you have true premises and a false conclusion if you come across that particular kind of thing we said that it is invalid argument and in the second method so that is due to semantic tableau method so this depends upon constructing a counter example so that means if you want to show that a given argument is valid then what you need to do is you have to deny the conclusion and then you need to construct a tree based on some kind of tree rules and then when you when you come across a situation where there is some there are some content contradictions in the branch then the branch closes that means you have established that not x is unsatisfiable that means x has to be valid x has to be true that is the second method which we have discussed and the third method is the syntactic kind of method so that is due to natural deduction method so where we employed some kind of basic principles of logic such as modus ponens modus tolens constructive dilemma etc and all and then we proved lots of theorems and all and then we also discussed something about reducing the given preposition logical formula into its corresponding conjunctive and disjunctive normal form any prepositional given prepositional logical formula can be reduced to its corresponding conjunctive normal form that is conjunctions of disjunctions or disjunction disjunctive normal form that is disjunctions of conjunctions so once you reduce the given formula into conjunctive and disjunctive normal forms we can talk about the satisfiability and then once you establish that it is unsatisfiable that means not x is unsatisfiable then obviously x has to be valid. So today we will be discussing another important and interesting method which is widely used in the context of automated theorem proving in the computer science so that is the name of this method is called as resolution refutation method so what we will be basically doing is like this so this is a method which works for only those formulas which are in conjunctive normal forms so it can also be called as a special case of formulating some kind of conjunctive normal forms from a given formula. So first we will talk about what we mean by the normal forms which we discussed in greater detail in the last few classes that is conjunctive and disjunctive normal forms are considered to be normal forms and then we will introduce some of the definitions such as literals, clauses etc and then we talk about what occupies the central position of this resolution refutation method that is the resolution principle and then once you state this resolution principle then we will talk about constructing some proofs based on the resolution refutation method and we will do some kind of bit of problem solving to know to well equip with this particular kind of method. So this is also considered to be one of the important decision procedure method as is the case with other decision procedure methods you can talk about whether or not a given well form formula is a tautology that is the valid whether or not the formula is valid or you can talk about when two groups of statements are consistent to each other are satisfiable to each other all these things one can come to know with the help of these particular kind of decision procedure method and as is the case with other methods resolution refutation method is also considered to be sound consistent and even sound and consistent. So this the basic idea is like this it was introduced by John Abraham Robinson in the context of automated theorem proving. So when we talk about predicate logic in resolution refutation method in the context of first order logic that is the predicate logic we will we will talk more about Abraham Robinson's direct work on resolution refutation method. So this method has been extended and it used in the computer science widely you know. So like semantic tableaux method it involves a refutation procedure that means you are trying to construct a counter example. So this in this method we try to show that a given well form formula is unsatisfiable so in the semantic tableaux method what we did we have denied the conclusion and then we have come up with a branch closure that means we have established the unsatisfiability of a given well form formula. So that means the original formula has to be a valid formula in the case of premises in a conclusion kind of format. So if you deny the conclusion then the conclusion leads to the branch closure that means not of X is false that means X has to be true. So this method assumes that a given formula is in the conjunctive normal form. So all of us know that any given formula can be either expressed in DNF that is the disjunctive normal form or it can also be expressed in terms of conjunctive normal form. So conjunctive normal form is a normal form in which only negation and disjunction occurs and conjunction of course there is no implication and double implication sign which exist in the conjunctive normal forms. So this conjunctive normal form is conjunctions of disjunctions if any clause that is C1 and C2 C3 etc where each C1 is a combination of some disjunctions and all. So in the disjunctions if formula contains a literal and its negation then obviously the disjunction is going to be true then if all the disjunctions are true then obviously the conjunctive normal form is going to be true and hence it is going to be valid. But here what we will do here is that given a formula in the conjunctive normal form so we will be applying resolution principle which we will be talking about in a while from now and then we will derive some kind of contradiction and