# Artificial Intelligence I

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2 pr(p F L W )=.2*.=.2 where L W stands for Wilma takes logic since what one arbitary student takes has no dependence on what another arbitary student takes. In all cases in probability theory if pr(p) is the probability that P is true, then the probability that P is false is -pr(p), so we can say that pr(p)=-pr( P). If we know in fact that 6 of the 2 students taking the programming course also take the logic course, then we can say that the probability that Fred takes the logic course if we know that he takes the programming course is 6 2 =.3. We write this pr(l F P F )=.3, where in general pr(q P) means the probability of Q when we know that P is true. So from this, if we don t know whether Q is true, but we know the probability of Q being true if P is true, we can calculate the probability of P and Q being true: pr(p Q)=pr(P)*pr(Q P). So pr(p F L F )=.2*.3=.6. If P and Q are independent then pr(q P)=pr(Q). Since is a commutative operator (i.e., we can swap its arguments around without affecting its meaning) pr(p Q)=pr(Q P), so pr(p)*pr(q P)=pr(Q)*pr(P Q). From this, it follows that: pr(p Q)= pr(p)*pr(q P) pr(q) This is known as Bayes rule after the philosopher who discovered it. In our example, it means that the probability that Fred takes the programming course if we know that he takes the logic course is.2*.3. =.6. Typically this rule is used to find the probability of some hypothesis P being true given that we know some evidence Q. Note that since P Q is equivalent to ( P Q), then using the rule for pr( P) above we can obtain pr(p Q) = pr(p)pr(q)-pr(p Q). We can also obtain a value for pr(p Q) in this manner, using the fact that P Q is just another way of writing P Q. A distinction should be made between pr(p Q) and pr(q P), which are two separate things. pr(p Q) is just the probability that P Q holds for some arbitary x, recalling that P Q is always true when P is false. A more general statement of Bayes theorem covers the case where we have some evidence E, and a range of hypotheses, H, H 2,, H n which could be drawn from it. The probability of any particular hypothesis is: pr(e H i )*pr(h i ) pr(h i E) = n (pr(e H k )*pr(h k )) k= The problem with this is it makes the assumption that the statistical data on the relationships of the evidence with the hypotheses are known, and that all relationships between evidence and hypotheses pr(e H k ) are independent of each other. In general this indicates that as the number of factors we are considering increases it becomes increasingly difficult to be able to account for the possible interferences between them.. Fuzzy Logic Fuzzy logic, developed by the computer scientist Lotfi Zadeh, is an alternative more intuitive approach to reasoning with imprecise knowledge. It does not claim to have the mathematical precision of probability theory, but rather to capture common-sense notions. The background to the notion of fuzzy logic is the idea that a truth value may apply partially, and that human reasoning often uses this idea of partial truth. For example, the sentence Fred is young might be taken as definitely true if Fred is 9 years old, definitely false is Fred is 5, but applying partially if Fred is 28. In standard logic we might have a sharp cutoff point, say 3, and we would then say for any person p with age below 3, young(p) is true, but for any person q above 3 young(p) is false. In fuzzy logic a numerical value between and is given to express the degree to which a predicate applies. This can be expressed in graphical form, where the graph on the left represents the classical logic approach (the x-axis represents age, the y-axis degree of truthfulness): AI Notes on reasoning with uncertainty 996. Page 2

3 This represents a situation where anyone under the age of 2 is reckoned as definitely young, anyone over the age of 4 is reckoned as definitely not young, but the predicate is partial or vague for those between the ages of 2 and 4. More precisely, where a predicate p is defined by the set of elements for each of which p(x) is true, in fuzzy logic a predicate p is defined by a set of pairs x,µ p (x), where µ p (x) is the degree of membership of x in the set which defines p. Note the difference between this and the probabilistic reasoning described above. When we said that pr(l F ) was.5 we did not mean that Fred was half taking the logic course, we meant that the probability that Fred was taking the course was.5. If in our example above the age of Fred is 3, if Fred is represented by f, and the fuzzy predicate young by y, then from the graph above µ y (f) is.5, meaning that Fred is indeed half young. Fuzzy logic also captures the idea of qualifiers or hedges to predicates in natural language, such as Fred is very young or Fred is fairly young. These might be represented in fuzzy logic by graphs which move the curbed part to the left and to the right of the scale respectively: "very young" "fairly young" The degree of membership of an item x in the union of two fuzzy sets A and B is the maximum of its degree of membership in A or B, while the degree of membership in the intersection is the minimum. So the truth value of P Q for two sentences in fuzzy logic P and Q is the minimum of the truth values of P and Q, while the truth value of P Q is the maximum of the truth value for P and Q. The truth value for P is the truth value for P subtracted from. So negation is dealt with in a similar way to probabilistic logic, but the connectives are not. As an example, let us consider reasoning about the set of subjects in a hypothetical Philosophy department. This set, called Ω, is the frame of discernment (set of all possible values x for an attribute). In our example, this set is: {Epistemology, Methodology, Aesthetics, Metaphysics, Logic, Politics, Theology} Suppose we have a predicate Interesting, representing how interesting each subject is. In classical logic we would simply gave to divide the subjects up into those that are interesting and those that are not. If we suppose that Aesthetics, Politics and Theology are interesting, we could represent the predicate Interesting by the set {Aesthetics, Politics, Theology} However, the reality is that we will not have a straightforward division into interesting and noninteresting subjects, but rather rate them on a scale of interestingness (note also that, of course, what AI Notes on reasoning with uncertainty 996. Page 3

