In the last class we discussed one of the important decision procedure methods that is natural reduction method earlier we also discussed truth table method and the semantic tab looks method. So in this lecture we will be talking about another different kind of decision procedure method which is called as reducing a given preposition logical formula into its conjunctive and decision to normal forms. So either you can reduce the formula into disjunctions of conjunctions that is DNF or conjunctions of disjunction CNF. So what we will achieve with this particular kind of reduction of the formulas into disjunction and conjunction of normal forms it is simple that so whenever you come across a conjunctive normal formula suppose X and something else P or not P etc and all we will be talking about two different kinds of formulas and all. So we will be reducing the given preposition logical formulas into either conjunctions of disjunction or disjunctions of conjunctions and all. So in the conjunctive normal form it is like C1 C2 C3 etc and all whereas each conjunct is nothing but a disjunction D1 D2 etc and all D3 D4 etc. So whereas in the case of DNF what we have is it is disjunctions of conjunctions and all. So we have conjunctions of disjunctions and disjunctions of conjunctions and all. So we go into the details of it in a while from now. So now observe this first kind of formula so in this formula the basic idea of reducing the formulas into conjunctive and disjunctive normal forms is this. So we know that in the case of conjunctive normal forms suppose in the case of this thing if a conjunction C1 and C2 C3 etc. So this is going to be true only when each conjunct is true and all otherwise it is going to be false. So that is the semantics of P and Q. So this reduction of well-formed formulas into conjunctive and disjunctive normal forms also follows this particular kind of idea. So that is TTF and F TFTF and then this formula is going to be false only in this case in all other cases it becomes T. So that means if you have conjunctions of disjunctions and all then if you come across in each conjunction you come across a literal and its negation and all then obviously the formula is going to be true. So this formula is going to be true this is true this is true every formula is going to be true provided when you have a formula and its negation is already there. So in the same way in the case of disjunctions in the case of DNF in particular whenever you have a formula a literal and its negation is there then it makes the whole formula false that means it makes the whole formula unsatisfiable. So this is the main idea that we will be using it so what essentially we will be doing in reducing the given well-formed formula into conjunctive and disjunctive normal forms and talking about the satisfiability and of course validity what we will be essentially doing is so we will try to reduce the given formulas which occur in this particular kind of format. So all the implications etc for example if you have a formula P ? Q ? R so what you will do is you will remove this implication by using the definition so that is a ? B is nothing but not here either the reduced formula will have only disjunctions and negations or the reduced formula from the given propositional logical formula which originally includes implication etc not so that we will reduce it to only conjunction and its negations and all. So that is what we will be doing it in CNF and DNF and in the process we will be able to talk about whether or not the formula is satisfiable etc so the basic idea of reducing the given formula into conjunctive and disjunctive normal forms is this suppose if X is any well-formed formula in the propositional logic which is of this particular kind of format X R P R ? P we know that P R ? P is obviously tautology and anything which is a disjunction of tautology have to be true only so because P R ? P is already true irrespective of whether X is true or X is false that is going to be true only so that means the given formula is obviously valid and all under whatever interpretation that you give it for X either it is T or F the variables that exist in X so where even though it X becomes true or X becomes false since P R ? P is already true so that it makes the whole formula true. So now the other idea is that disjunction is also considered to be associative because purely disjunctive well-formed formulas is valid if it contains a variable and it is negation and it is a unnegated formula like P R ? P is there and that is obviously tautology you know that makes the other formula also true so for example in this case ? P R Q it should be and in particular should not be R ?  R so this is considered to be valid because negation and unnegated forms are there here so that makes the whole formula true the first formula true and the second formula also is true so that makes the whole formula true so this is nothing but P ? Q so this is considered to be valid since it can be grouped to give simply Q R ? R if you simplify this formula you will get Q R ? R so the existence of propositional existence of propositional variable both negative and unnegated is a sufficient condition for validity so it is also considered to be necessary condition for validity you know so what you mean by validity any formula which is considered to be true a tautology is considered to be a valid formula you know so now observe this particular kind of thing and all for example if you have a formula like this one X 1 X 2 or not X 2 and another formula X 3 or X 4 or not X 4 and X 5 or X 6 or not X 6 something like that so now observe this particular kind of thing is a conjunction of all these formulas and all so it is conjunctions of disjunctions and all so that is why it is C and F so now each one each conjunct consist of set of disjunctions and all so now in this case this is already true X 2 or X not X 2 is true so now irrespective of what whatever is there here this makes the whole formula true so now in the same way a little and its negation is already existing here so that makes this thing true so that makes the whole disjunct true that means this is true this is true and even this is also it is also the case that X 6 and not X this is also true so all the conjuncts are true so that is a case here so all the conjuncts are true that is why the result in conjunct is also true C 1 and C 2 and C 3 is going to be true that means a given formula is a tautology so once you can show that it is a tautology you know obviously can that is as good as saying that a given formula is a valid formula so that is the reason why we reduce a given well form formula which is in the complex form which includes implication double implication etc into the normal forms which includes only conjunction and negation as its literals or disjunction or negation as the only things which you see in those normal forms so now existence of a propositional variable both negated and unnegated is a necessary and sufficient condition for inconsistency of pure conjunctive normal forms for example if you come across P and not P in the disjunctions in particular so this formula is going to be like this so in the case of this formulas for example if you come across in the disjunct in particular you come across a literal and its negation and all so that makes the whole conditional falls in a sense that P and not P is false so each disjunct will become false and a disjunction is going to be false only when both disjuncts are false in all other case it becomes T so in that way we will be able to know that when a given formula is also unsatisfiable and all so provided in a literal and its negation exist in that formula so a conjunct is true when it is all its conjuncts are true that is what is the case which is explained on the board on the left hand side whereas a disjunction is false when both the disjuncts are false so now what is a CNF so CNF stands for conjunctive normal forms that means a given well form formula you reduce it to conjunctions of disjunctions so now it is D1 and D2 and D3 where D1 D2 are disjunctions and all so disjunctions are represented by R and conjunctions are represented by and so if it is conjunct of a B then it is represented as A and B disjunct means A or B so now a well form formula in a preposition logic D1 and D2 and D3 and DN is said to be in conjunctive normal forms especially when X is considered to be an unnegated conjunction that means if it is a unnegated conjunction means it is P and Q etc and all if it is a negated conjunction it will lead to disjunction and all so that is why you have to rule out this possibility that X has to be an unnegated conjunction and every conjunct in the disjunction D1 D2 D3 etc. and all in X is an unnegated disjunction and all and the third condition is that every disjunct is considered to be a propositional variable it has to be either a propositional variable or it might occur in the form of negation of a propositional variable it should be in the form of P or it should not be in the form of not P. So in short X is a conjunctive normal form even only if it is a conjunction of disjunctions and no negation sign has an argument other than the singular propositional variable. So the idea here is this particular kind of thing first we are trying to talk about conjunctive normal forms so what is the conjunctive normal form it is the conjunctions of disjunctions D2 D3 etc Dn so D1 and D2 D3 Dn etc are considered to be conjunctions of disjunctions. So now the idea here is that this conjunctive normal form should have these things that formula X is an unnegated kind of conjunction for example if you have something like not D2 or