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

Combinatorial interpretation of measurements

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

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

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

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],

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

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

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

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

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

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

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 - ...

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

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) )

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.  

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 

  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 

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

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

terima kasih