4 one person finds interesting another may not, but for the sake of space we shall not explore this additional factor). Rather than impose an arbitary cut-off point on this scale, fuzzy logic would rate the subjects on a to scale. So perhaps Aesthetics has an interest factor of.9, Politics an interest factor of.7, Theology an interest factor of.8, Metaphysics and Methodology each an interest factor of.5, Logic an interest factor of.3, and Epistemology an interest factor of.. We can represent this by the fuzzy set: {<Epistemology,.>, <Methodology,.5>, <Aesthetics,.9>, <Metaphysics,.5>, <Logic,.3>, <Politics,.7>, <Theology,.8>} which we shall call I. The function µ I gives the interest factor of any particular subject, so µ I (Epistemology) is., µ I (Methodology) is.5, and so on. The cutoff point for the classical logic version was between.5 and.7. In fact there is a name for this: if A is a fuzzy set in Ω then the α-cut A α is defined by A α = {ω Ω µ A (ω) α} The set we suggested for the classical logic set for Interesting above might perhaps be the I.6 cut. Similarly, I.4 is {Methodology, Aesthetics, Metaphysics, Politics, Theology} (note that though we are restricting values of µ I (x) to single decimal places for convenience, this is not essential). Dealing with negation, Interesting(x) is µ I (x), so Interesting(x) is -µ I (x). For example, Interesting(Politics) is.3. Indeed, we can define a set I representing the fuzzy set of uninteresting things simply by replacing all the µ I (x) in I with µ I (x): {<Epistemology,.9>, <Methodology,.5>, <Aesthetics,.>, <Metaphysics,.5>, <Logic,.7>, <Politics,.3>, <Theology,.2>} Now let us suppose there is another fuzzy set, H, represent how hard each subject is: {<Epistemology,.7>, <Methodology,.6>, <Aesthetics,.4>, <Metaphysics,.9>, <Logic,.5>, <Politics,.2>, <Theology,.8>} Applying the rules of fuzzy set theory, the set H I is: {<Epistemology,.>, <Methodology,.5>, <Aesthetics,.4>, <Metaphysics,.5>, <Logic,.3>, <Politics,.2>, <Theology,.8>} while the set H I is: {<Epistemology,.7>, <Methodology,.6>, <Aesthetics,.9>, <Metaphysics,.9>, <Logic,.5>, <Politics,.7>, <Theology,.8>} So we now have a measure of the degree to which a subject fits into the class of Hard and interesting things represented by the set H I, and a measure of the extent to which it falls into the class of Hard or interesting things represented by the set H I. Note that whereas in classical logic, A A is always false, so the set representing A A is always the empty set, this is not so in fuzzy logic. The set I I, representing the class of things which are both interesting and not interesting is: {<Epistemology,.>, <Methodology,.5>, <Aesthetics,.>, <Metaphysics,.5>, <Logic,.3>, <Politics,.3>, <Theology,.2>} so it can be seen that the subjects which have the highest degree of membership of this class are those that are in the middle of the interestingnot-interesting scale. Clearly, the maximum degree to which any item can have a membership of the set A A is.5. Most of the expected properties of a boolean algebra apply, for example, de Morgan s rules: (A B) is equal to A B. We could define a symbol where as in classical logic A B is equivalent to A B, so for example, the set representing Hard Interesting is: {<Epistemology,.3>, <Methodology,.5>, <Aesthetics,.9>, <Metaphysics,.5>, <Logic,.5>, <Politics,.8>, <Theology,.8>} which could be seen as a measure to which the various subjects fit the notion that the hard subjects are the interesting one. In fact classical predicate logic could be considered just the special case of fuzzy logic where µ(ω) is restricted to be or. AI Notes on reasoning with uncertainty 996. Page 4

