 The last lecture where it is discussed in extensive detail about the resolution refutation method resolution refutation method is also considered to be one of the important decision procedure methods with which one will be able to know whether a given well-formed formula is tautology or valid or when two or three groups of statements are consistent to each other are satisfiable to each other etc all these things one can come to know. So resolution refutation method is also like semantic tableaux method where we looked for a counter example so here what we will do is given some set of clauses which are usually in disjunctions so given these clauses so we will try to prove an empty clause in all so after resolving these clauses in all so one will come up with a empty clause then that proof is called as resolution refutation method so that means you have used the resolution method and then you are refuting it in the sense that you have come up with an empty clause that is the contradiction so in this lecture what we will be doing is we will be studying some more examples based on resolution refutation method and then second in the second part of this lecture we will be talking about comparative evaluation of various methods that we have learnt so far so far we have learnt truth table method to start with this is considered to be the most simplest method and then truth table method we have used indirect truth table method so one need not have to construct all the rows and all it is more or less like a constructive method so especially when the number of rows increases number of variables increases and the number of rows increases then truth table method may not be an effective kind of decision procedure method maybe partly a computer can compute these things in a better way then especially when the number of row number of prepositional variables increases to 4 or 5 more than 5 things will become unmanageable in all for us because for checking the validity you need to check all the rows in which your true premises in a false conclusion that establishes that a given argument is invalid so then we talked about semantic table method and then we try to solve one single problem with the help of all these methods one is truth table method the second is semantic table method and the third one is reducing it into CNF and DNF and the fourth method is the semantic sorry the resolution method and we also use some kind of syntactic methods so that is the natural deduction method where we used conditional proof and the lecture add absurdum kind of proof so what I will be doing in the second part of this lecture is that I will take up one example then I will I will try to solve that problem using all these methods and then when the time arises I will talk about the importance the merits and demerits of these particular kinds of decision procedures that are available to us to start with in continuation to the last lecture here let us take some examples some more examples so that we can understand this resolution refutation method in a better way basically we it is a decision procedure method with which you can establish whether or not a given well formula is satisfiable or unsatisfiable or one can even come to know whether or not a given well formula is a valid or in the same way when you are talking about an argument you will come to know whether argument is valid or invalid so let us consider one simple example where we are trying to find out the inconsistent clauses inconsistent clauses in the sense that all these clauses that is p1 p2 p3 in the first example and the second clause is p1 and not p3 the third clause is not p1 and not p2 whether all these clauses after applying the resolution principle whether it lead to an empty clause or not is the one which we try to look for if you come across an empty clause that is considered to be these clauses are considered to be inconsistent otherwise it is considered to be consistent so now we have these clauses p1 p2 p3 usually this is expressed in this particular kind of format p3 and the second clause is p1 or p3 not p3 usually we write it in this way it is a set which consists of all these clauses and all and the third one is not p1 and not p2 so basically what we are trying to do is is that so these are the clauses C1 C2 and C3 so now what we will be doing is we will be applying the principle of resolution to it and we will find out the resolvent of these clauses now ultimately if we can reduce this empty clause this contradiction then it is considered to be inconsistent all these clauses are considered to be inconsistent so now apply the resolution principle for the first two things like this and this because you have a literal p3 and you have a literal not p3 applying resolution on a literal p3 taking into consideration the other things as formulas p1 p2 p1 etc what you will get is this p1 p2 and this will go away not p3 will go away then you have another p1 so now after applying the resolution principle you have to ensure that there is no redundancy and all for example in this case p1 appears twice enough so this is as good as same as p1 and p2 so now what we got is p1 and p2 so now once again you apply the resolution on these two things for example apply resolution on these two p1 p2 and this one so now you will write it write down here now you have not p1 and not p2 so that is the third clause in all so now applying simultaneously for example so applying resolution principle on p1 that leads to p2 so now once again applying this particular kind of thing resolution principle on these two clauses c1 and so now what we have here you have c1 c2 c3 and this is c4 for example and then c5 now this is as it is c3 only this is c5 so now once again applying a resolution on this one you will get these two on the solution you will get p1 p2 will go away because you have p2 and you have not p2 so that will go away and then so what is left is p3 so now here in this case so once you have this particular kind of thing so now you have p1 here and you have p2 here and you have not p1 and not p2 so now applying resolution on both p1 and p2 