all that is the empty set empty clause. So this we will talk in detail in a while from now so now this method is based on very few axioms it will not involve as many axioms as in the case of axiomatic method which we will be talking about it in the next few classes but it involves very few axioms and it is mainly dependent on resolution principle. So resolution method is used to test as is the case of semantic tableaux method it used to set test unsatisfiability of a set of clauses in the propositional logic so there are some clauses C1 C2 C3 etc and all these things combined together will lead to some kind of contradiction. So the resolution principle basically checks whether the empty class is contained or it is derived in the language yes so that is what it checks and all so the resolution method is based on the resolution principle. So resolution principle is like a kind of modus ponens this is also called as a cut rule so what is modus ponens we have a implies b and you have a and then this a gets detached you know what follows is b so this is what is modus ponens rule there is another rule which is nothing but it is an instance of modus ponens which is called as cut rule. So for example if you have a r ? and not a r ? so from this you can infer ? r ? so what happens here is is that you have a literal a and its negation is there in the second formula so these two cancels each other and then what remains is only these things ? ? so this is another kind of instance of modus ponens rule is called as cut rule it cuts a and not all fine all fine not all for closest and then it do not remain in the final in conclusion all what remains is ? or ? so this rule is called as cut rule this rule allows us to carry in a instead of in the case of modus ponens it will not allow this particular kind of thing a because it gets detached in the modus ponens rule but one of the advantages of this cut rule is used in the resolution principle is that in addition to this one we can add one more kind of preposition here so it comes up with some kind of extra preposition in this case ? so that is one of the advantages of this cut rule so now the resolution is restricted version of this particular kind of cut rule in which a must be a literal whereas ? and ? must be formulas so these this can be any formula and all but a that exist here has to be a literal now we need to talk about what we mean by literal what we mean by a clause and what we mean by a formula etc and when do you say that a given well-formed formula is in the conjunctive normal form you need to note that this method works only in case of the formulas which are already there in the conjunctive normal form so now here are some of the important notions before going into the details of this resolution repetition method so we need these important notions so let us go into the details of these important notions one by one first important thing is that we need to talk about a literal a literal is a prepositional letter simply it when you express it in the positive way it is P and you express it in the negative way in the negation of that one is not P usually in the language of logic what we write it is this thing P and it is complement so this is same as not P in some textbooks it is written as ? B P n so I am using this particular kind of symbol so it is negation of you can also write it as P bar so now this is what is considered to be a literal it is just like a prepositional variable or prepositional letter P Q or etc and all and it is complementary is not P or P bar so P is considered to be a positive literal and P bar or not P is considered to be a negative literal so now this is what we mean by a literal this is what we use in the resolution repetition method so this is a special instance of the cut rule that we have express it in the last slide now the second thing which is important in the context of resolution repetition method is a clause C is considered to be finite set of literals it is a combination of finite set of literals either it can be P or Q or R or it can be P bar or Q bar or R or it can be of course you can express the same thing as not P or not Q or R so literals combine together and form a clause so usually in the case of conjunctive normal forms these clauses will be in the form of disjunctions so what is a conjunctive normal form conjunctive normal form is like this C1 C2 C3 etc CN where each C1 is considered to be a disjunction a3 and a4 a5 a6 just for sake of understanding we are writing it like this so this whole thing is considered to be clause if you take only these the individual letters into consideration they are considered to be literals so if you express it like this is the positive thing and negative literal stands for this one or it can be even written as this not even this is exactly opposite of the literal even so this is what we mean by clauses in the case of CNF it is obvious that C1 and C2 C3 etc all consist of disjunctive literals I mean it consists of disjunctions now so now here is an important thing which you need to note an empty clause where it does not have any elements and all any literals so that is always considered to be false as it has no true elements so that is the important thing which one needs to know empty clause is always going to be false because it does not consist of any true elements so now a formula S is considered to be set of clauses now we need to talk about what we mean by a formula a formula S is a set of clauses C1 C2 CN etc the whole thing the conjunction of all these clauses will become a formula so that is going to be true as we all of