5 Recalling the suggestion that xa(x) might be considered as equivalent to A(n ) A(n 2 ) A(n m ) for n n m ranging over the objects in the universe, xs for S a sentence in fuzzy logic is simply the minimum µ value for any pair in the set representing S, while similarly since xa(x) was considered as representing A(n ) A(n 2 ) A(n m ), has the value of the maximum µ value for any pair in the set representing S. So xinteresting(x) has the value. and xinteresting(x) has the value.9. Truth Maintenance Truth Maintenance Systems (TMSs) are methods of keeping track of reasoning that may change with time as new items of knowledge are added. They are by their nature non-monotonic, as they are based on the idea that some facts may only be assumptions which are believed until disproved by further knowledge being learnt. Truth Maintenance Systems keep a record showing which items of knowledge are currently believed or disbelieved along with links to other items of knowledge on which this belief or disbelief is based. Because of this, if a new fact is learnt or an assumption changed the ripple effect it has on other beliefs based on the previous assumptions can be managed by tracing along the links. Although the name Truth Maintenance System has stuck as it was used to describe early versions of this sort of system, a better name that has been proposed and is also used is Reason Maintenance System, as these systems are more about belief and disbelief and managing the reaosn for belief or disbelief than about truthhood or falsehood. Truth maintenance systems are another form of knowledge representation which is best visualised in terms of graphs. The idea is that each item of knowledge which a system may believe is represented by a node in the graph. These nodes are labelled either in (for a node which represents something which is currently believed) or out (representing something which is not currently believed). Note the distinction between in and out representing beliefs, and true and false representing absolute values. There are several forms of truth maintenance system. For illustration we shall start off with a simple one. The links in the graph are directed and represent justifications, and are of two sorts, positive and negative. A positive link from node A to node B represents the idea that believing in A is a reason for believing in B, or A is a justification for believing in B. It is equivalent to the logic A B. A negative link from A to B represents the idea that believing in A is a reason for not believing in B, the equivalent to logic A B. For example, let us suppose we are representing the reasoning over whether we should put a roof on a pen in which we are going to keep some animal called Fred. We might then reason If Fred can fly, the pen needs a roof orfly Roof If Fred is a bird, Fred can fly or If Fred is an elephant, Fred cannot fly or If Fred is a bat, Fred can fly or If Fred is a canary, Fred is a bird or If Fred is a raven, Fred is a bird or This gives us the graph: Bird Fly Elephant Fly. Bat Fly Canary Bird. Raven Bird. AI Notes on reasoning with uncertainty 996. Page 5

7 Problems arise in this truth maintenance system if there are circular chains of justifications. For example, we might argue that Fred has feathers is a justification for assuming Fred is a bird, while if we know Fred is a bird we can assume Fred has feathers. This would lead to the structure: Feathered If this occurred as part of the justification graph, on what is said above we could say that neither Bird nor Feathered is just an assumption since both have justifications supporting them. However we have here what is known popularly as a circular argument if we have no other support for it (that is a positive justification pointing inwards to one of the nodes from a node which is not part of the circle) the entire argument is just an assumption, and the same would apply for a circular argument of three or more nodes. We might decide that if we have observed Fred laying eggs that is a justification for Fred being a bird, or if we have observed a feather fall from Fred that is a justification for Fred being feathered: Feathered Bird Bird Feather fall Eggs laid In this case if either Feather fall or Eggs laid becomes in, it is a support for the whole circular argument. Note that mutual negative justifications are acceptable, and in fact give the effect of negation: P P Here it is not possible for both P and P to be in without it causing a contradiction. It is possible for both to be out, representing the case where P is unknown. This can be generalised so that if there is a set of options out of which at most one can be true, the node for each option is linked to the rest by mutual negative links. If we want to show that something can only be one out of a bat, a bird and an elephant, and also only one out of a raven and a canary, we could modify our diagram to that below. Note how we have also introduced a separate node for the negation of Can Fly. Note that this graph does not rule out the possibility of a bird which is neither a raven nor a canary. In this case, Bird would be in, but both Raven and Canary would be out. Roof needed Can't fly Can Fly Bat Bird Elephant Canary Raven AI Notes on reasoning with uncertainty 996. Page 7

