Possibility Theory and its applications: a retrospective and prospective view D. Dubois, H. Prade irit-cnrs, Université Paul Sabatier 31062 toulouse france

Possibility Theory and its applications: a retrospective and prospective view

• D. Dubois, H. Prade IRIT-CNRS, Université Paul Sabatier 31062 TOULOUSE FRANCE

Outline

• Basic definitions

• Pioneers

• Qualitative possibility theory

• Quantitative possibility theory

Possibility theory is an uncertainty theory devoted to the handling of incomplete information.

• similar to probability theory because it is based on set-functions.

• differs by the use of a pair of dual set functions (possibility and necessity measures) instead of only one.

• it is not additive and makes sense on ordinal structures.

The concept of possibility

• Feasibility: It is possible to do something (physical)

• Plausibility: It is possible that something occurs (epistemic)

• Consistency : Compatible with what is known (logical)

• Permission: It is allowed to do something (deontic)

POSSIBILITY DISTRIBUTIONS (uncertainty)

• S: frame of discernment (set of "states of the world")

• x : ill-known description of the current state of affairs taking its value on S

• L: Plausibility scale: totally ordered set of plausibility levels ([0,1], finite chain, integers,...)

• A possibility distribution πx attached to x is a mapping from S to L : s, πx(s)  L, such that s, πx(s) = 1 (normalization)

• Conventions:

• πx(s) = 0 iff x = s is impossible, totally excluded

• πx(s) = 1 iff x = s is normal, fully plausible, unsurprizing

EXAMPLE : x = AGE OF PRESIDENT

• If I do not know the age of the president, I may have statistics on presidents ages… but generally not, or they may be irrelevant.

• partial ignorance :

• 70 ≤ x ≤ 80 (sets, intervals)

• a uniform possibility distribution

• π(x) = 1 x  [70, 80]

• = 0 otherwise

• partial ignorance with preferences : May have reasons to believe that 72 > 71  73 > 70  74 > 75 > 76 > 77

EXAMPLE : x = AGE OF PRESIDENT

• Linguistic information described by fuzzy sets: “ he is old ” : π = µOLD

• If I bet on president's age: I may come up with a subjective probability !

• But this result is enforced by the setting of exchangeable bets (Dutch book argument). Actual information is often poorer.

A possibility distribution is the representation of a state of knowledge: a description of how we think the state of affairs is.

• π' more specific than π in the wide sense if and only if π' ≤ π

• In other words: any value possible for π' should be at least as possible for π that is, π' is more informative than π

• COMPLETE KNOWLEDGE : The most specific ones

• π(s0) = 1 ; π(s) = 0 otherwise

• IGNORANCE : π(s) = 1,  s  S

POSSIBILITY AND NECESSITY OF AN EVENT

• A possibility distribution on S (the normal values of x)

• an event A

• How confident are we that x  A  S ?

• (A) = maxuA π(s); The degree of possibility that x  A

• N(A) = 1 – (Ac)=min uA 1 – π(s) The degree of certainty (necessity) that x  A

Comparing the value of a quantity x to a threshold when the value of x is only known to belong to an interval [a, b].

• In this example, the available knowledge is modeled by (x) = 1 if x [a, b], 0 otherwise.

• Proposition p = "x > " to be checked

• i) a > : then x >  is certainly true : N(x >  ) = (x >  ) = 1.

• ii) b < : then x >  is certainly false ; N(x >  ) = (x >  ) = 0.

• iii) a ≤  ≤ b: then x >  is possibly true or false; N(x >  ) = 0; (x >  ) = 1.

Basic properties

• (A) = to what extent at least one element in A is consistent with π (= possible)

• N(A) = 1 – (Ac) = to what extent no element outside A is possible = to what extent π implies A

• (A  B) = max((A), (B)); N(A  B) = min(N(A), N(B)).

• Mind that most of the time : (A  B) < min((A), (B)); N(A B) > max(N(A), N(B)

• Corollary N(A) > 0  (A) = 1

Pioneers of possibility theory

• In the 1950’s, G.L.S. Shackle called "degree of potential surprize" of an event its degree of impossibility.

• Potential surprize is valued on a disbelief scale, namely a positive interval of the form [0, y*], where y* denotes the absolute rejection of the event to which it is assigned.

• The degree of surprize of an event is the degree of surprize of its least surprizing realization.

