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 name "Theory of Possibility" was coined by Zadeh in 1978

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 distributionπ(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 senseif 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) = maxuA π(s); The degree of possibility that x A

N(A) = 1 – (Ac)=min uA 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: (AB) > (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

BA

πpreferred worlds

(in A)

Example (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 possibilistica 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) = minsS 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) = maxsS 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) = minsE µ(f(s)) (utility of the worst consequence in E)

criterion of Wald under ignorance– UOPT(f)= maxsE µ(f(s))

On a linear state space

u*

u*

π

pessimistic

prevision

optimistic

prévision

µo f

S

Pessimistic qualitative utility of binary acts

xAy, with µ(x) > µ(y): • xAy (s) = x if A occurs

= y if its complement Ac occursUPES(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 xg

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

PP = {P | P(Ei) ≥ ai for all i}.

PP 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: sAi mi (one point-coverage function)• Only in the consonant case can m be recalculated from π

1

F

3

possibility levels

1> 2

> 3

>…> n

2

4

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 shownthat:

(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

• 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

Elementary forms of probability-possibility transformations exist for a long time

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 PIPI(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.

• 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).

aL a + L

L

L

p

From probability to possibility : Continuous case

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

• A generic rule « if A then B » is modelled by (AB) > (Ac B).

• This is a constraint that delimits a set of possibility distributions on the set of interpretations of the language

• Applying the minimal specificity principle:(AB) = (ABc ) = (Ac Bc ) > (Ac B).

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 (∆), (AB) > (Ac B).

• Possibilistic inference∆ A B iff *(AB) > *(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 schoolIdea: the validity of a chain of uncertain

deductions is the validity of its weakest linkSyntactic inference K |(p ) is well-defined

Possibilistic logic

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

• Refutation and resolution methods extendK |(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

• Fuzzy logic (Pavelka)

– Formulas are non-Boolean

– Truth is many-valued

– Weighted formulas have crisp sets of models (cuts)

– Validity is Boolean

– degrees of truth are compositional

– represents real functions by means of logical formulas

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