Resolving Rule Conflicts

1
Resolving Rule Conflicts

• Assign explicit priorities to rules
• Specificity of a rules antecedents
• R1 IF car will not start
• THEN battery is dead
• R2 IF car will not start
• AND headlights will not work
• THEN battery is dead
• Order in which rule are entered in kbase
• timestamp, rule entered first given priority
• Order of rules in kbase
• Recency of facts entered
• real-time ES

2

• Certainty factors
• higher CF rule fires first
• rule that gives higher confidence to goal being
  pursued

3
Certainty Theory

• Representing uncertain evidence
• Representing uncertain rules
• Combining evidence from multiple sources
• eg. IF A and B THEN X cf 0.8
• IF C THEN X cf 0.7
• What is the certainty of X?
• -1lt CF lt 1
• -100 lt CF lt 100
• Used in
• rule conclusions, values of variables in premise,
  answers to user queries

4
Certainty theory

• Commutative property
• if more than one rule gathers information, then
  the combined CF value can not be dependent upon
  the order of the processing of the rules
• Asymptotic property
• the certainty model should incrementally add
  belief to a hypothesis as new positive evidence
  is obtained however, unless we encounter some
  evidence that absolutely confirms the hypothesis,
  we cannot be totally certain.
• Thus, confirming evidence increases out belief,
  but unless absolute certainty is found, cf
  approaches 1, but never equals 1.

5

• Rule IF E THEN H CF(Rule)
• CF(H,E) CF(E) CF(Rule)
• Example IF econ-two-years strong
• THEN likelihood-of-inflation strong
  CF 40
• Given econ-two-years strong with cf70
• CF(likelihood-of-inflation strong ) (40
  70)/100 28

6

• Conjunctive
• IF E1 and E2 and and En
• THEN H CF(Rule)
• CF(H, E1 and E2 and and En) minCF(Ei)
  CF(Rule)
• Disjunctive
• IF E1 or E2 or or En
• THEN H CF(Rule)
• CF(H, E1 or E2 or or En) maxCF(Ei) CF(Rule)

7

• Certainty propagation in similarly concluded
  rules
• R1 IF E1 THEN H CF1
• R2 IF E2 THEN H CF2
• (supporting evidence increases our belief)
• CFcombine(CF1,CF2)
• CF1 CF2(1 - CF1), when both gt 0
• (CF1 CF2) / (1 - min(CF1, CF2), when one
  lt 0
• CF1 CF2(1 CF1), when both lt 0.

8

• Premise with AND (conjunctive)
• Example IF economy-two-years strong
• AND availability-of-investment-capital
  low
• THEN likelihood-of-inflation strong
• CF of condition min(cf1, cf2)
• Premise with AND (conjunctive)
• Example IF economy-two-years poor
• OR unemployment-outlook poor
• THEN economic-outlook poor
• CF of condition max(cf1, cf2)

9

• Premise with both AND and OR
• Example IF has-credit-card yes (cf 80)
• OR cash ok (cf 90)
• AND payments ok (cf 85)
• THEN approval ok
• (has-credit-card yes AND payments
  ok) min(80,85)
• OR
• (cash ok AND payments ok) min(90,85)
• CF 80 85 - (8085)/100 97 (some systems use
  this)
• CF max(80,85) 85 (some systems use this)

10

• Example
• R1 IF weatherman says it will rain
• THEN it will rain CF 0.8
• R2 IF farmer says it will rain
• THEN it will rain CF 0.8
• Case (a) Weatherman and farmer are certain in
  rain
• CF(E1) CF(E2) 1.0
• CF(H, E1) CF(E1) CF(Rule1) 1.00.8 0.8
• CF(H, E2) CF(E2) CF(Rule2) 1.00.8 0.8
• CFcombine(CF1, CF2) CF1 CF2(1- CF1)
  0.80.8(1-0.8) 0.96
• CF of a hypothesis which is supported by more
  than one rule, can be incrementally increased by
  supporting evidence from both rules.