• He introduces a notion of conditional possibility

Pioneers of possibility theory

• In his 1973 book, the philosopher David Lewis considers a relation between possible worlds he calls "comparative possibility".

• He relates this concept of possibility to a notion of similarity between possible worlds for defining the truth conditions of counterfactual statements.

• for events A, B, C, A  B C  A  C  B.

• The ones and only ordinal counterparts to possibility measures

Pioneers of possibility theory

• The philosopher L. J. Cohen considered the problem of legal reasoning (1977).

• "Baconian probabilities" understood as degrees of provability.

• It is hard to prove someone guilty at the court of law by means of pure statistical arguments.

• A hypothesis and its negation cannot both have positive "provability"

• Such degrees of provability coincide with necessity measures.

Pioneers of possibility theory

• Zadeh (1978) proposed an interpretation of membership functions of fuzzy sets as possibility distributions encoding flexible constraints induced by natural language statements.

• relationship between possibility and probability: what is probable must preliminarily be possible.

• refers to the idea of graded feasibility ("degrees of ease") rather than to the epistemic notion of plausibility.

• the key axiom of "maxitivity" for possibility measures is highlighted (also for fuzzy events).

Qualitative vs. quantitative possibility theories

• Qualitative:

• comparative: A complete pre-ordering ≥π on U A well-ordered partition of U: E1 > E2 > … > En
• absolute: πx(s)  L = finite chain, complete lattice...

• Quantitative: πx(s)  [0, 1], integers...

• One must indicate where the numbers come from.

• All theories agree on the fundamental maxitivity axiom (A  B) = max((A), (B))

• Theories diverge on the conditioning operation

Ordinal possibilistic conditioning

• A Bayesian-like equation: A) = min(A), A) is the maximal solution to this equation.

• 

• (B | A) = 1 if A, B ≠ Ø, (A) = (A  B) > 0 = (A  B) if (A) > (A  B)

• N(B | A) = 1 – (Bc| A)

• • Independence (B | A) = (B) implies A) = min(), 

• Not the converse!!!!

QUALITATIVE POSSIBILISTIC REASONING

• The set of states of affairs is partitioned via π into a totally ordered set of clusters of equally plausible states

• E1 (normal worlds) > E2 >... En+1 (impossible worlds)

• ASSUMPTION: the current situation is normal.

• By default the state of affairs is in E1

• N(A) > 0 iff (A) > (Ac)

• iff A is true in all the normal situations

• Then, A is accepted as an expected truth

• Accepted events are closed under deduction

A CALCULUS OF PLAUSIBLE INFERENCE

• (B) ≥(C) means « Comparing propositions on the basis of their most normal models »

• ASSUMPTION for computing (B): the current situation is the most normal where B is true.

• PLAUSIBLE REASONING = “ reasoning as if the current situation were normal” and jumping to accepted conclusions obtained from the normality assumption.

• DIFFERENT FROM PROBABILISTIC REASONING BASED ON AVERAGING

ACCEPTANCE IS DEFEASIBLE 

• • If B is learned to be true, then the normal situations become the most plausible ones in B, and the accepted beliefs are revised accordingly

• Accepting A in the context where B is true:

• (AB) > (Ac B) iff N(A | B) > 0 (conditioning)

• • One may have N(A) > 0 , N(Ac | B) > 0 :

• non-monotony

PLAUSIBLE INFERENCE WITH A POSSIBILITY DISTRIBUTION

• Given a non-dogmatic possibility distribution π on S (π(s) > 0, s)
• Propositions A, and B

• A π B iff (A  B) > (A Bc)

• It means that B is true in the most plausible worlds where A is true

• This is a form of inference first proposed by Shoham in nonmonotonic reasoning

PLAUSIBLE INFERENCE WITH A POSSIBILITY DISTRIBUTION

Exa mple (continued)

• Pieces of knowledge like ∆ = {b f, p  b, p  ¬f}

• can be expressed by constraints

• (b  f) > ( b ¬f)
• (p  b) > (p  ¬b)
• (p  ¬f) > (p  f)
• the minimally specific π* ranks normal situations first:
• ¬p  b  f, ¬p ¬b
• then abnormal situations: ¬f  b
• Last, totally absurd situations f  p , ¬b p

Example (back to possibilistic logic)

• material implication

• Ranking of rules: b f has less priority that others according to *: N*(b f ) = N*(p  b) > N*(b f)

• Possibilistic base :

• K = {(b f ), (p  b), (p  ¬f)}, with  < 

Applications of qualitative possibility theory

• Exception-tolerant Reasoning in rule bases

• Belief revision and inconsistency handling in deductive knowledge bases

• Handling priority in constraint-based reasoning

• Decision-making under uncertainty with qualitative criteria (scheduling)

• Abductive reasoning for diagnosis under poor causal knowledge (satellite faults, car engine test-benches)

ABSOLUTE APPROACH TO QUALITATIVE DECISION

• A set of states S;

• A set of consequences X.