us know a conjunction is going to be true only when all its conjuncts are true that means all the disjunctions that exist are going to be true and all in that case C1 is true C2 is true and CN is true if all of its elements are true and an empty formula which is written as a empty set it does not consist of any clauses is always true it has no false element and all so now there is a minute difference between empty clause and empty formula and all formula does not consist of anything and all there is always going to be true because empty set is a subset of all sets and all so since it has no false element and all so obviously an empty formula is going to be obviously true but empty clause is going to be obviously false so these are the few things which we need to note so now the third thing is that an assignment a is consistent set of literals one not containing both p and not p for any prepositional letter p so suppose if you assign a value to a given formula so that has to be either t or f you cannot simultaneously apply both t and both f and all so that is what it says it has to be a consistent set of literals so now the notation that we make use of it in the resolution repetition method is this thing so when a formula a satisfies particular kind of yes set of formulas especially it is written as a double-turn style yes if and only if for all c that is for all c belongs to your language yes and c and a the intersection of c and a is not going to be empty if it is empty then it is going to be false and all as is the case of the second one empty clause always going to be false so you have to ensure that c and intersection of c and a has to be non-empty at least one valuation in which the formula is going to be true that is what we mean by satisfiability so the valuation induced by a makes every clause in s true so that there only the whole formula is going to be true because it is in the conjunctive normal form each c1 c2 c3 all these things have to be true so that you know your conjunctive normal form has to be true so now a formula is going to be unsatisfiable if there is no assignment a such that the formula is going to be true so these are the common things which we all already know so one has to ensure that there is no empty clause and empty clause is there is going to be the formula is going to be false but empty formula is always going to be true so now this is the resolution method so what are the various steps that are there that the things which we employ in the resolution method so what is that we are trying to resolve so now consider the two clauses c1 and c2 let us say these are the two clauses and first of all you need to note that this applies to only conjunctive normal forms it is only in the context of conjunctive normal forms you can talk about the resolution repetition method now consider the two clauses c1 and c2 containing literals L1 and L2 respectively and L1 and L2 are complementary to each other that means one letter is L another letter is literal is L bar that is not L these are the two literals that we have so we have two clauses c1 and c2 and the literals that exist in this c1 c2 or like this a literal its negation is there in the given formula so now the resolution procedure is as follows we will be solving some more problems so that you will understand this particular kind of method so now the first step that is involved here is that delete L1 from c1 and L2 from c2 yielding clauses c1 prime or c2 prime so then this is the first thing which we need to do first you need to eliminate this literals L1 and L2 so where c1 consist of a literal L and c2 consist of a literal not L so that is why L and not L leads to f and all so that is why it will go away annihilation kind of thing and all so now to form the disjunction of c prime of c1 prime and c2 prime so now then what you will do is delete if there are any redundant classes like PRP or P etc and all that reduces to only P this obtaining some kind of final the resulting clause c is called as the resolvent of c1 and c2 so that c is called as the resolvent of c1 and c2 so the clauses c1 and c2 are said to be parent clauses of resolve so let us consider some examples so that you will understand this method in a better way so the idea here is that it is based on the cut rule alpha or gamma not alpha or beta and then you get alpha gamma or beta so what is that we have done here so let us consider some examples in p1 p1 p2 p3 this is clause c1 and then you have c2 another clause like p1 p2 p3 so now these two gets resolved and all the first step that we need to do is to find out in literal and its negation and all so in this case we have not p2 here and then you have p2 here so that vanishes and all so now that is the first step that we need to make use of the first steps tells us that delete l1 from c1 the literal one literal here is what you need to delete is not p2 here this later and then in c2 we need to delete its corresponding positive literal that is p2 so now you have deleted that particular kind of thing now in the second step form the disjunction of these two things so this is now it is a changed formula C prime C1 prime or C2 prime so let us consider that this is C1 and this is considered to be C2 so now the literals got deleted and all now it has become C1 prime and this formula has been changed to C2 prime because we deleted the literals so now you form a disjunction of whatever remains here so that is p1 or p3 or p1 or p3 so that is what happens so obtaining some particular kind of clause C as a resolvent of C1 and C2 so now so this is considered this is the