11

• Case (b)Weatherman certain in rain, farmer
  certain in no rain
• CF(E1) 1.0, CF(E2) -1.0
• CF1 0.8, CF2 -0.8
• CFcombine(CF1, CF2) (0.8 (-0.8)) / (1 -
  min(0.8, 0.8) 0 (unknown)
• Case (b) CF(E1) -0.8, CF(E2) -0.6
• CFcombine(CF1, CF2) -0.64 - 0.48(1 - 0.64)
  -0.81
• Incremental decrease in certainty from more than
  one source of disconfirming evidence
• Case (c)
• CFcombine(CF1, CF2, CF3,.) 0.999 CFold
• Single piece of disconfirming evidence CFnew
  -0.8
• Cfcombine (0.999 - 0.8) / (1 - 0.8) 0.995
• Single piece of disconfirming evidence does not
  have a major impact on many pieces of confirming
  evidence.

12
Certainty Threshold

• CF-condition lt Threshold gt rule fails to fire
• Rule condition fails if CF(premise) lt Threshold
• But if rule fires, and after that
• CF(conclusion) lt Threshold
• the conclusion will still be asserted.

13
Interpreting CF values

• Definitely not -1.0
• Almost certainly not -0.8 Maybe 0.4
• Probably not -0.6 Probably 0.6
• Maybe not -0.4 Almost certainly 0.8
• Unknown -0.2 to 0.2 Definitely 1.0
• Acquiring CF from experts
• Prompting users for CF values

14
Example

• R1 IF the weather looks lousy (E1)
• OR I am in a lousy mood (E2)
• THEN I shouldnt go to the ball game CF 0.9
  (H1)
• R2 IF I believe it is going to rain (E3)
• THEN the weather looks lousy CF 0.8 (E1)
• R3 IF I believe it is going to rain (E3)
• AND the weatherman says it is going to
  rain (E4)
• THEN I am in a lousy mood CF 0.9 (E2)
• R4 IF the weatherman says it is going to
  rain (E4)
• THEN the weather looks lousy CF 0.7 (E1)
• R5 IF the weather looks lousy (E1)
• THEN I am in a lousy mood CF 0.95 (E2)

15

• Assume user enters following facts
• I believe it is going to rain, CF(E3) 0.95
• Weatherman says it is going to rain, CF(E4) 0.8
• Goal I shouldnt go to the ball game (H1)
• Step 1 Pursue R1 Pursue premise of R1
• weather looks lousy (E1) R2 and R4
• Step 2 Pursue R2
• CF(E1, E3) CF(E3) CF(Rule R2)
  0.80.95 0.76
• Step 3 Pursue R4
• CF(E1, E4) CF(E4)CF(Rule R4) 0.70.850.60
• Step 4 Combine evidence for E1
• CF(E1) CF(E1,E3) CF(E1,E4)(1 - CF(E1,E3))
• 0.76 0.60 (1.0 - 0.76) 0.90

16

• Step 5 Pursue premise 2 of Rule 1 (E2) R3 and
  R5
• Step 6 Pursue R5
• CF(E2, E1) CF(E1) CF(Rule R5) 0.950.9
  0.86
• Step 7 Pursue R3
• CF(E2, E3 and E4) minCF(E3), CF(E4)
  CF(Rule R3)
• min0.95,0.850.9 0.77
• Step 8 Combine evidence for E2 I am in a lousy
  mood
• CF(E2) CF(E2, E1) CF(E2, E3 and E4) ( 1 -
  CF(E2, E1))
• 0.86 0.77 ( 1- 0.86) 0.97
• Step 9 return to Rule R1
• From steps 4 and 9
• CF(H1, E1 or E2) maxCF(E1), CF(E2) CF(Rule
  R1)
• max0.9,0.970.9 0.87
• CF(I shouldnt go to the ball
  game)
• Conclusion I almost definitely shouldnt go to
  the ball game.

17
Controlling search with CF

• Meta rules
• Example
• IF CF(Problem is in electrical system) lt 0.5
• THEN PURSUE problem is fuel system