• A decision = a mapping f from S to X

• f(s) is the consequence of decision f when the state is known to be s.

• Problem : rank-order the set of decisions in XS when the state is ill-known and there is a utility function on X.

• This is SAVAGE framework.

ABSOLUTE APPROACH TO QUALITATIVE DECISION

• Uncertainty on states is possibilistic a function π: S  L

• L is a totally ordered plausibility scale

• Preference on consequences:

• a qualitative utility function µ: X  U

• µ(x) = 0 totally rejected consequence
• µ(y) > µ(x) y preferred to x
• µ(x) = 1 preferred consequence

Possibilistic decision criteria

• Qualitative pessimistic utility (Whalen):

• UPES(f) = minsS max(n(π(s)), µ(f(s)))

• where n is the order-reversing map of V
• Low utility : plausible state with bad consequences

• Qualitative optimistic utility (Yager):

• UOPT(f) = maxsS min(π(s), µ(f(s)))

• High utility: plausible states with good consequences

The pessimistic and optimistic utilities are well-known fuzzy pattern-matching indices

• in fuzzy expert systems:

• µ = membership function of rule condition
• π = imprecision of input fact

• in fuzzy databases

• µ = membership function of query
• π = distribution of stored imprecise data

• in pattern recognition

• µ = membership function of attribute template
• π = distribution of an ill-known object attribute

Assumption: plausibility and preference scales L and U are commensurate

• There exists a common scale V that contains both L and U, so that confidence and uncertainty levels can be compared.

• (certainty equivalent of a lottery)

• If only a subset E of plausible states is known

• π = E
• UPES(f) = minsE µ(f(s)) (utility of the worst consequence in E)

• criterion of Wald under ignorance

• UOPT(f)= maxsE µ(f(s))

On a linear state space

Pessimistic qualitative utility of binary acts xAy, with µ(x) > µ(y):

• xAy (s) = x if A occurs = y if its complement Ac occurs

• UPES(xAy) = median {µ(x), N(A), µ(y)}

• Interpretation: If the agent is sure enough of A, it is as if the consequence is x: UPES(f) = µF(x)

• If he is not sure about A it is as if the consequence is y: UPES(f) = µF(y)

• Otherwise, utility reflects certainty: UPES(f) = N(A)

• WITH UOPT(f) : replace N(A) by (A)

Representation theorem for pessimistic possibilistic criteria

• Suppose the preference relation a on acts obeys the following properties:

• (XS, a) is a complete preorder.
• there are two acts such that f a g.
•  A, f, x, y constant, x  a y  xAf  yAf
• if f >a h and g >a h imply f g >a h
• if x is constant, h >a x and h >a g imply h >a xg

• then there exists a finite chain L, an L-valued necessity measure on S and an L-valued utility function u, such that a is representable by the pessimistic possibilistic criterion UPES(f).

Merits and limitations of qualitative decision theory

• Provides a foundation for possibility theory

• Possibility theory is justified by observing how a decision-maker ranks acts

• Applies to one-shot decisions (no compensations/ accumulation effects in repeated decision steps)

• Presupposes that consecutive qualitative value levels are distant from each other (negligibility effects)

Quantitative possibility theory

• Membership functions of fuzzy sets

• Natural language descriptions pertaining to numerical universes (fuzzy numbers)
• Results of fuzzy clustering

• Semantics: metrics, proximity to prototypes

• Upper probability bound

• Random experiments with imprecise outcomes
• Consonant approximations of convex probability sets

• Semantics: frequentist, subjectivist (gambles)...

Quantitative possibility theory

• Orders of magnitude of very small probabilities

• degrees of impossibility k(A) ranging on integers k(A) = n iff P(A) = n

• Likelihood functions (P(A| x), where x varies) behave like possibility distributions

• P(A| B) ≤ maxx  B P(A| x)

POSSIBILITY AS UPPER PROBABILITY

• Given a numerical possibility distribution , define P() = {Probabilities P | P(A) ≤ (A) for all A}
• Then, generally it holds that (A) = sup {P(A) | P  P()} N(A) = inf {P(A) | P  P()}
• So  is a faithful representation of a family of probability measures.