simultaneously then you will get this particular kind of clause since you have a literal here and you have negation of this literal here and the same way not p2 here and p2 here so applying simultaneously the resolution principle on both the literals p1 and p2 you will left with only an empty clause so that means that given these three clauses are said to be inconsistent to each other because when it got resolved by using resolution principle it led to some kind of empty clause that is a contradiction so empty clause is always false that means there is no interpretation which makes this particular kind of formula true that means the formula is considered to be unsatisfying so now let us assume that another example where we are trying to check for the validity of a given argument so in the second example that you are seeing this slide so let us try to solve this particular kind of problem by using resolution repetition method so these are the clauses that we have a implies D the first one second one is not a or C or D not a or C or D and the third one is not C or not C or D and a not C or D and a and C and not D C and not D implies not E so these are the clauses C1 C2 C3 C4 etc actually these are not in the actual conjunctive normal form and all you should note that resolution repetition method applies only to those formulas which are in the conjunctive normal form suppose if the formula is not in the conjunctive normal form then we should ensure that we convert it into the appropriate conjunctive normal form so each and every formula that occurs in the preposition logic will have its own corresponding conjunctive normal form so converting any given formula into conjunctive normal form may not be that difficult so now this leads to a conclusion which is not D implies B so now in the resolution repetition method what you will do here is is that given these clauses C1 C2 C3 and C4 and then in addition to that you take the negation of this particular kind of formula so now let us write the converted into appropriate form and on so the first clause can be written in this way because a in place B is nothing but not A or B now the second clause can be written as say as it is it is already in the disjunctive form so we need not have to do much so now the third clause is not C or D so now we need to apply distributive law then it will become not C or D and not C or A so this is a third clause so now the fourth one can be written in this way so now this whole thing is taken as X and this as Y so now this is not of C and not D or not E so these are the clauses that we have C1 C2 C3 and C4 in this particular kind of format and the third one is is that so now what we are trying to do is is that you have listed out all the premises and all and now we take the negation of the conclusion so now negation of the conclusion is not D implies B so first of all not D implies B is same as not not D or B so now if you apply negation to this one so this will become not D and B because not D is D only negation of D is negation of D a negation of disjunction will become conjunction and the negation of B so this is the one which we have so now this can be written as the same as this so now this is the final clause and all not of D or B so now this particular kind of formula we further converted into an appropriate form then it will become same thing will become not C or not not D is D or not E so this formula is same as this one so so now what is that we are trying to do is is that we now we need to apply resolution principle on these clauses and then in addition to that if you take the negation of that one ultimately you need to generate this particular kind of thing the contradiction that is the empty clause if you can generate empty clause and that means you have result with using resolution refutation process you have showed that all these things X1 X2 C1 C2 C3 etc and the negation of this particular kind of formula is unsatisfiable so now what you need to do here is this so now you start applying the resolution refutation method on this particular kind of thing so so there are two ways to prove this particular kind of thing either you resolve this consequence sorry the terms C2 C3 etc now ultimately you will generate this particular kind of formula or you take the negation of the conclusion and show that it leads to contradiction so now so these two for example this will not go to anything not because you are not here now taking into consideration any one of these things so not C or D and not C or A so that means both the things are there so you do not have to write anything and all so in and you can skip this particular kind of thing because all these are premises only all the premises will be in the form of conjunction on C1 C2 C3 and ultimately it leads to some conclusion X so now so where we can apply this particular kind of thing so you have a here and you have not a here so now what you will do is not a or B and not C or E so now you have listed out C1 and C4 for example so now these two resolution you will get B resolution applying on this particular kind of literal A leads to B or B or not C these two you will get this one so now you apply resolution on these two things not C or D and B or not sorry not this one so where else we have literal and its negation is the one which you need to take into consideration so now B or so now this will become B or not C now here you have C here and you have not C here so now you write it like this not A or C or D so now these two applying resolution principle it is applying on C then the C will vanish so now what you have is B or not A or D because C vanish vanishes here because you have not C here and C here so applying resolution principle on the literal C you will generate this B or not A so one can apply resolution principle many times and all ultimately in the process you should be in a position to generate whenever you come across an empty clause you need to stop the proof procedure and then that is that will serve as a proof for this particular kind of dinner of the conclusion leads to contradiction so now so now observe these