one clause and this is another clause so this goes away and then you form the disjunction whatever remains here is considered to be a disjunction and then whatever remains here p1 or p3 is going to be another disjunction so this is C1 and C2 each letter is in the form of so sorry I am sorry for this so now what you need to do is in the second step so after removing the literal and negation you form the disjunction of all these things so that is p1 or p1 or p3 or p1 or p3 so now in the third step since we have p1 exist twice in all here so that is what is the third step delete the redundant literals from C prime so what are the redundant literals here p1 is used twice here even if you use thousands of times and all p1 or p2 p3 p1 or p1 or p1 it is always same as p1 so now this reduces to p1 or now p3 is again used twice and all so now it is simply p3 so now this is called as the resolvent of C1 and C2 so these two gets resolved and all and then we will get p1 or p3 so this is what we mean by the resolvent of two clauses so this makes use of this particular kind of rule which is called as cut so now let us consider some more examples so that you will understand this idea in a better way so as the method is very clear in one clause you have a literal L another clause you have another literal with a negative sign that means if you have p here and you have not p in the other clause and all so then what you need to do is you need to delay you need to delete the literals which are positive and negative and all then rest of the things can be grouped together with the help of disjunction that is the second step so now after grouping it into in the form of a disjunction and what you need to look for is whether this formula consists of any redundant literals and all like p or p or p etc if you the same letter occurs twice or twice and all it is same as p1 so that reduces to just simply literal p1 if p1 or p1 or p1 is same as p1 so now the clauses C1 and C2 that resultant clause after formulating in this way is called as a resolvent of these two clauses so that is what is considered to be the case the same kind of thing can be applied in simplifying digital switching circuits and all provided they are in conjunctive normal form so now there are some definitions one needs to follow before going into the examples and all let us talk about some more definitions. So from clauses C1 C2 what are this C1 or C2 C1 consist of some kind of disjunctions now p1 or p2 p3 etc so they are of this particular kind of form C1 consist of a literal L union some kind of form C1 prime so there is a first thing and C2 consist of its negative literal L bar union C2 prime so now from these two you can infer C1 prime union C2 prime which is what we call it as a resolvent of C1 and C2 so we may call C1 and C2 as parent and C as the chain whatever you get it as an outcome of the C1 and C2 is what is the resolvent that is C the chain and say that we resolved the literal L anyway we are annihilate this literal gets annihilated now because L and not L leads to F so that goes away so we resolved on that particular literal L so this is what we mean by a resolvent so now resolution deduction process is like this so a resolution deduction this also can be considered as a proof as natural deduction here what you are reducing is the resolvent in all a resolution deduction or a proof of C given a formula S that means you got a C from S and all reduce S C from S it is a finite sequence like C1 C2 to Cn and ultimately leads to C of clauses such as that each C i that means each step in your proof is a member of S or a resolvent of clauses C i and C j so there are two resolvents and all then either it should be it should belong to a member of it has to be a member of S that means it has to be some axiom or some theorem already which is brought in S or it has to be a resolvent how did we get this resolvent by using this resolution principle that is an instance of cut rule so only in this case you call it as that means you have to obtain all the steps of your proof as a result of applying the resolution principle or it has to be already member of S that means it has to be theorem or some other axiom or something like that so the resolution repetition method involves very very very few axioms and then mostly depends upon the resolution principle each time you will be applying resolution principle you will be getting the corresponding resolvent and all that adds to that clause adds to another clause will lead to another resolvent etc in that sense this proof is also considered to be an effective proof because an effective proof is a proof which ends in finite steps in finite intervals of time so now a deduction of empty set empty clause from S is called as resolution refutation of S suppose if you have a form clauses C1 and C2 and then you got an empty set so then what you deduced is an empty set so if there is such kind of deduction we say that S is considered to be refutable and we simply write it as a contradiction or empty clause is derived from S that is S singleton style R with respect to resolution repetition we got the contradiction so the empty clause empty clause is always going to be false. So now some examples we will let us talk about some examples so that we can understand this thing this method in a better way so we have two three two formulas PR and not Q not R so we can conclude P not Q in this particular kind of way so now let us talk about some formulas and all not Q R R C1 this is a conjunctive normal form and all so now let us say PR Q R R another one is just for the sake of understanding