From confidence sets to possibility distributions

• Consider a nested family of sets E1  E2 …  En

• a set of positive numbers a1 …an in [0, 1]

• and the family of probability functions

• P = {P | P(Ei) ≥ ai for all i}.

• P is always representable by means of a possibility measure. Its possibility distribution is precisely

• πx = mini max(µEi, 1 – ai)

Random set view

• Let mi = i – i+1 then m1 +… + mn = 1

• A basic probability assignment (SHAFER)

• π(s) = ∑i: sAi mi (one point-coverage function)

• Only in the consonant case can m be recalculated from π

CONDITIONAL POSSIBILITY MEASURES

• A Coxian axiom (A C) = (A |C)(C), with * = product

• Then: (A |C)(A C)/ (C)

• N(A| C) = 1 – (Ac | C)

• Dempster rule of conditioning (preserves -maxitivity)

• For the revision of possibility distributions: minimal change of when N(C) = 1.

• It improves the state of information (reduction of focal elements)

Bayesian possibilistic conditioning

• (A |b C) = sup{P(A|C), P ≤ , P(C) > 0}

• (A |b C) = inf{P(A|C), P ≤ , P(C) > 0}

• It is still a possibility measure π(s |b C) = π(s)max(1, 1 /( π(s) + N(C)))

• It can be shownthat:

• (A |b C) (A C)/ ((A C) + (Ac C))

• N(A|b C) = (A C) / ((A C) + (Ac C))

• = 1 – (Ac |b C)

• For inference from generic knowledge based on observations

Possibility-Probability transformations

• Why ?

• fusion of heterogeneous data
• decision-making : betting according to a possibility distribution leads to probability.
• Extraction of a representative value
• Simplified non-parametric imprecise probabilistic models

POSS PROB: Laplace indifference principle  “ All that is equipossible is equiprobable ” = changing a uniform possibility distribution into a uniform probability distribution

• POSS PROB: Laplace indifference principle  “ All that is equipossible is equiprobable ” = changing a uniform possibility distribution into a uniform probability distribution

• PROB POSS: Confidence intervals Replacing a probability distribution by an interval A with a confidence level c.

• It defines a possibility distribution
• π(x) = 1 if x  A,
• = 1 – c if x  A

Possibility-Probability transformations : BASIC PRINCIPLES

• Possibility probability consistency: P ≤ 

• Preserving the ordering of events : P(A) ≥ P(B) (A) ≥ (B) or elementary events only (x) > (x') if and only if p(x) > p(x') (order preservation)

• Informational criteria:

• from  to P: Preservation of symmetries
• (Shapley value rather than maximal entropy)
• from P to : optimize information content
• (Maximization or minimisation of specificity

From OBJECTIVE probability to possibility :

• Rationale : given a probability p, try and preserve as much information as possible

• Select a most specific element of the set PI(P) = {:  ≥ P} of possibility measures dominating P such that  (x) >  (x') iff p(x) > p(x')

• may be weakened into : p(x) > p(x') implies  (x) >  (x')

• The result is i = j=i,…n pi

• (case of no ties)

From probability to possibility : Continuous case

• The possibility distribution  obtained by transforming p encodes then family of confidence intervals around the mode of p.

• The -cut of  is the (1)-confidence interval of p

• The optimal symmetric transform of the uniform probability distribution is the triangular fuzzy number

• The symmetric triangular fuzzy number (STFN) is a covering approximation of any probability with unimodal symmetric density p with the same mode.

• In other words the -cut of a STFN contains the (1)-confidence interval of any such p.

From probability to possibility : Continuous case

• IL = {x, p(x) ≥ } = [aL, aL+ L] is the interval of length L with maximal probability

• The most specific possibility distribution dominating p is π such that L > 0, π(aL) = π(aL+ L) = 1 – P(IL).

Possibilistic view of probabilistic inequalities

• Chebyshev inequality defines a possibility distribution that dominates any density with given mean and variance.

• The symmetric triangular fuzzy number (STFN) defines a possibility distribution that optimally dominates any symmetric density with given mode and bounded support.