something like that for example let us say D1 is P1 P2 and let us say P3 or P4 etc. Now let us assume that these are the only two formulas that are there and so in the pure sense this is not considered to be a CNF conjunctions of disjunctions not because it consists of negation of some kind of formula and all negation of disjunction and all so that means this formula has to be reduced it will be reduced to not P3 and not P4. So if it consists of negation of some kind of variables then that is not considered to be a conjunctive normal form. So you have to ensure that all these three conditions are satisfied and all X is an unnegated conjunction and all and every conjunct in X is an unnegated disjunction that is the thing which needs to be followed and every disjunct in a proportional variable or the negation of it should have to it is either a proportional variable or it should be a negation of that particular kind of variable. So let us consider some examples and all before that a well formed formula is considered to be a degenerate conjunction it appears like we have used these three rules and all like X is an unnegated conjunction so that is satisfied in this particular kind of formula P or not P or Q actually this is a disjunction but it looks like it is also it can also be considered as a conjunction also I mean conjunct C1. So X is an unnegated conjunction there is no negation of a formula which is not of X and Y etc and all which occurs there in this formula P or not P or Q and every conjunct in X is an unnegated disjunction here we do not have any this negation of disjunction kind of thing occurs here and is not of P or P etc will not figure out here second rule is also followed and every disjunct is a propositional variable or the negation of a propositional variable either P is the case the first one you have P and the second case you have not P so literal and it is negation is there and all this so every disjunct is a propositional variable or the negation of a propositional variable and it satisfies this particular kind of conjunct which is written in the form of a disjunct satisfies all the things and all it is in that sense this is called as degenerated kind of conjunction so usually it is a disjunct in all but the whole term P or not P or Q can still be treated as if there is only one such kind of formula which exists there here it can be called as a degenerate case of a conjunction so why because we can still we can write like this of course this is P is a conjunct P or P or not P or Q maybe if you want to extend it you can write it as P and P or not P and not P which does not make any sense and all but still you can write it in that way Q and Q etc Q and Q is same as Q etc or you can even write it as PR in various ways you can write it and all but this is an example of degenerate case so what essentially we are trying to talk about is this particular kind of things CNF so what is a CNF first of all it is a conjunction of disjunctions and all this is conjunctions of disjunctions and all so D1 D2 D3 etc so now each one is called as the degenerated case of this one is PR not P or Q because it follows all the three rules of the one which we have talked about so now some kind of convention which we follow for coming up with CNF and DNF of a given well formed formula so the conventions are like this no redundant disjuncts occur in any conjunct in a CNF conjunctive normal form so that means if you come across a formula such as PR P or PR P something like that it is simply represented as P however number of times you it occurs in all in the formula three times four times it does not matter but it is treated as P only because you are not saying anything more than P here so redundant disjuncts will vanish in all so now we will just write it as PR P so it is like in the Boolean logic we have one R1 that is 1 plus 1 is 1 only so the same thing which is followed here also so the only thing which we are talking about here is that instead of disjunction we are talk instead of plus operation we are talking about disjunction so that is the first thing the convention which we follow and the second thing is no redundant conjuncts occur in the conjunct in normal forms just like P and not Q and suppose if you have another P and not Q it is reduced to P and not Q even if it occurs 100 times in all but it is still treated as P and not Q the third thing is that within disjunctions that is D1 D2 D3 etc and all variables appear in the alphabetical order if it does not appear in the alphabetical order you need to use distribution laws or associative law and make it in some particular kind of order so now particular order with the only unnegated occurrence of a variable preceding the negated occurrence of a variable it is like so that means unnegated ones should come first whereas the negated ones come later so now for example observe this particular kind of thing not PR R or not Q or Q or P and etc and all so here negative forms came first and then unnegated form came later etc and all we need to reorder this particular kind of thing suppose if you follow this particular kind of convention like ABC etc and some order we will follow so now this will change to so the first disjunct it is not PR R or not Q or Q or P changes to PR Q or not Q or not Q or R and so we are reshuffling this particular kind of thing and then we are putting it in some kind of order after P Q follows Q or follows only for our convenience you know so these are the three thing conventions which we follow so that you know we will avoid a lot of redundancy etc so now how to convert a given well form formula into its corresponding conjunctive normal form in the same way you can apply the same procedure to reduce the given well form formula into disjunctive normal forms where a disjunctive normal form is disjunctions of conjunctions so first what you need to do is we need to use some definitions to remove the occurrences of implication double implication and R so that you know it will be converted into conjunctions you know so then you use De Morgan's laws and the law of double negation that is not not P is equivalent to P to remove all the negations which are there outside the brackets and all for example if we have not of X and Y it will become not X or not Y the same way not of X or Y will become not X and not so now then you have to use the double negation to ensure that no variables preceded by more than one negation sign that we need to ensure and then of course we need to follow some other kinds of rules such as distributive law PRQ or R is nothing but PRQ and PRQ or PRR actually this should be PRQ and R or you can use P and Q or R if it is PRQ and R it will become PRQ and PRR if it is P and Q or R it will become P and Q or PR P and R P and Q or P and R so these are the distribution laws which we apply to this one so that in given well form formula is converted into appropriate conjunctive normal form. So now let us try to convert this particular kind of formula into the conjunctive normal form so P ? Q ? P is what is given so now first thing we need to do is to eliminate that implication how do we eliminate this implication by using a simple definition of material implication that is P ? Q is nothing but not PRQ so that is a reason why the first conditional P ? Q ? P will become not PR again Q ? P is written as not Q or P that is the definition that we have used in the first step. So now this further simplifies to you can use associative law and then you can regroup or reorder it in such a way that it will become PR not PR not Q so this is kind of case which we have talked about earlier that is a degenerated case of the special case of conjunctive normal form there is a degenerate case where you have all the rules three rules are applied on this particular kind of thing. So now this PR not PR not Q is finally considered to be in that particular kind of conjunctive normal form so this as such in it appears to be not to be in the conjunctive normal form but you can write it in this way that PR not PR not Q and the same thing which you can write it PR not PR not Q so now this is nothing but this is like C1 and C2 where each C1 is nothing but a degenerate. So in this case PR not PR not Q because PR not P is already true and all this because we know that it is a tautology so anything whether Q is false or Q is true that is going to be true only so it is in that sense it is considered to be a valid formula also of course this is indeed a valid kind of formula in the preposition logic because it is an instance of paradox of material implication in the last class we showed that using natural deduction we showed that P ? Q ? P is a theorem or it is considered to be a valid formula. So what essentially we did here is that given a formula we reduce this we eliminated this implication by using De Morgan's laws and the definition of material implication etc. and all and we reduce the formula into a formula which consists of only disjunction or negation so these are the only things which you will come across in the final formula. So some more examples which we take into consideration how to reduce a given formula into conjunctive normal form so conjunctive normal form is conjunctions of disjunction each conjunct these combination of different formulas which are in the form of disjunction. So what is given to us is P ? Q ? Q is what is given to us so now what essentially one needs to do is in this formula we have implication and conjunction is there so now we need to eliminate this implication how do you eliminate this implication use the definition so that is a ? B is nothing but not here so this is P and P ? Q that is taken as X I mean that