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: Plausibility

:

It is possible to do something

(physical )

It is possible that something occurs

(epistemic) Consistency : Permission:

Compatible with what is known

(logical)

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 π S to L :  s, π (normalization) x (s)  x attached to x is a mapping from L, such that  s, π x (s) = 1

Conventions:

π x (s) = 0 iff x = s is impossible, totally excluded π x (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 = µ OLD •

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(s 0 ) = 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) = max u  A π(s); The degree of possibility

that x



A

N(A) = 1 –  (A c )=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

p

( x) = 1 if x



[a, b], 0 otherwise.

Proposition p = "x >

• i) a >  : then x >  

" to be checked

is certainly true : N(

x >

 ) =  (

x >

 ) = 1.

• ii) b <  : then x >  • iii) a ≤  is certainly false ; N(

x >

 ) =  (

x >

 ) = 0.

≤ b: then x >  N(

x >

 is possibly true or false; ) = 0;  (

x >

 ) = 1.

Basic properties

 (A) = to what extent

at least one

consistent with π (= possible) element in A is N(A) = 1 –  (A c ) = to what extent no element outside A is possible = to what extent π implies A  (A  B) = max(  (A),  (B)); N(A  N(B)). B) = min(N(A),

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

legal reasoning (1977).

considered the problem of • "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:

π x (s)  L = finite chain, complete lattice...

•

Quantitative

: π x (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) N(B | A) = 1 – = 1 if A, B ≠ Ø, =  (A  B) if  (A) =  (A) >  (A   (A  B) > 0 B)  (B c | 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) >  (A c ) 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) >  (A c  B) iff N(A | B) > 0 (conditioning) • One may have N(A) > 0 , N(A c | 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  B c ) 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

A B š preferred worlds (in A)

Example (continued)

• Pieces of knowledge like ∆ = {b  f, p  can be expressed by constraints  (b  f) >  ( b  ¬f)  (p  b) >  (p  ¬b)  (p  ¬f) >  (p  f) b, 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 p *: N*(b  f ) = N*(p  b) > N*(b  f)

Possibilistic base :

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

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 X when the state is ill-known and there is a utility function on X.

S • 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  – µ(x) = 0 totally rejected consequence U – µ(y) > µ(x) – µ(x) = 1 y preferred to x preferred consequence

Possibilistic decision criteria

•

Qualitative pessimistic utility

(Whalen)

:

U PES (f) = min s  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): U OPT (f) = max s  S – min(π(s), µ(f(s)))

High utility: consequences



plausible states with good

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 – U PES (f) = min s  E consequence in E) µ(f(s)) (utility of the worst criterion of Wald under ignorance – U OPT (f)= max s  E µ(f(s))

On a linear state space u* u* š µo f pessimistic prevision optimistic prévision S

Pessimistic qualitative utility of binary acts xAy, with

µ(x) > µ(y): • xAy (s) = x if A occurs = y if its complement A c occurs U PES (xAy) = median {µ(x), N(A), µ(y)} • •

Interpretation:

If he is not sure about A it is as if the consequence is y: U PES (f) = µ F (y) If the agent is sure enough of A, it is as if the consequence is x: U PES (f) = µ F (x) Otherwise, utility reflects certainty: U PES (f) = N(A)

WITH U OPT (f) : replace N(A) by



(A)

Representation theorem for pessimistic possibilistic criteria

• Suppose the preference relation following properties:  a on acts obeys the • • (X S ,  a ) is a complete preorder.

• there are two acts such that f  a  A, • if f > a  f,  x, y constant, x h and g > a h imply f  a y  g > a g.

 h • if x is constant, h > a x and h > a xAf  yAf 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 U PES (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) = e n •

Likelihood functions

(P(A| x), where x varies) behave like possibility distributions P(A| B) ≤ max x  B P(A| x)

•

POSSIBILITY AS UPPER PROBABILITY

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

From confidence sets to possibility distributions

Consider a

nested

family of sets E 1 a set of positive numbers a 1 …a n  E in [0, 1] 2 and the family of probability functions …  E n P = {P | P(E i ) ≥ a i for all i}.

