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
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