this thing PR not Q R not R so now so these are the three clauses and all so now we will be applying resolution method on this particular kind of formula so this is considered to be a formula so now each one is considered to be a clause these are considered to be clauses mostly they are all combination of disjunctions and all because it is a conjunctive normal form a conjunctive normal form is a conjunctions of disjunctions each one is disjunct so now we need to note that resolution method applies only to the conjunctive normal forms that means one has to convert a given formula in preposition logic into the conjunctive normal form then only you can talk about the resolution repetition method okay so now one may use this resolution principle twice or even twice also depending upon the need so now these two what are the clauses that you have P and you have not P so this gets resolved and all not P so now what remains here these two applying resolution principle in this way leads to not Q R R because we are resolving on P then Q not Q R R is a disjunction of these things whatever is left in this particular kind of form so this is the one which we have so now since R occurs twice and all so we need to get away from the redundancies and all so now this formula will become not Q R Q now for example if you are trying to resolve this formula not so now it is so now suppose if you are trying to resolve this particular kind of thing then you are resolving again on P only here so then you will get this formula Q R R not Q Q R R R not Q R not R so now original formulas are C 1 C 2 C 3 etc and all and now we are getting its corresponding formulas some other letters which we can use and then you can say this particular kind of thing so now you can apply again resolution principle on these two and then you can say that since you have Q here and not Q here this goes away and then in the same way not Q here and Q here so first time when you apply the resolution principle to this one it will become R since you have not Q also here this also goes away and then what remains here is this thing R Q R R not Q this becomes these two will resolve into this particular kind of thing or not Q again you write the disjunction of these things whatever remains here R R not Q not R disjunction because Q not Q is gone and all so whatever is remained is Q so this is what we have R R not Q R not R R Q so now so what is that we are trying to say is this you can apply this resolution method n number of times till you obtain a contradiction we do not get a contradiction now the given formula is going to be true so now what has happened here so this is not Q or Q and again if you apply resolution principle on this one you will get it is not Q is here and Q is here it goes away then you will get R R Q or now we can remove the redundancy and all because Q occurs twice here R also occurs twice here so now this will become R R Q R not so in this disjunction since you have a literal and its negation is already there here so then obviously whether or not whether Q is true or false this this resolvent is always going to be true so that means the Q has to be even if it is true or false it does not matter because you already have literal its negation here so the disjunct is always going to be true that clause is also always going to be true so in this way one can find out resolvents and all so now let us consider another simple example where how we can apply particular kind of resolution principle and all so here is an instance it is like this a natural language sentence is given to us so that is if this apple is sweet then it is good to eat obviously sweet apple obviously will like to eat and all eat now the second statement is like this if it is good to eat then I will eat it therefore if this is this apple is sweet then I will eat it so there are these are the three sentences corresponding to three different formulas and all the clauses so it is a combination of C1 and C2 and C3 so in an argument all the premises P1 P2 P3 lead to some kind of conclusion so so here the first two are considered to be premises and whatever follows after therefore is considered to be the conclusion so now we would like to know whether this formula is going to be true or false so in that case what you will do here is that we employ resolution refutation method so what is a resolution refutation method what in that method what you will do is you take the premises as it is and you deny the conclusion you take the negation of the conclusion and if you can derive empty clause then the original conclusion holds and all that means the negation of the conclusion is unsatisfiable then obviously the original conclusion in undeniable form is going to be valid so now let us consider these two things not A or B so that is the first thing if the apple is sweet then it is good actually that is in this particular kind of format so this is nothing but not A or B by definition it will be not here B and the second statement is also like this B implies C and now the conclusion is like this if this apple is sweet then I will eat it so that means A if this apple is sweet and I will eat this so now the corresponding definitions we can write like this this is each one is considered to be a conjunction C1 and C2 and obviously something so it is not B or C and then we have not A or C so now in the resolution refutation method what we will do is so we will take the combination of this thing not A or B and not B or C and you take the negation of this particular kind of thing and see whether it leads to empty clause or not not here