8 The truth-maintenance system described above was a deliberately simplistic one given to demonstrate some of the ideas behind truth maintenance. It suffers in particular from the fact that the outness of some node cannot be used as a justification for the inness of another. This means, for example, that we cannot have a full negation effect, since we cannot use P becoming out as a justification for P coming in, rather both P and P stay out (representing P is unknown) unless a change of assumption is made. Doyle s Truth Maintenace System The original Truth Maintenance System, as proposed by Jon Doyle, had a rather more complex and hence more powerful system of assumptions and justifications. In Doyle s TMS a justification is not a single belief, but a set of beliefs, or rather two sets of beliefs. For a justification to be valid, all of its first set of beliefs must be in, and all of its second set out. For this reason, these two lists are referred to as the inlist and outlist respectively. A belief is in if at least one of its justifications is valid, but unlike the system described above there is no concept of negative justifications forcing a belief to be out. A belief is an assumption if it relies on a justification with a non-empty outlist, that is it is in because of the outness of some other beliefs. An assumption may be justified by its own negation being out, similar to the circular belief structure described above. A justification with both inlist and outlist empty is always valid, so any belief it justifies will always be in. Such a justification is referred to as a premise justification. A graphical representation of the set of beliefs and justifications requires a graph with two different sorts of nodes. Such a graph, illustrating the situation described above is shown below. Beliefs are represented by elliptical nodes, and justifications by square nodes. For convenience in discussion, the justifications are numbered. Roof needed 4 Can't Fly 2 Can Fly 5 Bat Elephant Bird Not Bird 7 8 Canary Raven The resulting graph is similar to that shown before as in this case each justification has a single belief. In this graph, however, a justification supported by a belief being out can be used to make another belief in, for example making Can t Fly out makes justification valid, and therefore the belief Can Fly becomes in. Can Fly is justified by any of justifications, 5 or 6 being in. In practice there would have to be some further beliefs and justifications not shown, for example Elephant would have to be justified by the outness of a node Not Elephant and vice versa. Unlike our previous system, this one can correctly deal with default reasoning. Previously we noted that we would like to say that we will assume that a bird can fly providing it is not an abnormal non-flying bird, such as an ostrich, or one whose wing is broken. Adding this to our system creates a AI Notes on reasoning with uncertainty 996. Page 8

9 justification for Can Fly which depends both on the inness of Bird and the outness of ab (meaning Fred is a non-flying bird ). Such a set of beliefs is shown below: Roof needed 4 Can't Fly 2 Can Fly 5 Bat 3 Elephant 5 6 Bird 9 Not Bird ab Ostrich Canary Raven Broken wing 3 Unbroken wing 4 Here, we can start off just by assuming Fred is a bird by having Bird in, supported by Not Bird being out, while if Canary, Raven and Ostrich are also out we are not making any assumptions as to the sort of bird Fred might be. We will also assume that Fred does not have a broken wing, so Broken Wing is out. This means there is no justification for ab, so we can make ab out, making justification 6 valid, and hence Can Fly is in. We have successfully shown that from the assumption that Fred is a bird and without any further assumptions, we can draw the further assumption that Fred can fly. If, however, we then add the assumption that Fred is an ostrich or Fred has a broken wing by making either Ostrich or Broken Wing in, we have a valid justification for ab. This means that 5 becomes a valid justification supported by ab, while 6 becomes invalid since one of its outlist is in. Can t fly becomes in, supported by justification 5, while Can fly goes out because 6 is invalid, and too is invalid since s outlist now also has an in in it; we assume that Bat is also out, so the remaining possible justification for Can Fly, 5, is invalid. Note that, correctly, the simple assertion that Fred is an ostrich will lead to justifying both that Fred is a bird, and that Fred cannot fly, while the assertion that Fred is a canary or Fred is a raven will lead to justifying both that Fred is a bird and that Fred can fly. As described so far, this truth maintenance system does not have any way of removing assumptions that lead to contradictions, or indeed of recognising contradictions. Two facts which are contradictory may both be in so long as they both have valid assumptions. The way contradictions re dealt with is to have explicit contradiction nodes. If a contradiction node becomes in, it is noted and at least one of the assumptions that led to it becoming in has to be retracted. As an abstract example, consider the following diagram: AI Notes on reasoning with uncertainty 996. Page 9

11 C B 2 A D if C and D are in and there are no further justifications for A and B, with A in, 2 is a valid justification, making B in, but then is invalid making A out. But with A out, 2 is invalid so B is out, which makes valid putting A in and so on. Further Reading Books on probability are more likely to be found in the Mathematics section than in the Artificial Intelligence section of the library. Two books which are particularly on the application of probability theory to artificial intelligence are: F.Bacchus Representing and Reasoning with Probabilistic Knowledge MIT Press 99. R.E.Neapolitan Probabilistic Reasoning in Expert Systems Wiley 99. There has been a recent growth in interest in fuzzy logic, and it has found a variety of applications. Three books discussing it are: A.Kandel Fuzzy Techniques in Pattern Recognition Wiley 982. C.V.Negoita Expert Systems and Fuzzy Systems Benjamin/Cummings 985. B.Souc ek Fuzzy Holographic and Parallel Intelligence Wiley 992. Doyle s original paper on his Truth Maintenance System, as well as many other key papers on issues in non-monotonic reasoning are found in the collection M.L.Ginsberg (ed) Readings in Nonmonotonic Reasoning Morgan Kaufmann 987. Another book on non-monotonic reasoning is: W.Lukaszewicz Non-monotonic Reasoning Ellis Horwood 99. A book on truth maintenance systems is: B.Smith and G.Kelleher (eds) Reason Maintenance Systems and their Applications Ellis Horwood 988. AI Notes on reasoning with uncertainty 996. Page

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