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 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
- 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 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:
- 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) = 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
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 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
- 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: sAi 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 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 - 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 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
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 - 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 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
- 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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