means not of X or Y Y is same as Q here so that is what is the case in the what is happening in the second case so now once you use the definition so now if there is any negation outside the bracket and all you need to push it inside by using demagans laws so now in the second step the negation is outside the bracket and now we push it inside negation of conjunction will become disjunction negation of a formula P you will become not P ? P ? ? ? P R Q that is the first one R Q is the case this is not still in conjunctive normal form that means not conjunctions of disjunction but you have to reduce it further using further applying demagans rules and whenever you have ? ? P X I now use the notion of double negation so the first formula ? P R ? ? P R Q will become now P ? Q and Q remains the same so now it is not not P now use distributive law X R Y and Z will become X R Y and X R Z that is what we have done here not PR P and not PR not Q or Q and Q is remains the same so now once you have done this thing what you need to do is you have to reorder the formulas in such a way that there is some order which is maintained the convention we need to follow P preceded by Q etc so now this Q ? PR P and ? PR ? Q and the Q if you use distributive law it becomes Q R ? PR P and then Q R ? PR ? Q so now you drop the brackets and all so now this again you can use some kind of reordering it will become you can use associative law here it will become PR ? P and Q so you have to ensure that unnegative term comes first followed by that negation of its term and then followed by that anything goes in the same way in this case also ? PR Q R ? Q that means first unnegative term came first and then the negated term comes later so this is what we have reduced a given formula P and P ? Q ? Q is reduced to this one so what is so great about reducing this particular kind of formulas so now observe this particular kind of formula this is like a X and Y kind of thing so X and Y when it is going to be true when both conjuncts are true so now in this case if you observe the first conjunct that is that involves PR ? PR Q PR ? P is obviously total so anything whether Q is false or Q is true that is going to be true only that means the first conjunct is automatically first disjunct is automatically true so now coming back to the second one here also we have a literal Q and its negation is there so that means is always true now irrespective of whether not P which occurs there in this formula second formula whether it is true or false that is going to be true only so that means the entire conjunct C1 and C D1 and D2 both are true so that is why the given formula is true that means a given formula is a tautology so how this happened and all especially because each disjunct has a literal and its negation occurs in this particular kind of disjunct in a conjunctive normal form so that makes this particular kind of formula true that means tautology and hence the given formula is a valid kind of formula so now a well form formula is in CNF especially when it has this form it is a conjunctions of disjunctions first you need to write talk about the connective that occurs there so that is a conjunction of what conjunction of disjunctions that is why D1 and D2 to DN where n is greater than 1 whereas a is considered to be a well form formula which is in disjunctive normal form the other way around the dual of this one if it is a disjunctions of conjunctions so so how you need to read this particular kind of formula sometimes it will be confusing for us in the case of conjunctive normal forms so what appears here is it is a conjunctions of disjunctions so now DNF is disjunctions of conjunctions C3 say conjunctive normal form is going to be true when all the disjuncts in that occurs in that conjunctive normal form is going to be true so when this happens especially when you have a formula like a literal and its negation occurs in this particular kind of formula then each disjunct will become true then that makes the conjunctive normal form true and hence valid in the same way DNF in the case of DNF each conjunct has to be suppose if each conjunct is false then that makes the whole formula unsatisfiable enough so for showing the validity you will use CNF for showing the unsatisfability you will be using the DNF so DNF disjunctive normal forms are nothing but disjunctions of conjunctions first you need to read the sign here this is a disjunction of what the conjunctions in the same in the conjunctive normal form first you write the conjunction here conjunctions of what disjunctions D1 D2 D3 etc so now there are some remarks which needs to be discussed here so a well form formula is in disjunctive normal form if it is an unnegated conjunction of variables that means there is no formula which occurs like this that it is not of P or Q etc and all whereas individual letters will have negation and you can have P or you can have not P also so either negated are unnegated the procedure for finding DNF is exactly same as that of CNF that means you remove the implication sign first by using the definition and then you push the negation inside and use De Morgan's laws and distributive laws etc then ultimately you will have in the case of DNF's you will have disjunctions of conjunctions in all and if every conjunction in DNF contains some variables and negated and unnegated then the given well form formula is said to be inconsistent so that means in any formula like this thing they are all conjunctions and all so like X1 X2 X3 etc so they are all conjunctions so now it so happened that you have X1 and not X2 in any one of this in all this conjunction also so that means this this makes this formula this makes this C2 falls in the same way C1 is also having that particular kind of thing a conjunction and a literal and its negation occurs in each conjunct in all that makes every conjunct falls and since since all the PRQ is going to be falls when both are falls that means every conjunct is falls then this is going to be inconsistent or unsatisfiable kind of formula. So now the question comes to us is there may be some kind of perfect normal forms a CNF a conjunctive normal form is said to be a perfect normal form especially if every conjunct contains as a disjunct every propositional variable negated and unnegated that occurs in the whole well form formula for example if you take example will serve a purpose here so PRR and PR0Q so this is an imperfect kind of normal form so what one needs to do here is that we need to find a perfect normal form corresponding to this imperfect normal form so how do we find it there is a procedure for finding it so if in any conjunctive normal form D is a conjunct that means D is a conjunct which does not contain some prepositional variables pK then D is represented as DR pK and DR pK in if in any disjunctive normal form C is a disjunct which does not contain some prepositional variable pK so how do we reduce an imperfect normal form into a perfect normal form so the first one is this thing PRR and PR0Q for example let us take into consideration this thing so now this formula PRQ and NOTQ is written in this sense PRQ or PR0Q so now this is p is same as dropping the inconsistent disjuncts and all what is the inconsistent disjunct Q and NOTQ if you drop that particular kind of the Q and NOTQ is obviously false and all and PR false leads to P only so now P leads to P is nothing but PRQ and PR0Q dropping the inconsistent disjuncts so now PRQ and PR0Q that is the first one and then the second one PRQ and PR0Q are all so this is another kind of thing in perfect form that is translated into this PRQ or all using the distributive law and you can write it in this way PR0Q or all so now again this reduces to P PRQ or PR0Q or all so now P is equivalent to PRR and R so it is R and R is reduced to F so PRF is P only now PRR and PR0R so now PR0Q or R and PR0Q or NOTR so now we have after doing all these things we have something called a perfect normal form so what is that we have done here is that we have reduced the given formula into corresponding kind of perfect forms in perfect forms have been translated into perfect kind of forms so something is in imperfect conditional in perfect well-formed formula can be reduced to perfect kind of formula PRR and PR0 so now so this can be written as this one so the first thing P it can be written as this one using disjunctive normal form this is nothing but PRQ now this we are trying to translate it into perfect kind of form and all so now PR you are adding this particular kind of thing Q and NOTQ this is nothing but PRQ and PR0Q so this is using distributive property PRRQ and PR0Q and all this is first is PRQ and PR0Q so now this reduces to just P so now what we mean by P is this one PRQ and PRQ so what exactly we are doing here is is that since it is not in perfect normal form and all conjunctive normal form where this is D1 and D2 so we are trying to reduce it into perfect kind of normal form so now using this particular kind of equivalence we substitute it in this particular kind of thing so now this P is substituted here so now what will happen PRQ and PR0Q RR so this is the first disjunctive so now this will become this one this is PRQ RR and this is and PR0Q RR so this is the second disjunctive so how did we get this particular kind of thing first we included this particular kind of thing PRQ and NOTQ this is same as P only we are not just doing anything new way so this is nothing but PRQ and PR0Q so now the same thing whatever we generated here this we substituted in this one so this is what we have written here our distinction are so now this is what we get so now the same thing you