two things so here we have not D and anyhow not B so that is what we have so not of D or B means not D not B so now applying the for the first thing for the not D you will get your applying resolution principle on D you will get B or not so this is what you get now once again applying a resolution principle on not B which is there already then this B and not B goes away and then you will get not so now we need to see whether we will generate a from some other things like so now you have not C or A and then you have not A so this leads to a particular kind of thing so B or not C not A or C or D so now we need to apply on this thing so not A or C or D not C or D and then somehow we need to take these clauses in such a way that you generate this empty clause so now instead of for taking this into consideration now you take this B or not C and not B so these two you will generate not C so this is what you generated from not B B or not C so now apply this particular kind of thing now not A and not C or A these two and under resolution you will get not C not C or A is not serving our purpose so somehow we need to get this particular kind of D or A B or not A not B you will get not so get this one is which we are trying to look for not A or B not here C or D so not C or D C or A etc. Somehow we did not still generate this particular kind of contradiction you take into consideration this is not not D or B not D and not B and not of D or so this and for example in this case not C or D and then not D this will you will generate not C and this one not C and not A or C or D you will get these two on resolution you will get this particular kind of applying resolution on C you will get not A or D so ultimately what is happening here is that we have applied result resolution n number of times and all somehow we are not able to generate contradiction C and not D or E not of C and D so if you can generate contradiction out of these four clauses and all then your proved not D implies B negation of not D implies B leads to unsatisfiability and hence you are said to have shown that true premises and a false conclusion that is an invalid kind of argument and all so somehow this needs to be worked out in greater detail I will come back to this problem in a while from now so what we will be doing is we will take up one simple problem and then we try to see how this problem can be solved in three different ways you know that is using the different decision procedure methods that we have learnt here. So let us consider some one more example and with which we will come to know how to solve this particular kind of problem by using various kinds of method so the one which is explained on the board in the first problem so ultimately what one needs to solve here is there so you apply resolution on in N number of steps and all and then resolve those things and ultimately so that has to be contradictory with this particular kind of thing so then you are said to have achieved your particular kind of task. So here what we have done simply is that we started applying resolution principle on different kinds of conjuncts and all ultimately we generated some kind of formulas so somehow we are not able to get to our conclusion that is the empty clause suppose if you do not get this particular kind of empty clause after applying the resolution exhaustively apply this resolution on all these steps and all then the argument is considered to be invalid. So now let us consider another example so we will see how we can apply this resolution principle in solving this particular kind of problem not a or b yeah this is what is creating problem here so I made a mistake here so that is the reason why I am looking for the solution here actually this is not of a implies B. So now this will become a or not not a means a or B so now as you can clearly see here so now we need to apply again and all so this problem changes in all so if it is simply a implies B and not a or C or D then there is no way in which you can generate a contradiction here that means the argument is going to be invalid that means not D implies B does not follow from this one so the actual problem which I could have taken for proving the validity of this given argument is in this sense so this negation is missing here if you take the negation sign here then by applying resolution principle then maybe one can come up with a kind of repetition so let us try to see this try to solve the same kind of problem by taking into consideration this negation sign and then we try to solve the same problem with the help of other methods so actual problem is this one negation of not not a implies B and not a or C and then not C or E and F not C or E and F this problem is different in all so forget about this thing so now let us come back to a different kind of problem this this problem will try to solve it by using different kind of methods so let us consider a simple example so this problem is the one which we have already seen so this is like this here is a puzzle in which there was a robbery in which lots of goods was stolen and the robbers left in a truck and it is known that nobody else could have been in all other than ABC that means it is simply represented as A or B or C and C never commits a claim without his participation that means C implies A and B does not know how to write that means B requires the company of A and C so this is represented as B implies B and A or B and C so now we need to find out whether A is innocent or guilty now this problem we try to solve it by using various kinds of methods first we will start with the simple resolution refutation method so in that method what we will be doing is we will be listing out all the all the conjunctions in all and we will start applying the resolution refutation method on this particular kind of thing so then if we can generate an empty clause from these things and that means we are said to have achieved our particular kind of thing so what is that we are trying to do here in this example is like this so here are these conjunctions all A R B R C so now what I will be doing here is that