From possibility to probability

• Idea (Kaufmann, Yager, Chanas):
• Pick a number  in [0, 1] at random
• Pick an element at random in the -cut of π.
• a generalized Laplacean indifference principle : change alpha-cuts into uniform probability distributions.
• Rationale : minimise arbitrariness by preserving the symmetry properties of the representation.

• The resulting probability distribution is:

• The centre of gravity of the polyhedron P(
• The pignistic transformation of belief functions (Smets)
• The Shapley value of the unanimity game N in game theory.

SUBJECTIVE POSSIBILITY DISTRIBUTIONS

• Starting point : exploit the betting approach to subjective probability

• A critique: The agent is forced to be additive by the rules of exchangeable bets.

• For instance, the agent provides a uniform probability distribution on a finite set whether (s)he knows nothing about the concerned phenomenon, or if (s)he knows the concerned phenomenon is purely random.

• Idea : It is assumed that a subjective probability supplied by an agent is only a trace of the agent's belief.

SUBJECTIVE POSSIBILITY DISTRIBUTIONS

• Assumption 1: Beliefs can be modelled by belief functions

• (masses m(A) summing to 1 assigned to subsets A).

• Assumption 2: The agent uses a probability function induced by his or her beliefs, using the pignistic transformation (Smets, 1990) or Shapley value.

• Method : reconstruct the underlying belief function from the probability provided by the agent by choosing among the isopignistic ones.

SUBJECTIVE POSSIBILITY DISTRIBUTIONS

• There are clearly several belief functions with a prescribed Shapley value.

• Consider the least informative of those, in the sense of a non-specificity index (expected cardinality of the random set)

• I(m) = ∑  m(A)card(A).

• RESULT : The least informative belief function whose Shapley value is p is unique and consonant.

SUBJECTIVE POSSIBILITY DISTRIBUTIONS

• The least specific belief function in the sense of maximizing I(m) is characterized by

• i = j=1,n min(pj, pi).

• It is a probability-possibility transformation, previously suggested in 1983: This is the unique possibility distribution whose Shapley value is p.

• It gives results that are less specific than the confidence interval approach to objective probability.

Applications of quantitative possibility

• Representing incomplete probabilistic data for uncertainty propagation in computations

• (but fuzzy interval analysis based on the extension principle differs from conservative probabilistic risk analysis)

• Systematizing some statistical methods (confidence intervals, likelihood functions, probabilistic inequalities)

• Defuzzification based on Choquet integral (linear with fuzzy number addition)

Applications of quantitative possibility

• Uncertain reasoning : Possibilistic nets are a counterpart to Bayesian nets that copes with incomplete data. Similar algorithmic properties under Dempster conditioning (Kruse team)

• Data fusion : well suited for merging heterogeneous information on numerical data (linguistic, statistics, confidence intervals) (Bloch)

• Risk analysis : uncertainty propagation using fuzzy arithmetics, and random interval arithmetics when statistical data is incomplete (Lodwick, Ferson)

• Non-parametric conservative modelling of imprecision in measurements (Mauris)

Perspectives

• Quantitative possibility is not as well understood as probability theory.

• Objective vs. subjective possibility (a la De Finetti)

• How to use possibilistic conditioning in inference tasks ?

• Bridge the gap with statistics and the confidence interval literature (Fisher, likelihood reasoning)

• Higher-order modes of fuzzy intervals (variance, …) and links with fuzzy random variables

• Quantitative possibilistic expectations : decision-theoretic characterisation ?

Conclusion

• Possibility theory is a simple and versatile tool for modeling uncertainty

• A unifying framework for modeling and merging linguistic knowledge and statistical data

• Useful to account for missing information in reasoning tasks and risk analysis

• A bridge between logic-based AI and probabilistic reasoning

Properties of inference |=

• A |=π A if A ≠ Ø (restricted reflexivity)

• if A ≠ Ø, then A |=π Ø never holds (consistency preservation)

• The set {B: A |= π B} is deductively closed

• -If A  B and C |=π A then C |=π B

• (right weakening rule RW)
• -If A |=π B and A |=π C then A |=π B C

• (Right AND)

Properties of inference |=

• If A |=π C ; B |=π C then A  B |=π C (Left OR)

• If A |=π B and A  B |=π C then A |=π C

• (cut, weak transitivity )
• (But if A normally implies B which normally implies C, then A may not imply C)

• If A |=π B and if A |=π Cc is false, then A  C |=π B (rational monotony RM)

• If B is normally expected when A holds,then B is expected to hold when both A and C hold, unless it is that A normally implies not C

REPRESENTATION THEOREM FOR POSSIBILISTIC ENTAILMENT

• Let |= be a consequence relation on 2S x 2S

• Define an induced partial relation on subsets as

• A > B iff A  B |= Bc for A ≠ 

• Theorem: If |= satisfies restricted reflexivity, right weakening, rational monotony, Right AND and Left OR, then A > B is the strict part of a possibility relation on events.