substitute it into the second one you will get so you will generate the other one PR so now what you will do here is so now PRQ RR and PR0Q RR in the same way you can write P so now P as so this is same as P only so now this can be written as PRR so now you substitute this P value whatever we got it here into the second one so now this becomes for P you substitute this one PRR and PR0R this is the first thing the whole thing which we got it for P is nothing but this one the whole thing are not so now this is as it is in all and so now this will become PRR R0R and PRR PRR PR0R so PRR R0R this is the first thing and then you have to use this PR0R so now this seems to be in the perfect kind of normal form so initially we have just PRR and PR0Q so now you are if you are trying to write it in the normal form and all we need to ensure that all the disjuncts that occur here will have all these letters and all whereas in this case Q is missing here whereas in this case R is missing so this is not in a this is not in a perfect form and all a perfect form includes all the disjunction of whatever disjunct occurs in the given formula so for that reason what we have done is simply this that we have just taken one formula into consideration which is an equivalent of this particular kind of thing P for example if you have X R P and not P is same as X only because P and not P is obviously false and all X are something which is false is X only it is like X plus 0 is X so now essentially what we have done here is that this is an imperfect normal form in the sense that the disjunct D1 does not consist of Q the variable Q here in the same way in the case of disjunct 2 we do not have R here so we need to transform this thing in such a way that if you want to have it in a perfect normal form then we need to make use of this particular kinds of transformations and ultimately you will see here this is the final translated version of the same formula PRR and PR not Q that is like this PR Q or R and PR not Q or R and followed by that we have PRR not R and PR not R and not R but now here so can we say that this particular kind of formula is valid or not so for validity what needs to be the case is that all conjuncts have to be true so that means each disjunct have to be true to make this formula true so we can talk about such kind of thing for only these two formulas PR not Q or R at least a literal its negation is there not its negation some negated formula or negated formula is there but when it comes to this particular thing PR not PR Q or R if this any disjunct will become false it is going to make the whole conjunct falls had it been the case that if it has at least one not R here and then if you have some P and not P or something like that is here then it makes all the conjuncts all the disjuncts true and it makes the conjunctive normal formula true. So this is the way in which one can convert a an imperfect normal form that is PR R and PR not Q into this particular kind of format and all where we have not changed the entire structure of the formula and all but we made use of some definitions and all like P is nothing but PR Q and not Q and in addition to this particular kind of thing definition and pushing the implication inside etc and all so what we are trying to do is we are making use of some laws of logic also so one is the first one is law of identity P and something which is a tautology is obviously P only or identity law is not identity law but PR the contradiction is obviously P only it is like in the Boolean logic PR 0 P plus 0 is P only and the domination law if you add anything any contradiction to a given formula contradiction dominates and all so that is given statement is going to be contradiction and the domination law with respect to disjunction if you add any tautology to a disjunction disjunct that is going to be true only the tautology dominates over here in the first case contradiction dominates there an idempotent laws P and P is nothing but P PR P is also P only and commutative laws as we know P and Q is nothing but Q and P which happens which occurs for even disjunction also PR Q is nothing but Q or P on associative law P and Q and R or PR Q or R the same as PR now brackets Q or X so these are some things which we already know and the De Morgan's laws are as usual not of P and Q will become not P and not Q and absorption law is simply seems to be little bit surprising for us something which is PR P and Q will become P so now some more examples how to convert not of P implies Q or R implies P into its corresponding conjunctive disjunctive normal forms the first step what you will do is you write we remove the implication so then P ? Q will become not PR Q that is followed by negation of that particular kind of formula or R ? P is written as ? R P so now in the first disjunct first disjunct you have to push this negation inside then it will become P and ? Q and R ? R P remains the same so just let me work it on the board and all so that you will get to know this thing in a better way just one formula which I work on it the rest of the formulas one can transform it into the appropriate DNF so the formula that we are trying to reduce is this thing not of P ? Q implies not R P so now we are trying to reduce it into CN the CNF is like this dis conjunctions of disjunctions D3 etc so now the first thing which you do is you have to eliminate this implication and all so now how do we eliminate this thing so we have a definition P ? Q by definition is nothing but not P R Q so this is what you write it here and then you remove it in the next step you know so now this is not R P so now in the second step you remove this particular kind of implication so now this will become the whole thing is treated as X and this as Y X ? Y is nothing but not X or Y so now we substitute for X this is not of not of not P Q or not R P so now this is now not not of for this one will become this one not P R Q or not R P so now you can use distributive law and all then you will get not P R so not P R Q or not R R not P R Q R not of P ? Q and then R P not of not not of not P ? Q yes here thing is that not of not P R Q it will become and negation of a disjunction will become conjunction of this is as it is so now this needs to be changed like this not P and Q or not R R P so now this step will become this thing so now you apply this thing not P and Q P and Q and not R R not P and Q and not P and P that happening so now it is not P and Q or so now this will become not P R so now this this thing we take into consideration and you remain keep it like this only not R or P and Q and is the one which is coming here so now Q R not R P so now this is in the disjunctive form D1 and this is D2 now it is a conjunction of D1 and D2 that is why it is called as conjunctions of disjunction that is why CNF conjunctive normal form so now whether or not this formula is satisfiable or not this is not in not in a perfect form and all now we need to reorder it and then you can write it like this so not P R P now this comes first unnegated term comes first followed by that you have a negative term and then not R and in the same way Q R not R you write it in this way P R Q R not so now in this case you have P R not P which is already true is a tautology so now irrespective of whether not R is true or false is going to be T so now even in this case also P R Q R not R a negative term and the unnegated term is there here so that means it ensures us that whether or not P is false or P is true this whole formula is going to be T on so that means the whole formula is going to be true that means this formula is going to be valid form so in that sense we can talk about validity of a given formula especially in the form in the case of CNF so in this class what we have seen is that a given well form formula we reduce it into its corresponding conjunctive or disjunctive normal form conjunctive normal form is nothing but it is a conjunction of disjunctions and disjunctive normal form is disjunctions of conjunctions so the basic idea of reducing this conjunctive given formula into conjunctive and disjunctive normal forms is that the movement you see the structure of this formulas you can immediately come to know that using the semantics of disjunction semantics of conjunction you will come to know especially in the case of CNF if each D1 is true then the conjunctive normal form is going to be true in the case of DNF's it is a disjunctions of conjunctions if each conjunction is false then you can show that a given formula is unsatisfiable and so this is the advantage of reducing the given formulas into given well form formulas into its corresponding disjunctive or conjunctive normal forms and sometimes it might appear that these formulas will not appear in a perfect form like all the variables will not occur in a given formula so we need to do a little bit of transformation without disturbing the truth value of it just like you know for example you can replace P with P or Q and Q Q and not Q so that is P or Q and P and not Q and all. So like this you know one can transform in perfect normal forms into perfect normal forms one example we have seen in this particular kind of class in the next class what you are going to see is that we will be making use of this conjunctive and disjunctive normal forms especially in solving some kind of puzzles as well as will be making use of this conjunctive normal forms and disjunctive normal forms in analyzing some simple switching digital circuits. So we will talk about analysis of as an application of this conjunctive and disjunctive normal forms we will be talking about the analysis of simple digital switching circuits.