you have to deny this conclusion and if this leads to an empty clause then taking the negation of the conclusion leads to unsatisfiability that means the conclusion taking the negation of the conclusion leads to contradiction that means negation of X is unsatisfiable that means if negation of X leads to some kind of empty clause then obviously X has to be valid or X has to be true so now so given this particular kind of problem be translated into its corresponding clauses the first one is translated as not A or B the second one B ? C is translated as this one and the third one is this so now as a combination of all these things it should lead to unsatisfiability and all because we have taken the negation of the conclusion so now we need to apply resolution principle on this particular kind of thing so now so you have to little bit you have to change it in all so this is nothing but A not not A is A and negation of disjunction is conjunction and this will become not C negation of C means negation of see on so now this is not in a proper disjunctive disjunction and all so somehow you need to use De Morgan's laws and you need to convert it into the corresponding thing so this is negation of negation of A or suppose this is same as this particular kind of thing okay so now what we will be doing here is this so first you resolve these two things on B so what are the literals that exist here you have B here and you are not B here so these two vanishes and all and then rest of the things what you need to do is you need to take the disjunction of whatever remains here so the literal and its negation goes away and then whatever is there here is this one so these two after applying the resolution principle you will get not A or C so now you have not of not A or C so now this is exactly opposite to this one if it is X and it is not X and all so now this leads to empty clause so now what is that we have derived we derived empty clause by taking the negation of the conclusion that means this is considered to be a resolution refutation method procedure for finding out that denial of the conclusion leads to an empty clause and all if we do not deny this conclusion it would not have led to this empty clause so it is in that sense in the process of constructing counter example we have come up with an empty clause so that is why the original conclusion holds and all here so that is negation of this one original conclusion leads to unsatisfiability that is empty clause so that is why X has to be valid or X has to be true what is X here X is this particular kind of thing that means this conclusion follows from the premises in all the explanation of this one is like this you consider not A or B and not B or C if B is true then obviously not B has to be false so B is to substitute in not B and not B will become false and the second clause in particular that is not B or C because not B is already false in order for this statement to be true it depends upon the value of C if the C is also false and all obviously the not B or C is going to be false so that means now that is the case in all C has to be true in that case so now if you take B as false and all false then obviously not B is going to be true then in the first clause that is not here B B is already false then the truth value of not here B depends upon what value not a takes if not a takes a value F and all the whole disjunct is going to be false that means not he has to be true so that means only one of the things have to be true so that is either B has to be true or B not B has to be true so now the explanation for the above is like this if X is true either not here C is true obviously then not here C must be true if not here C is false then X cannot be true so essentially here in this case we cancel B and not B so now these are some of the examples of a resolvent a resolvent of two clauses let us say C1 and C2 which consists of some disjunctions usually this C1 and C2 consist of at least one literal L and then the other clause you will have exactly the negation of that one is a literal L bar is there in the other clause then the resolvent is defined in this sense resolvent of C1 C2 is nothing but C1 – L union C2 – that negative so it is like in this case so these let us say you have not A or B and not B or C not B or C so now here it is C1 and this is C2 so now what is the resolvent of C1 and C2 they are like this so now it is C1 – this literal whatever is there here negative not a union C2 this is C2 only – this is B C2 B so now in this sense this will become a simple formula if you remove this B you have A here C1 – B is here union means here it use it as disjunction union is same as disjunction whereas intersection is same as conjunction so now not B is removed from here then it is C now this is considered to be resolvent of C1 C2 so that is what it essentially says now the resolution principle which is which occupies the central position of your resolution method a resolvent of two clauses C1 C2 is considered to be a logical consequence of C1 and C2 so that means each resolvent of any two clauses is automatically considered to be a logical consequence of this one for example if you take these two into consideration the logical consequence of this one is its corresponding result it is in that sense in your proof each time when you applying this resolution principle you are coming up with a resolvent and that particular kind of resolvent is considered to be an a logical consequence of these two formulas the C1 and C2 so now one can come up with a better proof with the help of resolution tree proofs and all