• So a consequence relation satisfying the above properties is representable by possibilistic inference, and induces a complete plausibility preordering on the states.

A POSSIBILISTIC APPROACH TO MODELING RULES

MODELLING A SET OF DEFAULT RULES as a POSSIBILITY DISTRIBUTION

• ∆ = {Ai  Bi, i = 1,n}

• ∆ defines a set of constraints on possibility distributions (Ai  Bi) > (Ai  ¬Bi), i = 1,…n

• •(∆) = set of feasible π's with respect to ∆

• •ne may compute  : the least specific possibility distribution in (∆)

Plausible inference from a set of default rules

• What « ∆ implies A  B » means

• Cautious inference

• ∆  A  B iff
• For all (∆), (AB) > (Ac B).

• Possibilistic inference

• ∆  A  B iff *(AB) > *(Ac B) for the least specific possibility measure in (∆).
• Leads to a stratification of ∆ according to N*(Ac B)

Possibilistic logic

• A possibilistic knowledge base is an ordered set of propositional or 1st order formulas pi

• K = {(pi i), i = 1,n} where i > 0 is the level of priority or validity of pi

• i = 1 means certainty.

• i = 0 means ignorance

• Captures the idea of uncertain knowledge in an ordinal setting

Possibilistic logic

• Axiomatization:

• All axioms of classical logic with weight 1

• Weighted modus ponens {(p ), (¬p  q )} | (q min(,))

• OLD! Goes back to Aristotle school

• Idea: the validity of a chain of uncertain deductions is the validity of its weakest link

• Syntactic inference K |(p ) is well-defined

Possibilistic logic

• Inconsistency becomes a graded notion inc(K) = sup{, K |- (,)}

• Refutation and resolution methods extend K |(p ) iff K {(p 1)} |- (,)

• Inference with a partially inconsistent knowledge base becomes non-trivial and nonmonotonic K |nt p iff K | (p ) and  > inc(K)

Semantics of possibilistic logic

• A weighted formula has a fuzzy set of models .

• If A = [p] is the set of models of p (subset of S),

• |(p ) means N(A) ≥ 

• The least specific possibility distribution induced by |(p ) is:

• π(p )(s) = max(µA(s), 1 – )

• = 1 if p is true in state s

• = 1 – if p is false in state s

Semantics of possibilistic logic

• The fuzzy set of models of K is the intersection of the fuzzy sets of models of {(pi i), i = 1,n}

• πK(s)= mini=1,n {1 – i | s pi]}

• determined by the highest priority formula violated by s

• The p. d. πK is the least informed state of partial knowledge compatible with K

Soundness and completeness

• Monotonic semantic entailment follows Zadeh’s entailment principle K |= (p, ) stands for πK ≤ π(p a)

• Theorem: K | (p, ) iff K |= (p )

• For the non-trivial inference under inconsistency: {(p 1)}  K |nt q iff (q  p) > (¬q  p)

Possibilistic vs. fuzzy logics

• Possibilistic logic

• Formulas are Boolean
• Truth is 2-valued
• Weighted formulas have fuzzy sets of models
• Validity is many-valued
• degrees of validity are not compositional except for conjunctions
• Represents uncertainty

Example: IF BIRD THEN FLY; IF PENGUIN THEN BIRD; IF PENGUIN THEN NOT-FLY

• • K = {b  f, p  b, p  ¬f}
•  = material implication
• K  {b} | f; K  {p} |  contradiction

• using possibilistic logic:  < min(,)

• K = {(b  f ), (p  b ), (p  ¬f )}

• then K  {(b, 1)} | (f ) and K  {(b, 1)} |nt f

• Inc(K{(p, 1), (b, 1)} = 

• K  {(p, 1), (b, 1)} | (¬f, min(,))

• Hence K  {(p, 1), (b, 1)} |nt ¬f