Welcome back to the lecture series on pattern recognition. In the last class we have been discussing about decision boundaries, decision regions, discriminant functions following the discussion on normal distribution. And for the task of pattern classification we took a very simple example of a minimum distance classifier or nearest neighbor classifier in which we had simply taking the distance of the sample x from the mean without worrying about the class priors and the class distribution functions. And we also discussed that this is not a correct method to estimate or to perform the task of classification because you will get good amount of errors in this process. And there are lots of examples to also show that which we will worry about it later. So to incorporate those we will bring in the classic Bayes theorem. So let us look at the slide for the classic Bayes theorem. So Bayes theorem the Bayes decision rule has been discussed in an earlier class or the Bayes theorem is given by the expression as you see on the top. So we are trying to classify a sample x this is what the capital X indicates a feature vector with an arrow and the W i belongs to indicates the class i. So what Bayes rule or the Bayes theorem says that it requires 3 inputs. The 3 terms which you see on the right hand side 1, 2 and 3 are all given here and this is the output. What are the individual terms which are here? Now there are different books which will actually use these terms in different ways and I am going to use as much of the common terms which are used to describe these corresponding terms within the Bayes theorem which is used for later on used for class assignment. So what are they? This is what you get p of x the denominator term is basically called the unconditional density function or evidence. Some books will casually call it evidence but I think we will stick to the word called unconditional density function or unconditional probability. It is the probability distribution of a feature X in the entire population that is what it means. If you are taking color as a feature to discriminate between flowers and fruit I want to find out how much of red flowers are there. It is not a question of trying to identify whether a flower is a tulip or a separate category of a flower or the classical Rajniganda whatever you want to whatever color it has or the rose. How many flowers have a color red? So if color red or if color is a feature the redness of the color for that particular sample that can be calculated from a huge set of samples. If you want to take some other example let us say the height of a particular person. You take a group of individuals. You may try to categorize individuals based on different patterns such as say language, dress, color whatever the case may be, food habits but let us say I want to take height as a feature and use it for a classification whatever the case may be but height is a feature. So if you take a group of 100 individuals you would like to find out how there are how many people with the height let us say exactly 5 feet or 5 feet 6 inches that will indicate the unconditional density function because you are sampling the data without worrying about the class under which category the data falls that is p of x. p of omega i or w i as a symbol indicates you can use it as a w i as the prior probability that random sample is a member of the class c i look at the index is the same. The prior probability that a random sample is a member of the class c i this is sitting in the numerator of the expression here of the base what is called as the casually the class prior. Let us take an example of a 2 class problem discriminating between apples and oranges. I give you a bag of fruits and there are 10 fruits 6 of them are apples and 4 of them are oranges I repeat there are 10 samples 6 of them are apples 4 of them are oranges. So I can compute p of w 1 for apples p of w 2 for oranges very simple answer what should be p of w for apples the prior for apples it will be 6 by 10 or whatever it is 0.6 and the other will be of course 1 minus that will be 0.4 very simple so that is an example of class prior you need to find out because actually what happens is in the case of fruit classification the fruits are sometimes seasonal so it is good to find out for a particular season what is there certain type of seasonal fruit which is available in large quantities say mango in summer or could be jackfruit or some other fruit which could be apples could be typically more or cheaper let us say at least in winter oranges also are seasonal fruit but of course these days we are having almost fruits all through this is an except mangoes probably not available all that okay. Let us go back the other term which is the most significant term in the numerator so we have discussed two of these the class prior and unconditional probability the term here which is the most significant one is called the class conditional probability or likelihood given a particular class wi how many times as this feature x occurs okay it is the likelihood of obtaining feature value x given that the sample is from a class wi it is equal to the number of times or occurrences of x if it belongs to class wi that means if you pick up apples as a class how many fruits will you have which are the color red well that probability could be higher you would like to compute how many red fruits will you have have if you are given only oranges now you know how many times an orange can become red or some other color let us say which is not orange which is a black or green well oranges if it can be green but let us take some other color red or black so you can compute the likelihood or class conditional probability as it is called for a particular feature given a particular kind so given all these three terms on the right hand side a very simple probabilistic estimate will give you the left hand side quantity which is called the measure condition or posterior probability and this is the probability for any fixture vector x being to be assigned for a class wi so you assign x to a wi if this measured condition probability is higher for a particular class the rule is the same the rule for assignment is the same whichever the corresponding probability posterior probability under base satisfies the maximum likelihood you will assign it but how to compute the likelihood that is done using this formula so on the numerator you have just remember class conditional probability you have a class prior and you have an unconditional can you repeat this with me on the numerator you have class conditional likelihood then second this one is class prior denominator is unconditional probability using these three you compute the base theorem so assuming you know this formula let us go back to our discussion on decision boundaries and decision regions to be obtained under the base paradigm so now look back you remember base this is the expression which we just discussed now okay now consider a discriminant function gi which we have been talking about in the last class a log of the numerator of the expression numerator of the base expression you see these two terms if I take the log this is what I will get and I am ignoring the denominator what was the denominator here which we are ignoring we discussed that just a few minutes back it is the unconditional probability which we are ignoring and we are taking the two numerator terms and taking a log of that expression and within that what we are bringing in now is this term class conditional probability or casually also called the likelihood correct class conditional probability or likelihood this is the class prior okay this is called the class prior okay and this is called the class conditional probability or likelihood is given by this famous expression we have seen this couple of classes back also in the last class beginning this is the what is this expression multivariate Gaussian density function so that means in a redimensional space this is the distribution look in the expression you have remember there is a vector sign which I have kept consistent here but after some time I may not keep this vector notation I have tried to be consistent as much as possible throughout my slides actually ideally this should also have a vector sign but this is a vector mean vector this is sample vector covariance matrix here normalizing term here so using this expression using this expression if we incorporate inside the log can we write the expression of gi will work this out in the board so what did we have we had the base rule I am just writing the two numerator terms can you just tell them what they are the first is the class conditional probability or likelihood what is that multiplied by class prior correct so that is that is my base so what I am ignoring here is the denominator term and what I am writing now is gi of i of x my first discriminant function in my class okay so what I am writing I am taking a log of this basically okay simple log of so it will be log of this plus log of the class prior very simply and this particular term I am writing here this I am saying will be a let us say this is a multivariate Gaussian density function so this will be root over determinant of sigma i for the corresponding ith class correct so there is a symbol here the subscript i then 2 pi to the power d correct the square root will be part of this correct it will be overall this is the normalizing term basically that is the idea then this is the main thing exponential minus will be there here correct divided by 2 x okay I should be careful here that x is a vector mind you I had this vector sign inside my slides for the sake of convenience I am ignoring that but if you are copying you can put the vector sign here because both of these are vectors then you have the covariance matrix inverse of the covariance matrix and then x minus mu i correct there will be i here both okay so what is this this is the this is the overall so if you substitute that take the log so this there will be a this factor which will come out plus the log prior at the end the interesting part will be this factor here so if you check it out I think it will be minus d by 2 log 2 pi here into this can you tell me what I will get here minus 1 by 2 log determinant of okay and the log in the exponential will be cancelling out so there will be a minus sign here so it will be a minus half by this one no sorry x minus mu i transpose covariance inverse let keep getting used to this term plus correct so this is what you get this is your gi this is your gi okay to be very precise let us correct it here this is x just take out if there are any mistakes of notations here okay now if you look at this this is what we get in the slide as well let us look back in the expression to see that whatever we have derived as you see here it is the same correct look at the expression at the bottom as given here and the one which is now in the slide at the bottom here the only thing in the slide is that I have brought in this covariance term at the beginning the rest of it is basically the same organization slightly different man that is all if you look into this expression here these can be visualized to be constant terms which are outside they are not function of x this is the one which only varies if you change x of course if you change i that means go from one particular class to the other there will be another term which could be also changing as well as the class prior unless the class prior themselves are constants or uniform across different classes that means you have same number of arranges as the number of apples when you go for go to the market of the stores grocery if you want to purchase two types of flowers you have as your number of rows as number of tulips or something else that is okay in that case class priors also could be same or different covariance matrix also could be same or different across different classes but given a particular class i when you change the sample of the instance which is being tested the only term which is varying is this one and this is the one actually which gives you the distance d the distance d which we have been talking about since about one or two classes back the distance d distance of a sample from the class mean nearest neighborhood classifier is actually a special case of this expression with the covariance term suppressed you only have these two you get that d which we talked about and we will see that again as a special case so this is the distance plus these are some constant terms which do not vary with x but if you go from one particular class to the other yes there will be some changes which will be taking place and we will discuss that but what may happen also is that the covariance term in the class priors also could be same as a special case across classes so in fact the concentration henceforth will be mainly on this particular term let us go back to the slide so to recapitulate what we have just done now that within the purview or using the Bayes theorem incorporating the multivariate Gaussian function as the class conditional probability for a particular class we have got a discriminant function expression for gi which contains the class priors the class pdf the class pdf in turn contain the covariance matrix which is very important which we have just discussed about now in terms of distance about a couple of classes back and of course a few constant terms which is again based on the dimensionality of the problem and the covariance matrix the determinant of the covariance matrix and if you look back into the expression in the slide as given here it is the term which on the left hand side of this expression for gi of i of x is the one which dictates your distance from or which actually dictates your classification the job of classification and also it is giving a distance