I will be trying to solve this problem by using one of the some of the methods that we have already discussed so far so the first C 1 clause is like this and second clause is C and A sorry C R A and the third one is B implies B and A R B and C so now this is already in the disjunctive form this is also in a this consists of disjunctions and now we need to convert this thing into appropriately into corresponding disjunctive form so now this whole thing is taken as X and this is why so now this will become not be R the whole thing B and A R B and C so now you need to apply associative law here so now this will become not B R B and A that is the first one and the second one is not B R is R not B R B and C so now these are the things which we have and then we can further expand it in this way using distributive law here so now this will become not B R B and not B R A so this is the first one if you apply distributive law this is the thing now R now the other one that is not B R B not B so this is not B R B and and not B R C so this is the one which we have so so what are the clauses now for us so these are the clauses that we have first one is A R B R C and not C R A and the whole thing so now in this case not B R B is obviously is going to be true only so you need not have to worry much about it is it is written as top always true and all so now whatever is left here is these things not B R A R again here not B R B is always true and all so we need not have to worry much about it so and then not me or C so that means not B R A and then the other clause is not B R C so now one can do it in various ways and all applying resolution principle on this one whether you will get a as an outcome or not that is a problem that is a question that we are trying to answer here so each resolvent is considered to be a logical consequence of the above two clauses so now so these are the clauses that we have C1 C2 and C3 so now the conclusion here for us is that now we are trying to find out whether or not a is guilty so assuming that a is guilty suppose if you take into consideration that not a whether that is inconsistent with this thing or not is on which we are trying to see so now A R B R C and this one so now we need to apply the resolution principle on this one so now these two applying resolution on little C you will get not B A R not B so because not C will go away so what is left is A R not B so you have to add those two things it will become a R not B so now applying resolution on these two things you will get applying resolution on the literal B you will get A R C R A so now in order to avoid redundancy here so you will simply write a occurs twice here so you will just simply write A R C so now this is what has come as an outcome of these resolutions so now you apply resolution on these things not C R A applying resolution on the clauses A R C and the clause that are that is there here not C R A so now the C and not C will vanish then you will get A R A so now this A R A is in order to avoid redundancy not we simply write A so what is that we got we listed out all this the information that is there here is the first clause AR B RC and not C R A and then the third clause is this one so we simplified this thing into this particular kind of format by using distribution and associative law and here is what we got so we got it because of this particular kind of thing not B or B is always true so you not have to worry much about it so what is left here is not B R A and not B RC that is what we have written here is the two clauses and applying resolution on C 2 and C 4 that means this is a logical consequence of these two clauses C 2 and C 4 we got AR not B and again applying C 1 and let us say this is C 5 and all resolution on C 1 and C 5 we got AR not AR C and we already have this particular kind of thing not C R A so that is what is already there and now again once again you apply resolution principle on this particular kind of thing concerned with the literal C whenever you have a literal C and negation is there it cancels annihilates then what is left here is AR A so AR A in order to avoid redundancy we got AR. So A is the one which we got it as an outcome of all these propositions that means A is in the original interpretation A means A is guilty if it is not here if you somehow you generated not A as a result of applying the resolution here then not A means A is innocent so now the same particular kind of thing which can be this can be solved by using the semantic tablux method as well we can show that whether or not is a guilty or not so now in the semantic tablux method what you need to do is this thing so these are the three propositions that we have so now we are trying to say that A is assuming that A is guilty so now we are assuming that suppose let us say A is the conclusion from this one and there are two ways to solve this particular kind of problem using the semantic tablux method semantic tablux method always look we look for the counter example that means if you assume that A is guilty in the beginning then what you need to do is you have to consider not A and then you constructed semantic tablux tree and then you need to generate a contradiction. So now you list out all the premises not CRA and B implies B and A are B and C so I am comparatively evaluating checking this particular kind of problem with various methods so that you will come to know the merits of these methods so now right now I am trying to solve the same problem by using semantic tablux method so now you list out all the premises in all so now the problem tells us that these are satisfiable that means all these statements are consistent to each other at least under some interpretation this formula is going to be true x1 x2 x3 so now you expand use treat rules and all to expand this particular kind of thing so now so you apply this one x implies y then it will become not x and y so this will become not B and this is a B and A are B and C so now so this is over now this can be further simplified into this thing so either it is B and A or it should be B and C now this we checked it and all