P

is always representable by means of a possibility measure.

Its possibility distribution is precisely

π x = min i max(µ Ei , 1 – a i )

Random set view

F 1  2  3  4 possibility levels 1 >  2 >  3 >… >  n • Let m i =  i –  i+1 then m 1 +… + m n = 1

A basic probability assignment (SHAFER)

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

Only in the consonant case can m be recalculated from π

CONDITIONAL POSSIBILITY MEASURES •

A Coxian axiom

with * = product  (A  C) =  (A |C) * (C), Then:  (A |C) = (A  C)/  (C) N(A| C) = 1 –  (A c | C) Dempster rule of conditioning (

preserves

s

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

It is still a possibility measure

π(s | b C) = π(s) It can be shown that:  max(1, 1 / ( π(s) + N(C)))  (A | b C) = (A  C) / (  (A  C) + N (A c  C)) N(A| b C) = N (A  C) = 1 –  / ( (A N c (A | b  C) C) +  (A c  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

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

•

POSS



principle PROB: Laplace indifference

“ 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

p

(x) >

p

(x')

if and only if

p(x) > p(x') (order preservation

) :

Informational criteria

: from



to P: (Shapley value rather than maximal entropy) from P to



:

Preservation of symmetries 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

PI most specific element

(P) = {

p(x')

 :  of the set ≥ P} of possibility measures dominating P such that p

(x) >

p

(x')

iff

p(x) >

• • may be weakened into :

p(x) > p(x')

implies p

(x) >

p

(x')

The result

is p i =  j=i,…n p i (case of no ties)

From probability to possibility : Continuous case

• • The possibility distribution encodes then family of confidence intervals around the mode of p.

p obtained by transforming p

The



-cut of

p

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

confidence interval of any such p. contains the (1

- 

)-

From probability to possibility : Continuous case

• I L = {x, p(x) ≥  } = [a L , a L + L] is the interval of length L with maximal probability • The most specific possibility distribution dominating p is π such that  L > 0, π(a L ) = π(a L + L) = 1 – P(I L ).

 p a L L a + L L

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

( 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:

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

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

cardinality of the random set) , in the sense of a non-specificity index (expected 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  p i =  j=1,n min(p j , p i ).

• 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

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

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

|= p •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  -If A |= π B and C |= π B and A |= π A then C |= π B (

right weakening rule RW

) C then A |= π B (

Right AND

)  C

Properties of inference

|= p • 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 C c is false, then A

(rational monotony RM)

 C |= π B

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 2 S •Define an induced partial relation on subsets as A > B iff A  B |= B c for A ≠  x 2 S •

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  (A  B) >  (A c  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:  (A  B) =  (A  B c ) =  (A c  B c ) >  (A c  B).

MODELLING A SET OF DEFAULT RULES as a POSSIBILITY DISTRIBUTION • ∆ = {A i  B i , i = 1,n} • ∆

defines a set of constraints

distributions  (A i  B i ) >  (A i  on possibility ¬B i ), i = 1,…n 

•

 (∆) = set of feasible π's with respect to ∆ • O 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) >  (A c  B). •

Possibilistic inference

∆ |= * A  B iff  *(A  B) >  *(A c  B) for the least specific possibility measure in  (∆). Leads to a

stratification

of ∆ according to N*(A c  B)

Possibilistic logic

• A possibilistic knowledge base is an ordered set of propositional or 1st order formulas p i • • K = {(p i  i ), i = 1,n} where  i priority or validity of p i  i = 1 means certainty.

 i = 0 means ignorance > 0 is the level of

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(µ A (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

{(p i  i ), i = 1,n} • π K (s)= min i=1,n violated by s {1 –  i | s  [ p i ]} determined by the highest priority formula

The p. d.

π K

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 π K ≤ 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

: K = {(b then K   f {(b, 1)} |  ), (p (f  b   ) and K ), (p   ¬f  )} {(b, 1)} | nt • Inc(K  {(p, 1), (b, 1)} =  • K  {(p, 1), (b, 1)} | (¬f, min(  ,  )) • Hence K   < min(  ,  ) {(p, 1), (b, 1)} | nt ¬f f