measure because all of these the rest of it are not functions of x so when you vary x within this expression it is this term which is changing and actually it gives a value of distance many cases may arise due to the varying nature of the covariance matrix sigma it could be diagonal with equal and unequal elements there may be off diagonal terms positive or negative we have already seen some of these variations when we are looking at the isocontour plots in 2d for a particular Gaussian example you remember two classes back we had talked about asymmetric Gaussians in fact in the last class we have talked about asymmetric Gaussians oriented Gaussians what did it depend on it was the fact that if we had off diagonal term 0 or not whether the diagonal terms one was equal to the other or not these are all the factors which dictated or whether the Gaussian was asymmetric or not whether it was oriented or not the same thing is going to happen with the diagonal off diagonal terms of the covariance matrix is going to dictate how this particular expression in fact it is an expression of a distance already which we have got using a discriminant functions and this will give rise to dr's which in turn will give rise to db's let us look at some special cases of this particular function remember this expression of gi which have just derived it in the class today let the discriminant function for the ith class be the same gi of x so assuming class priors are same for two different classes i or j in fact if it is a two class problem but if it is arbitrary c different classes all the class priors are same that is what we are assuming then the only term which remains actually is this probability function and then based on this we can write a simple expression like this which you have done this is the term and we can write that this constant term here which you have actually broken into two parts or broken into two parts in the last slide is given by some constant q why it is constant because not a function of x anymore and this is a constant multiplied by a distance vector well distance is usually a scalar okay distance usually a scalar okay so this is a scalar quantity mind you di square which is a norm of a distance what is basically taken and this k is simple 1 by 2 it is a simple constant 1 by 2 this factor and this di is basically given by this expression which is going to be the most important part of the discussion for the rest of this part of the course on distance measures with multivariate Gaussian functions for classification for decision boundary estimation. So the classification is now going to be a because why this term is going to be only there for gi this is a constant the classification is now influenced by the square of distance in hyper dimension space of x from mi weighted by the inverse of the covariance matrix and we are going to examine this in little bit more detail in the rest of the class today and henceforth and this term has already been introduced earlier may not be explicitly this is actually the Mahalanabes distance the distance from x to mi y in featured space weighted by the covariance matrix so it is a weighted distance if you had just taken x minus mi which was the nearest neighbor classifier or the nearest neighbor classifier or minimum distance classifier then you did not pay attention to the class as well as the class PDFs or the class conditional probability density functions in such a case the Mahalanabes distance takes the simple role of a squared Euclidean distance but if it is weighted by the covariance matrix specifically its inverse to be precise then you are talking it as a Mahalanabes distance criteria for a quadratic term okay so look into the slide this is the one which I am going to discuss now henceforth so it is all ball ball down to the covenants matrix and the distance to the mean the rest of the class priors they are relevant in certain examples but we will bring them back when we discuss a few things together little bit later on so for a given x sum for some arbitrary i is equal to m the value of g well it is the same small g I have written in capital here is largest when the square of this distance is the smallest for a class i is equal to m remember if you go back to the previous expression we are talking about a constant minus this k remember there is a minus half for the k so if the d value goes down the g i value will be maximum remember the class assignment rule assign it to the class for with the discriminant function becomes the maximum so maximum value of g i if you look back to the expression the maximum value will be assuming it to be constant if this to be maximum this has to go down the distance has to go down the distance must be minimum for this to be maximized so that is what is written here so g m must be largest where this is the smallest the distance is the smallest for a particular class i is equal to m and for that particular class m assign x to that particular class based on the nearest neighbor rule the simplest case of course is that the covenants matrix is equal to an identity matrix i this criteria of course yields have just been talking in the last few minutes that this yields our equivalent distance norm the nearest neighbor classifier or the minimum distance classifier this is equivalent to just taking the mean of a particular class for which x is the nearest for all mu and the distance function is very simplest there is just substitute an identity matrix here you will have these two terms which actually yield you the square norm of the distance from the in d dimension space of course is here going to be your d square but remember here all this d indicated as a vector but you are taking the square norm so it is basically the scalar quantity here so in all vector notations this is how it can be expanded this is how it can be expanded you can write you can write expression in this particular form and then this g of ix remember this is d square by 2 by this because there is a constant here which I am bringing in here for normalization of normalization and this is the this is very simply substituting this overall by 2 you will get this expression and what how can I write this in terms of this where look at this what term w i 0 which is taking care of this term the w transpose is this multiplied by x it seems I am ignoring this term if I ignore this term I can write this expression as something like this I can do this only when well I would not say x is negligible that is not the basic idea I am ignoring this term which is called a class invariant term it is a class invariant term why because if I keep changing the value of i this term does not change why am I trying to do this remember I am interested in classification discriminant function I have formed now I am trying to find out each discriminant function produces a decision region two discriminant functions for two different values of subscript i and j they will create two different the dr's or decision region there is a decision boundary between this pair i and j we will have a decision boundary between this pair i and j so if you move from class i to j to compute this g of i and look back into the expression what will change with this expression here for the same sample x if you change i to j this is the term which is going to change not this so hence this is called a class invariant term because it does not change between classes you are the same value of x because that is the test sample which you are trying to classify after you have learned the discriminant function you move from class 1 to class 2 to class 3 or class a to class b to class c the first quadratic term which you see here in this expression this does not change across classes this does not have a subscript i it is the other two terms which have so I am ignoring this for the sake of comparing gix across classes across classes then I am having this and I am writing this the only thing which you need to worry about is that I have probably taken g i to be minus of this because the sign has been reversed here okay so we what we are doing here is as if gix will be a constant minus this part which is given here you can see that there is a negative sign which has been introduced here this negative sign has vanished so do not get this flip is not automatic it is the way you have taken the g i because I want to maximize some function by minimizing the distance so this will help you in doing this this is actually called in a very simple sentence I have linearized the discriminant function this will actually give you linear decision boundaries this is also very casually called a correlation detector but that is not our main aim we will say that this is the simplest form of a linear discriminant function the linear discriminant function can also be written actually is this the expression which we had in the last slide if you go back let us see here this is what we had it is a sample x multiplied by a weight vector plus a bias term this is basically a bias term which is big which depends on the class mean I repeat again that this is called the bias term which depends on the class mean and this is the weight vector which also depends on the class mean weight vector depends on the class mean and the bias term everything is depending on the class mean because the covariance term is ignored where did we ignore the covariance term you have taken the covariance matrix equal to identity matrix that is why we could write this in the first place and only which whatever remains is just the mu that is why we are able to get a linear expression now of this so the linear discriminant function for a separable class is given by this which we have seen in the previous slide and the W vector is it d dimensional vector depending upon the dimensionality of the problem which is a vector of weights used for class I and we have the expression of W which is simply the class mean so this is it is a d dimensional vector the same thing which holds good here this function leads to decision boundaries that are hyper planes in higher dimension and I have talked about this in the last class that it is a point in 1d line in 2d planar surfaces in 3d and of course hyper planes in higher dimensions we will examine this linear decision boundary in very great detail and the simplest case of a 3d which is a plane of course in 2d it will become a line it will become a plane passing through the origin and the expression gets to the simple what has been written here is now basically this expression the W matrix is given by the 3 components here do not confuse this W is with the class class levels which we have used probably earlier and the 3 dimensional space so x is the 3 dimensional vector as simple dot product here simple dot product here is what has been written in this particular case this is a plane passing through the origin but of course that is a special case you may have a line also passing through the origin or somewhere in 2d space plane in 3 dimensional space hyper planes in n dimensional space once these weights are learnt is it just simple and it is always okay that I pick up these weights class means assign it to my w i's assign the w bias term w i 0 based on the me or me i transpose and the classic job of the classifier is done well in the case of linearly separable problems where problems can be separable by a linear hyper plane a line in 2d or a plane in 3d this will work we will see that with an example now we will see that with an example now we will take examples from geometry in 2 dimensional space and understand the significance of this linear decision boundary which will lead us later on to an important concept of linear discriminant functions lda linear discriminant analysis or flda a little bit later on not an immediate extrapolation right now because what we will do is we will learn the importance and significance of linear decision boundaries then bring in the which one we have ignored to get this linear decision boundary we have ignored the in the gi expression of gi we have ignored one important factor what was that factor the covariance matrix we will bring in that covariance matrix we will see the significance of that covariance matrix which will might which will in fact bring non linearity into the picture of the decision boundary not only due to the diagonal elements but off diagonal elements and see some examples of those and then again come back to linear decision boundary using the official linear discriminant analysis or flda criteria lda which will lead us to a the popular method of supervised classification called lda a linear or fissures linear discriminant criteria analysis that is going to be overall the discussion hence forth in the next few lectures we will stop here


I'm well aware that everything that I just shared with you all right now are things that you already know. But the question and the determining factor on how well you guys recover is not in knowing these things, it's in doing these things. There are many franchise organizations on this call right now that have seen me speak. I've spoken at your events, I've spoken at your conferences, I've led workshops for you. You've heard this stuff before. And my question to you is how well have you been doing it? See, there's always going to be a gap between what we know and what we do. It's called a performance gap. And the key to us recovering with optimism and poise is making sure that we're doing everything in our power as leaders to close that gap not only for ourselves and for our team but for our franchisees. We need everyone to execute on a daily basis and do the things that we know we're supposed to do. And I'll end with the quote that is hanging up in the Penn State Training Center for the Football Program and that is are the habits you have today on par with the dreams you have for tomorrow.