so now we need to look into this particular kind of thing so now wherever the open branch is there you need to write this particular kind of information so since this is a branch it will be like this not C or A and here also you write the same information and every branch which is open you need to write this particular piece of information so now each time you finish with checking with this formula you need to see whether a literal and negation exist in a branch so this is considered to be one branch is considered to be one branch like this so now in this case we do not have we do not still have a literal and negation all the branches are open so now apply rule for this particular kind of thing a RBRC so then this particular kind of information needs to be written on every branch and on like B and C so this information a RBRC we have written it under whatever branch which is open so now so this is a RBRC a B or C B and C like this B and C so now we need to inspect all these branches and all wherever a literal and negation occurs the branch closes so now this brand goes like this a0 C0 B so that branch is open so now you have B here and not be at this branch closes here and C and not see this closes here itself and now this branch is open now you have B here and not be here this branch closes and this is also open so now you have a0 C and a B and there is no C here that means this branch is also open and this is also open but you have here you have C and not seen this closes it is a big branch in all so sometimes some resolution refutation method might be better than the semantic tableaux method because the number of branches are more here so in the resolution refutation method we have seen in the last slide that that will simplify our task you know so now here this is also open these also open and all the open branches suggest us that that makes this formula true and all that means if you observe this particular kind of open branch this says that under the interpretation in which a is true C is false sorry this branch closes here itself so need not have to worry about it because your BC here and then not see here this closes here itself so you need not have to worry much about whatever follows so now in this case all the branches are open so now open branches tells us the answer and all that satisfies this particular kind of formula so now any open branch which you take and take into consideration the first one is like this AC and not be so that is the first thing and the other open branch you will find it here a not be and we do not know whether about C in this particular kind of branch so this is the first one second one and third one and the third one is a not see and be also another kind of solution and then the other one is AB so this is the one which we have and other things are AB again the same thing which you need not have to write again another thing which satisfies this formula is AC and B and the other branches are like this ABC and BC we do not know about a here so now the last one is this thing ABC which is already we have listed so all these things are considered to be the interpretations which are going to satisfy this particular kind of formula that means when A is true C is false B is false that makes these three sentences true and in the same way when A is true B is false that also satisfies this particular kind of formula so now in all these cases you will observe that you have turned out it turned out to be the case that we got only AM so that means that usually A is considered to be guilty so our interpretation is that when I write simply a that means a is guilty when I write simply not a that means a is considered to be innocent so now the same kind of thing can be solved by using truth table method in the but in the truth table method the problem is that you have three variables that means number of entries would be eight eight rows will be will occur in the truth table so now in the truth table what you need to look for is a row in which all these statements are considered to be true that means conjunction of C1 C2 C3 whenever that is true and all so that row corresponds to our particular kind of answer the same thing which you might get it so a particular kind of example it can be done in n number of ways and all so so there in the natural deduction what you will do here is that from A or B or C and not CRA and all these things so what essentially you do here is that you list out all the premises in all so now we need to assume something in all here suppose if you start with a is guilty in all so that is the original conclusion so you need to assume something other then you take the negation of this thing into consideration and then see whether that leads to contradiction or not so for example in this case you list out all these things and all A or B or C is CRA etc and all so what essentially one needs to do is by taking all these premises and all ultimately you should be able in a position to deduce a in that case a follows from these things with using resolution refutation method we showed that a will come as a logical consequence by applying the resolution principle so in the natural deduction method what you will be doing is you will be using the principles of logic like modus ponens modus tollens etc and all and then you will generate something like a and all from these things so let us see whether we can deduce a from this particular kind of premises so now for so you have a or B or not see and you have not see are a so from these two things you will get of course A or B or C and not C or A so from this you will get a or B or B or A so in order to avoid redundancy you will write it only once and all so this will be a or B simply A or B so this rule is which we have discussed earlier this rule is called as cut rule so that says that alpha or gamma not alpha or beta suppose if you have a formula like this then so this will become gamma or beta this is called as cut rule which is also the rule responsible for the resolution principle the same rule which we apply here then you will get a or B so 1 and 2 this is a kind of modus