1. CHAPTER 8 Decision Making CS267 BY GAYATRI BOGINERI (12) & BHARGAV VADHER (14) October 22, 2007

2. Outline of the chapter  Introduction  Core and Reduct  Simplification of Decision Table  Decision Algorithm  Degree of Dependency

3. Introduction Definition: For given KR system such that K = (U,A), we can define decision table as follows T = (U,A,C,D) where U = universe A = set of actions C = condition attributes D = decision attributes C,D is subset of A This decision table is also called ‘ CD – decision table’.

4. Core and Reduct Core is a condition attribute value which is indispensable that means it discern the value of decision attribute. Reduct is a decision rule which must satisfy following conditions 1. The rule must be true or consistent. 2. Predecessor of rule must be independent.

5. Simplification of decision table It means removal of unnecessary conditions to make the decision. We will show it with the following steps listed. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

6. Let take the example of optician’s decision of whether the patient is suitable for contact lens or not. We have set of condition a, b, c and d. which is shown below a. Age 1. young 2. pre-presbyopic 3. prebyopic b. Spectacle 1. myope condition 2. hypermetrope attributes c. astigmatic 1. no 2. yes d. tear production rate 1. reduced 2. normal

7. Based on given condition the optician has to take one out three of the following decision. 1. hard contact lens 2. soft contact lens decision attributes 3. no contact lens So we have total of 24 (3*2*2*2) possible combination of given conditions which are shown with their decision in tabular form below.

8. UABCDE ---- 111221 212221 321221 431221 511122 612122 721122 822122 932122 1011113 1111213 1212113 1312213 1421113 1521213 1622113 1722213 1822223 1931113 2031123 2131213 2232113 2332213 2432223

9. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

10. Check dependency of condition attribute  decision attribute every set of condition attribute must have unique decision attribute. If the above shown condition is true for all decision rule then we say that dependency condition attribute  decision attribute is valid. Logically it is represented as below {a,b,c,d} == > {e}

11. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

12. Reduction of condition attribute, if any. Now check for each condition attribute whether it is dispensable or indispensable with respect to decision attribute. Suppose we are checking for condition attribute a in a given example. Remove a and look for the rest of the condition attribute.(shown in figure) After removal of a we have two pairs of decision rule which are inconsistent. 1. b2c2d2  e1 (rule 2) b2c2d2  e3 (rule 18 and 24) 2. b1c1d2  e2 (rule 5) b1c1d2  e3 (rule 20) So condition attribute a is indispensable that means not removable. Checking every condition attribute we found that every attribute is indispensable so no one is removable. So now we can say that given table is e-independent.

13. After removal of condition attribute a UBCDE ---- 11221 22221 31221 41221 51122 62122 71122 82122 92122 101113 111213 122113 132213 141113 151213 162113 172213 182223 191113 201123 211213 222113 232213 242223

14. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

15. Find core values of all decision rules. We have to find out the core values of each decision rule in order to decide the reducts of that rule, because core is an intersection of all reduct of same rule. Consider the condition attribute and rules associated with decision attribute 1 that is e1. If we want to find out the core of first rule that is a1b1c2d2  e1 First remove A and check for rest of attribute, you find that b1c2d2  e1 (rule 1) and b1c2d2  e1 (rule 3) which are inconsistent rule under decision attribute e1. So attribute A of rule 1 is not core. Similarly B is also not core but C and D are core attribute for rule 1. UABCDE 111221 212221 321221 431221

16. UABCDE ---- 1--221 21221 3-1221 4-1221 5--122 6--122 72-122 8--122 9-2122 10---13 11---13 12---13 13---13 14---13 15---13 16---13 17----3 18222-3 19----3 20311-3 21---13 22---13 23----3 24322-3 Core values of decision table

17. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

18. Find reducts of each decision rule. We have to add to the core value of each decision rule, such value of condition attribute of that rule so the 1. Predecessor of rule is independent. 2. The rule is true. Suppose we are checking for first rule that is a1b1c2d2  e1 We already know that first rule has two core that is C and D. i.e. U A B C D E 1 _ _ 2 2 1 So here we have three possibility of reduct which are 1 X 1 2 2 1 i.e. Rule b1c2d2  e1 possibility for 1 1 X 2 2 1 i.e. Rule a1c2d2  e1 reducts of 1 X X 2 2 1 i.e. Rule c2d2  e1 first rule Now check for every rule whether it is true or not.

19. UABCDE ---- 1x1221 1’1x221 ---- 21x221 3x1221 4x1221 51x122 61x122 6’x2122 ---- 72x122 82x122 8’x2122 ---- 9x2122 10xxx13 11xxx13 12xxx13 13xxx13 14xxx13 15xxx13 16xxx13 ---- 17xxx13 17’222x3 ---- 18222x3 ---- 19311x3 19’xxx13 ---- 20311x3 21xxx13 22xxx13 23xxx13 ---- 24322x3

20. 1.Check dependency of condition attribute  decision attribute 2.Reduction of condition attribute, if any. 3.Find core values of all decision rules. 4.Find reducts of each decision rule. 5.Combine and assign unique rule number to each rule.

21. Combine and assign unique rule number to each rule. In the last step we have to categorize the reducts which have the same decision rule. And then assign a unique number to each of these decision rule, and it is your minimal decision table.

22. So after categorize the rules it will look like this UABCDE ---- 1’,21x221 1,3,4x1221 ---- 5,61x122 7,82x122 6’,8’,9x2122 ---- 10-17,19’ 21,22,23 xxx1 3 17’,18222x3 19,20311x3 23’,24322x3

23. And finally our minimal decision table will be as follows UABCDE ---- 11x221 2x1221 31x122 42x122 5x2122 6xxx1 3 7222x3 8311x3 9322x3

24. Decision algorithm For making decision algorithm from decision table follow two steps. 1. replace all the value into the minimal decision table. 2. combine all the rules as per particular decision attribute. From given minimal decision table we have the following rules. a1c2d2  e1 b1c2d2  e1 a1c1d2  e2 a2c1d2  e2 b2c1d2  e2 d1  e3 a2b2c2  e3 a3b1c1  e3 a3b2c2  e3

25. Now combine the rules as per decision attribute, we will have decision algorithm By combining rules we have (a1 V b1)c2d2  e1 (a1 V a2 V b2)c1d2  e2 d1 V (a3b2c1) V ((a2 V a3)b2c2)  e3. Which is our decision algorithm.

26. Degree of dependency Degree of dependency can be calculated with the formulae number of unaffected rule with lack of this attribute degree of dependency = ---------------------------------------------------------------------- total number of rules degree of dependency of a = 19 / 24 i.e. 5 rule affected (2,5,18,20,24) degree of dependency of b = 3/4 i.e. 6 rule affected degree of dependency of c = 1/2 i.e. 13 rule affected degree of dependency of d = 1/4 i.e. 18 rule affected So attribute d is the most significant one in our decision table, because with the lack of attribute d, 18 rules are affected.

27. Q & A