Stevenson falls 79-75 to Arcadia here at Owens Mills Gymnasium, Glenn Clark with head coach Gary Stewart. Coach, your team came out with a lot of fire to start the second half and they built up a nine-point lead. Why were you unable to sustain that sort of energy and continue to drive that way and put it away in the second half? We're just not good enough right now. We're not making the necessary plays. We gave up six threes the entire game, but we give up two threes and back-to-back possessions. You know, that hurt us. We're doing the same thing over and over again and we're getting the same result. It's frustrating. I know the guys are frustrated coaching staff as well. We'll keep hammering at it, but the effort's there. We just got to clean up some things. It seemed like you were going with some different combinations of players in the second half and different groups of guys out there. Were you trying to find something? What was the strategy with that? I saw Kobe was out there for maybe a little bit longer than typical and a couple of guys that were out there, maybe that were a little bit unexpected. Well, we were going through some things with flu and injury, so we had to extend some guys a little bit longer. Then some match-ups. We put Jordan in there because he's one of our better interior defenders and we're having trouble guarding on the low block. We got a little bit of foul trouble there, so that necessitated a move to Jordan. Ultimately, you mentioned the effort. Look, these have been a lot of close games this season that you've been on the wrong end of. What is it ultimately going to take for your team? Presumably, they're going to continue to play close games as you go on to come out on the positive end of some of these games in the late stretch. Well, it's very simple. It's an easy thing.  You've got to come out on the other side of both of those, and tonight we didn't. Coach, I know it's another tough week ahead starting Wednesday night. You had the Lebanon Valley. What are you expecting? Yeah, I'm not really focused on the opponent. I'm focused on ourselves. We've got to take care of the ball. We've got to rebound the ball better, have to have better defensive communication. So I'm really looking forward to practice on Monday and trying to get better at those things. Coach, I appreciate you coming over and taking a few minutes for us. Thank you. He's Coach Gary Stewart. I'm Glenn Clark for GoMustangSports.com.


Hey guys, I just wanted to give a brief overview of what Deseret Bookshelf Plus app looks like as you Log in to the app. This is kind of what it's going to look like It offers you audiobooks and ebooks that you can access for your mobile device I just want to take you through a few of the many resources you have so this my favorite thing is the audiobook You'll see up on the top There's a little blue headphone that means there's something you can listen to as an audio book And if it has the orange plus in the bottom right, that's something that's available to you So as I'm scrolling through all these books, these are all the books that are available to me Sherry Duke collection tennis shoes among the me-fats power of everyday missionaries to the rescue biography of Thomas S. Monson the Washington hypothesis. Just want to browse through some of these titles There's Brad Wilcox Infinite omen amazed by Grace McSherry Duke. There's tons of John by the way books So it actually speaking of that if we go up here, and we were just to search for John by the way Just look at all these titles that are We go to titles John by the way. These are all his books and again if you see the blue Headphones in the top right. That's an audio book audio book audio book The other ones are e-books But basically every book that he has is available in audio booking e-book format, which is pretty awesome So I know there's a lot of John the by the way fans And the list goes on I can scroll through forever this goes on and on and on If we went back here And did a search on like Gerald Lund. I know there's a lot of people who love Gerald Lund books as well It's you know, it's two titles again Got work in the glory The indented Another work in the glory um brave So are the eagles And there's just on and on and on and again all of these books are available to you for It's 6.99 per month if you have a platinum membership um And in the article I wrote on Mormon light.org Kind of breaks down what the difference is If you go back, let's just go back go let's go through these books Joseph Smith, uh the prophet by Truman Madsen Orphan keeper dragon watch so it has some books that are good for kids as well like youth I'm tennis using on the nefats is a great series power everyday missionaries Washington the hypothesis and the linking the hypothesis by tim ballard who's founder of operation underground Railroad is amazing books here in the force of the lord is another Gerald Lund one Lots of Gerald Lund who's full of doubt As you see i'm just scrolling through the book after book after book after book These are all audio books that you can access now something cool about desert book self plus As you pay the monthly fee and then you can just download a book Download as many as you want you don't need to do one at a time So for example, if I wanted to download this john by the way one I would hit read with plus rich excellent audio books you can read it and it just starts As you can see it starts downloading the first chapter And then it downloads the second chapter so it'll download two chapters for you at a time And then if I wanted to download the third one, I just click on this button on the right This is the selected chapters being downloaded. So I said, okay And it already downloaded the next two for me and then as I play Um I can actually change the speed Because when I drive In my car, I like to listen to it a little quicker You can listen to up to two times speed, which is really fast, but Um, again, just a quick demo. I'll I have other tips of the quote ball I'll sold it to you in a different video within my post But there's just hundreds and hundreds and hundreds over 500 audiobooks at this time And again, you can download I'm again, this is like hanksmith. This is a brand new book Actin oh, this is the one. Sorry Um And look at hanksmith real quick I know a lot of people Enjoy hanksmith Look at hanksmith his new book It just came out It is available here Yeah, be happy his audiobook. You just download that right here. Anyway Um over 500 audiobooks this time over 1500 ebooks personally for me. I just look at it for the audiobooks For the 699 a month. It's Unbelievable Now you can listen to it in your car Listen to by working out You can have your kids listen to it. There's a lot of fun stories for kids So just a brief demo again on desert bookshelf plus app


Hello y'all, what's up guys? This is Chris Pike. My friends call me Bixie. Back in action today, I want to talk to you about YouTube's recent update to their AI or altered content policy. On March 18th they came out with a blog post that goes into detail around this and while the blog post is pretty good, I was using the heartbeat tool the other day and I came across a video by TubeBuddy and I actually found about 16 moments from his video that I thought were really well done and that you guys should definitely check out. So if you're a YouTuber and you're uploading content that has AI in it, that's one thing but there's some weird edge cases out there where if you've got like VFX or maybe you've got a voiceover or even a voice translation and you have to label it, it just gets a little funny here. So let's go into TubeBuddy's video. We're going to look at some of the moments as he explains it pretty good. So here we go. All right, so if you're following along just head over to the website here h.ki or hkey and then when I get there you'll see here we've made a few updates to the UI on the left side or the UX but I'm just going to go ahead and click on my heartbeat and then from there I'm going to click on moments and when I do that I'm going to have a whole bunch of moments here selected for this video here which is the TubeBuddy video. So the first one here is called YouTube update and they're going to discuss well the details of it so let's click it off here. YouTube just dropped another update and it's a little bit of a mixed bag. There's this new rule around using AI in visual effects in videos. Everything from AI voiceovers to using VFX for realistic scenes. On one side this is going to make YouTube a better place but on the flip side