so this resolution tree proofs looks like this the definition of this one is like this so the same thing which will be what we will be trying to do is put it in a some structural format so then it will it will look like the semantic tableaux method but it is not that one but it looks like a tree and then ultimately what you generate is in the tree what if you get its contradiction that is empty clause then the original conclusion is unsatisfiable I mean negation of the original conclusion unsatisfiable that means the actual conclusion holds so now resolution tree proofs are like this a resolution tree proof C from s in a label binary tree T is having this particular kind of properties so the root of the tree is labeled as C so that is what we what is that we are trying to deduce in all the root and your branches in all so in the semantic tableaux method what we have is the given formula occupies the root but it we have considered the upside down kind of tree where the root is there in the upstate upside it occupies the upside position whereas the other formulas which you come across are there in the nodes so now the root of the tree is going to be your clauses and all whatever clauses that you have taken into consideration and the resolvent is considered to be logical consequence it is corresponding nodes etc it is like these branches so now the leaves of T are labeled with elements of s and if any non-leaf node sigma is labeled labeled with C2 and its immediate successes are labeled as any other letter other than sigma 0 sigma 0 and sigma 1 are labeled with C0 and C1 etc then C2 is considered to be a resolvent of C0 and C1 so C has a resolution tree proof from s if and only if there is a resolution deduction from deduction deduction of C from s so now let us consider some kind of a resolution tree proofs and all so now here is the case and all just I will draw one simple I will prove some I will provide a simple proof resolution tree proof for given the problem so that is you have these formulas P and R and Q R so these are the clauses that we have when I write P, R that means P and R when I write Q, R that is Q and R then not P T and then you have s not T so now we have another kind of formula which is not s and all not s is here so now so these two so now what occupies the root kind of here so that is the clauses that we have different types of clauses C1 C2 C3 C4 etc so now these two I think it should be written in this way P R okay leave it like this P so now these two in resolution you will get P R and Q not for example so now these two after resolution you will get this clause P Q and you already have a clause Q of course these are all the things which are already there here first formula is B R R the second Q or not a not P T s are not a not s and even not Q is also there and these two the resolution you will get Q and not Q is here so what you will get is P so now observe these two things not P T here and s and not T here so these two resolution you will get not PRC not PRS so now you have not s here and not P here these two after resolution you will get not P so now this P and not P leads to you can call it as box or sometimes you write it as this particular kind of symbol so this is called as a contradiction so so this will serve as some kind of resolution refutation proof for a given formula now for example if you take this 1 2 3 4 maybe 5 and then you have taken into consideration not s as your conclusion now actual conclusion is yes and all yes but you have taken into consideration not s so now after applying resolution principle 2 3 twice thrice etc and all and then if you take this not not s into consideration that the denial of the conclusion and that leads to contradiction now so that means in this case maybe s is going to be your conclusion so the denial of the conclusion leads to some kind of empty clause so in this way one can find out a proof for a given kind of formula for example if you are asked to show that a given formula is given argument is valid etc and what you do is you deny the conclusion and it and you construct using the resolution principle and you will form you will formulate you will come up with a contradiction that is the empty clause in that case I mean the denial of the conclusion leads to unsatisfiability that means the actual conclusion holds so in this class we discussed about resolution repetition method so where it applies to only conjunctive normal forms whenever you have two conjuncts C1 C2 and these two gets resolved into another kind of conjunct especially when you have a literal and its negation is there in the other closet so now it has his own this method has its own advantages that is resolution method is considered to be sound that is if there is a resolution refutation of s then s is considered to be we are saying effectively that a given s is considered to be unsatisfiable and this corresponding lemma is that if the formula s is a combination of C1 and C2 that is satisfiable and C is a resolvent of these two clauses then obviously C is also considered to be satisfiable so and this resolution refutation method is also considered to be complete that is if you can somehow show that s is considered unsatisfiable then obviously there is a resolution refutation of s so like any other deduction decision procedure methods this resolution refutation method is also considered to be consistent sound etc so now we will be talking more about this particular kind of resolution refutation method in the context of predicate logic there we will talk about this particular kind of method in greater detail.