1. Looking for Plausibility
Wan Ahmad Tajuddin Wan Abdullah
Faculty of Computer Science and Information Technology &
Department of Physics, Faculty of Science
Universiti Malaya
Kuala Lumpur, Malaysia
ICACSIS2010
Bali, Indonesia

2. Combinatorial interpretation of measurements

4. probability P(X)  N(X) P(X  Y) = P(X)P(Y)
P(X  Y) = P(X)+P(Y)-P(XY)
problem – diminished value with alternatives

5. possibility cf. fuzzy logic pos(XY) ≤ min(pos(X),pos(Y))
pos(XY) = max(pos(X),pos(Y))
– grounding?

6. plausibility normalized scale from 0 to 1:
pl(X) = : X is not all plausible
pl(X) = : X is fully plausible
not exclusive
pl(X) + pl(Y) > 1 though X  Y = 

7. not self-dual (cf. possibility) pl(X) ≠ 1-pl(X)
when something is not at all plausible, it does not mean that it is fully plausible that that something is untrue
negative values for plausibility
require that
pl( X) = - pl(X)
now
pl(X)  [-1,1],

8. abduction
plausible – “seemingly or apparently valid, likely, or acceptable” (Wiktionary)
A T C
hypotheses rules conclusions
Deduction: A, T  C
Induction: A, C  T
Abduction: C, T  A

9. Bayes P(q |x) = P(x |q ) P(q ) / P(x) priors unknown
conditional likelihood prior
probability
priors unknown

10. Dempster-Shafer “belief masses” of sets of propositions
 bel(X), pl(X)
bel(X)  P(X)  pl(X)
pl(X) = 1 - bel(X)
combinations - recalculate
assignment of belief masses problematic
their association with probabilities misleading

11. biconditionality
tendency for implications to be interpreted going the other way
Bird(x)  Have_feathers(x),Fly(x)
like
P(q |x) = P(x |q )
non-rational
explain analogy and induction
related to abduction

12. Collins-Michalski
types of inferences: mutual implication, mutual dependency, generalization, specialization, and similarity/analogy
large number of parameters to model uncertainty
supports notion that plausibility has abductive basis

13. Connectionist connections
Si = f(hi)
hi = J(1)iSi + ∑j J(2)ij Sj + ∑ j,k J(3)ijk Sj Sk + ...

14. J(2)ij = J(2)ji Symmetrical connections Minimization / optimization
cf spin system
Minimization / optimization
Lyapunov / energy function
E = - ∑i J(1)i Si - ½ ∑ij J(2)ij Si Sj

15. Bird(x)Have_feathers(x),Fly(x). Fly(Tweety). Have_feathers(Tweety).
1515
Logic Programming
Bird(x)Have_feathers(x),Fly(x).
x Bird if x Have_feathers and x Fly.
Fly(Tweety).
Tweety Fly.
Have_feathers(Tweety).
Tweety Have_feather.
Have_fur(Sylvester).
Sylvester Have_fur. 
Bird(Tweety)
Horn clauses – at most 1 logical atom in consequent

16. Logic Programming on Little-Hopfield networks
1616
Logic Programming on Little-Hopfield networks
Logic programming ~ minimization of “logical inconsistency”
A ← B, C. A v ¬B v ¬C
D ← B. D v ¬B
C ←. C
EP = ⅛(1 - SA) (1 + SB) (1 + SC)
+ ¼ (1 - SD) (1 + SB)
+ ½ (1 - SC)
3rd order bipolar neural network
E = - ⅓ Σi Σj Σk Jijk(3)SiSjSk - ½ Σi Σj Jij(2)SiSj - Σi Ji(1)Si
Si := sign(Σj Σk Jijk(3)SjSk + Σj Jij(2)Sj + Ji(1) )

17. Plausible plausibility
abductive interpretation:
pl(Ai)  fraction of propositions in C forced correct
0 if Ai does not have any effect on C
1 if Ai forces all of C correct.

18. Rules for combination of plausibilities?
For Ai and Aj forcing disjoint subsets of C
and having no joint effects ,
pl(Ai  Aj) = pl(Ai) + pl(Aj)
If T contains
Ai  Ci
Aj  Ci
then the value is less
Ai Aj  Ci
then the value is more
singleton C’s - involve some kind of probability 

19. for disjoint hypotheses,
pl(AiAj) = pl(Ai) + pl(Aj),
pl(Ai  Aj) = pl(Aj) - pl(Ai)
cf synaptic weight assignment in connectionist logic with the addition of a clause 

20. Connectionist logic:
force the correct values for C and for hypothesis
final network state with least value for
number of clauses unsatisfied
~ number of errors in C
~ plausibility
– Hebbian learning

21. conclusion conceptualization need to refine:  combination rules
 probabilistic relationship
- application

22. terima kasih