this is going to bum out a lot of creators and there it is guys so in a nutshell it looks like YouTube is going a little over the top here and they're going to be labeling everything and I get why they're doing it. Sometimes you want to they want to crack down on people that are doing it illicitly like putting in fake backgrounds to you know do financial videos or you know they're trying to sell you something or you know they're doing fake updates to health videos where you know if you work out once every four years you're going to be you know 300 pounds of solid muscle etc etc so I get it let's go into the next moment here where he talks about some of the details and here you go. On March 18th YouTube launched this new feature in Creator Studio. Now creators have to give their viewers a heads up when realistic content content a viewer could easily mistake for a real person place scene or event is made with altered or synthetic media including generative AI. This is very vague I'm going to state that right here it's very vague. Now when uploading a video to YouTube creators will be asked do any of the following describe your content makes a real person appear to say or do something they didn't say or do altered footage of a real event or place generates a realistic looking scene that didn't actually occur. So there's a big issue there right if you put if you upload a video of you you know out walking and then all of a sudden you take away you know you take the background out and you put a different background in where instead of in New York you're in Rome I mean do you have to label that now even though it's you know clearly you know kind of funny or not intended to deceive people I don't know it's it's going to get weird so anyway so let's go forward here a little bit where he talks about generated labels digitally generated and then this label will appear that says sounds or visuals were significantly edited or digitally generated will appear above your video to inform viewers yes so if you've done any psychology and you're familiar with labeling theory you might be able to see where this is going where when you label something well it doesn't always have the uh good effect you think it might for a less sensitive subjects this label stays in the description but for touchy subjects like health or elections they'll be right there on the video telling folks that the content might not be authentic it should therefore be treated with caution so for touchy subjects like like elections for example financial advice people's trying to sell you financial packages or courses or health where you're doing you know fitness type related things and all the crazy claims people make with the fitness world yes this this makes perfect sense to me let's get forward though we're going to go to minute 45 uh where he talks about vfx and translating this is where it gets a little dodgy you're using ai voices to narrate a factual script or you're translating videos into another language yep now you'll need a label for that and it doesn't stop there if you're tweaking footage adding special effects are using ai to whip up scenes that look super real you're in this boat too crazy so for all of you motion graphics types out there and you cinematographers if you're using green screen footage and stuff like that wow you're gonna have to label your videos i mean i don't know this this seems like it could be going a little bit into the overkill side of things but anyways let's discuss whether or not these should be labeled we've got stuff like a bogus meteorite crash or visual effects to make city locations our illusions look legit do you really need a label for those given these labels are all about alerting viewers about potentially dodgy content it's a little bit puzzling yet youtube kids that not all artificially made content needs the warning so that's one of the good signs it seems like youtube is aware that they may be doing a little bit of overkill here but i mean sometimes you got to throw the baby out with the bathwater i suppose let's keep going a little further here for unrealistic content for this out they've got another checklist for content that they do okay not to disclose this includes generated ai used for productivity like generating scripts content ideas are automatic captions fine that's one thing we all use automatic captions you could even automatic caption our own voices and i mean that's clearly done with ai that's yeah this this could get messy but let's skip let's go forward like i said let's talk about the unrealistic content and see where they land on this side of the debate clearly unrealistic content such as an animation or someone writing a uniform through a fantastical world color adjustment or lighting filters special effects like background blur or vintage effects like come on right like if someone like youtube has movies on on youtube right whole movies right and if there's a funny scene or an animated movie that they got to label the whole movie as ai altered content like come on let's just get it's getting a little crazy here so anyways let's go forward here to uh the content checks that the box does uh and uh yeah here we go well you'll see it's been bugging me there's loads of content out there that technically checks the box for disclosure right but they're not trying to cause harm or intentionally deceive anybody exactly like take these channels big muscles which uses an ai voice for some solid fitness tips and then there's heizek from azerba job his youtube education channel is one of my go-tos for genuinely helpful in-depth videos even though it's powered by an english 11 labs voice yeah so there you go i mean all of these videos are going to get that label i don't know it's going to get a little messy here i'm going to finish this one off though with uh his he discusses that this is the first edition first iteration of this from youtube and they they're looking for feedback and no doubt they're going to change the policy up based on that so here we go skipping forward to 353 first edition i had a chat with a youtube employee earlier this week and they're actually looking for feedback from creators like you and me they want to know what part of this update is right what's wrong what's missing what's confusing i'll pin my personal feedback below the subscribe button drop your thoughts in the comments and i'll make sure that youtube gets the message loud and clear so there you go guys what do you think about this of course i recommend you watch the whole video there it's really really good these are some of the moments that i enjoyed out of it although a lot of this video had a lot of great stuff in it personally i get why youtube's doing it but it seems like they're being a little heavy-handed based on my interpretation so far and the fact that i have to put this in a lot of my videos is really annoying so anyways guys there you go let me know what you think in the comments below thanks for watching


What happened to Mr. Sweari? He took his clothes in the morning. He used to fight with a underground gun here. He told him to get a gun, and he already took a hit. He was beaten up by both masters. He took a trace of the gun.  Today when Sweari fought with him, he was going to war with our army. He also told him to get a gun. వాలులె మాలూరాబీనినినాస్లినారినాలెసాడ్రిమాలెన్లెత్తాపాన్వని. నమారెరియోవని ఇనార్వనిలిబెలినినిపోచేనులినిని instrumental classroom  apparat nai Sahib  получить d dak heirw gost san gandshal thaisar le o lawa ja sok market, ab thayawarab esshaa le ma Laboratory Diagoks. Bal ch적 hi chapta bhibaral yahao nai pan conditions to itha towar sigh Lately, there has been houses filled with statistics made to address that crisis for the young people. It's been four years now, but the fitness is ashame. I should not speak City birth, I order the parents to help посмотрite nether brother in-law's house. In this situation, there is no nuestros numbers, And nobody is worried, The kids don't feel well. And now theьогоt has made the trend, that many infants might ask on their change or coisa like that. Many children do not study or haveっぱl – , hints a insects , Ṣa dhya nabārāṅti Ṣāvṭa vimariya or Ṣa dha nanda vimariya Ṣa uṭo bārāṅti pAdhaṅti, on dha jāya kaptey ho kisāra yān Ṣa dhya Ṣakār yāl hospital Ṣikilwākh yāl bhāvsi yāl yāl Ṣakār Ṣakār yāl Ṣikashī ṭhābārāṅti Ṣakār yāl Ṣakār yāl Ṣakār yāl Ṣakār yāl Ṭhār yāl Fixing more pleasure... People say there are so many problems, director's done good work. बीश गर क्या र� Modya, man Attar इत तटटट तटटट बी और पछके लिज बी दा जंगी का लैजी चाजनोदी