ponens rule it is a special kind of rule you can use any rule which preserves the truth that you can use here so now so now this means something is A or B is true means definitely some one of these things should be true and so assuming that B is true here irrespective of whether A is false or A is true let us assume the B is true and all so now these two three and five modus ponens you will get B and A or B or C so how did you get this one three and five modus ponens you get this so now some of we have to reduce A as an outcome of this particular kind of so now from this since this is true and all B and A and B are C is already true so that means even B and A is also true because of this particular kind of thing X and X or Y is going to be true when both X and Y are true or even in the case in which one of the disjunct is false also then that also it leads to this thing so now since B and A is the case so now from this you can eliminate this conjunction, conjunction elimination rule which we use and then from this you will get this one conjunction elimination rule you will get a that means we have reduced A from these three premises you know using a kind of proof method which is called as natural deduction method so now you have to note that in this particular kind of method each step is considered to be the assumptions are obviously considered to be true and then if that has to be true this also has to be true each step is considered to be true and hence the final last step of your proof which is considered to be a theorem so that also turned out to be true so this is another way of showing that in this particular kind of problem A is said to be guilty in all in this particular kind of way so there is another way of doing this particular kind of thing within the natural deduction method so now what you do here is this particular kind of thing so now assuming that let us assume that A is guilty in all so now for the sake of argument you take into consideration not A that means you are assuming that A is not guilty so now we are trying to come up with some kind of contradiction so now so on the fourth step since not C or A and you have A here so this option is ruled out what you get is not see so now in the second principle so how did you get this one not not a is denial of conclusion so now you got from 2 to and 4 you got this not see so now since not a and not see are true you have not a and not see is also going to be true so now seven step not a and not see so now these two so from this particular kind of thing not see are a is already true that means a is also considered to be true and from this particular kind of thing we can take into consideration assumption that a is also true so now you have you have come across a situation where you are not a is true and a is also true so that means this leads to a contradiction that means the negation of this particular kind of formula a assuming that a is considered to be not guilty needs to this particular kind of contradiction so in this lecture what we have done is simply this that we started with some examples of resolution refutation method in that method what we essentially show is if I want to show that a given argument is valid then what you need to do is given these clauses there are the premises you add the conclusion to it the denial of the conclusion and then you will be deriving an empty clause if you can derive empty clause from that one that means denial of this particular kind of formula the denial of the conclusion is unsatisfiable that means the original conclusion holds so that is one way of showing that given argument is valid in case using resolution refutation method so then in the second part what we have seen is we have taken up one simple problem that exists in the prepositional logic and then we showed with various methods the methods that we have with which we have shown are semantic tableaux method and we also shown the same thing by using a syntactic method that is a natural deduction method with which we showed that in fact a is considered to be guilty that is what we have deduced from a given set of premises that is a natural deduction method the same kind of thing which we did not show in this class is that we can show the same thing with the help of truth table method that means you list out all the premises and then you assume that not here and then you will come across a row in the truth table in which your true premises and a false conclusion so in that case you can show that this argument is invalid that means not a leads to contradiction that means a has to be true so in this sense one can solve the same problem by using any one of these decision procedure methods it is all up to our convenience which method to employ so in some cases natural deduction method might be simpler or in other case semantic tableaux method may be better or in some simple cases to begin with truth table method face better than other methods are in some other cases resolution refutation method especially when you have conjunctive normal forms then you can straight away apply the resolution principle and simply get your answer because each resolvent is considered to be a logical consequence of the conjunctions that you are taking into consideration that conjunction is nothing but each conjunction will have its own corresponding disjunctions so that is what is in a conjunctive normal form so in the next class what we will be doing is that we will be taking up the axiomatic propositional calculus in that what you will be doing here is that we will be taking into consideration the primitive axioms and then we will be using as many little as many minimal kind of principles and all like only modus ponens is the one which we use and we use transformation rules and then we will reduce all the valid formulas within your logical system so in the next class we will be taking up the axiomatic propositional calculus there we will study two axiomatic systems one is due to Russell and Whitehead that is in the Pinkipia Mathematica and the second one is due to Hilbert and Ackerman Hilbert Ackerman axiomatic system.