Possibility Theory

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Possibility theory

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Possibility theory is a mathematical theory for dealing with certain types of

uncertainty and is an alternative toprobability theory. ProfessorLotfi Zadeh

first introduced possibility theory in 1978 as an extension of his theory of

fuzzy sets and fuzzy logic. D. Dubois and H. Prade further contributed to its

development. Earlier in the 50s, economist G.L.S. Shackle proposed the

min/max algebra to describe degrees of potential surprise.

Formalization of possibility

For simplicity, assume that the universe of discourse is a finite set, andassume that all subsets are measurable. A distribution of possibility is a

function from to [0, 1] such that:

Axiom 1:

Axiom 2:

Axiom 3: for any disjoint

subsets Uand V.

It follows that, like probability, the possibility measure on finite set isdetermined by its behavior on singletons:

provided Uis finite or countably infinite.

Axiom 1 can be interpreted as the assumption that is an exhaustive

description of future states of the world, because it means that no belief

weight is given to elements outside .

Axiom 2 could be interpreted as the assumption that the evidence from

which was constructed is free of any contradiction. Technically, it

implies that there is at least one element in with possibility 1.
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Axiom 3 corresponds to the additivity axiom in probabilities. However there

is an important practical difference. Possibility theory is computationally

more convenient because Axioms 1-3 imply that:

forany subsets Uand V.

Because one can know the possibility of the union from the possibility of

each component, it can be said that possibility iscompositionalwith respect

to the union operator. Note however that it is not compositional with respect

to the intersection operator. Generally:

Remark for the mathematicians:

When is not finite Axiom 3 can be replaced by:

For all index setsI, if the subsets are pairwise disjoint,

Necessity

Whereasprobability theory uses a single number, the probability, to describe

how likely an event is to occur, possibility theory uses two concepts, the

possibility and the necessity of the event. For any set U, the necessitymeasure is defined by

In the above formula, denotes the complement ofU, that is the elements

of that do not belong to U. It is straightforward to show that:

for any U

and that:

Note that contrary to probability theory, possibility is not self-dual. That is,

for any event U, we only have the inequality:
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However, the following duality rule holds:

For any event U, either , or

Accordingly, beliefs about an event can be represented by a number and a

bit.

Interpretation

There are four cases that can be interpreted as follows:

means that Uis necessary. Uis certainly true. It implies that

.

means that Uis impossible. Uis certainly false. It implies that

.

means that Uis possible. I would not be surprised at all ifU

occurs. It leaves unconstrained.

means that Uis unnecessary. I would not be surprised at all ifU

does not occur. It leaves unconstrained.

The intersection of the last two cases is and

meaning that I believe nothing at all about U. Because it allows for

indeterminacy like this, possibility theory relates to the graduation of a

many-valued logic, such as intuitionistic logic, rather than the classical two-

valued logic.

Note that unlike possibility, fuzzy logic is compositional with respect to both

the union and the intersection operator. The relationship with fuzzy theory

can be explained with the following classical example.

Fuzzy logic: When a bottle is half full, it can be said that the level of

truth of the proposition "The bottle is full" is 0.5. The word "full" is

seen as a fuzzy predicate describing the amount of liquid in the bottle.
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Possibility theory: There is one bottle, either completely full or totally

empty. The proposition "the possibility level that the bottle is full is

0.5" describes a degree of belief. One way to interpret 0.5 in that

proposition is to define its meaning as: I am ready to bet that it's

empty as long as the odds are even (1:1) or better, and I would not bet

at any rate that it's full.

Possibility theory as an imprecise probability theory

There is an extensive formal correspondence between probability and

possibility theories, where the addition operator corresponds to the

maximum operator.

A possibility measure can be seen as a consonantplausibility measure in

DempsterShafer theory of evidence. The operators of possibility theory canbe seen as a hyper-cautious version of the operators of thetransferable belief

model, a modern development of the theory of evidence.

Possibility can be seen as anupper probability: any possibility distribution

defines a unique set of admissible probability distributions by

This allows one to study possibility theory using the tools ofimprecise

probabilities.

Necessity logic

We callgeneralized possibility every function satisfying Axiom 1 and

Axiom 3. We callgeneralized necessity the dual of a generalized possibility.The generalized necessities are related with a very simple and interesting

fuzzy logic we call necessity logic. In the deduction apparatus of necessity

logic the logical axioms are the usual classical tautologies. Also, there is

only a fuzzy inference rule extending the usual Modus Ponens. Such a rule

says that if and are proved at degree and , respectively, then wecan assert at degree min{,}. It is easy to see that the theories of such a

logic are the generalized necessities and that the completely consistent

theories coincide with the necessities (see for example Gerla 2001).
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Possibility theory

Possibility theory is an uncertainty theory devoted to the handling of

incomplete information. As such, it complements probability theory. It

differs from the latter by the use of a pair of dual set-functions (possibility

and necessity measures) instead of only one. This feature makes it easier to

capture partial ignorance. Besides, it is not additive and makes sense on

ordinal structures. The name Theory of Possibility was coined in (Zadeh

1978), inspired by (Gaines and Kohout 1975). In Zadeh's view, possibility

distributions were meant to provide a graded semantics to natural language

statements. However, possibility and necessity measures can also be the

basis of a full-fledged representation of partial belief that parallels

probability (Dubois and Prade 1988). Then, it can be seen either as a coarse,

non-numerical version of probability theory, or as a framework for reasoning

with extreme probabilities, or yet as a simple approach to reasoning with

imprecise probabilities (Dubois, Nguyen and Prade, 2000).

Basic Notions

A possibility distribution is a mapping from a set of states of affairs S to

a totally ordered scale such as the unit interval . The functionrepresents the knowledge of an agent (about the actual state of affairs)

distinguishing what is plausible from what is less plausible, what is the

normal course of things from what is not, what is surprising from what is

expected. It represents a flexible restriction on what the actual state of affairs

is, with the following conventions:

means that state s is rejected as impossible;

means that state s is totally possible (= plausible or

unsurprizing).

If the state space is exhaustive, at least one of its elements should be the

actual world, so that at least one state is totally possible (normalisation).

Distinct values may simultaneously have a degree of possibility equal to 1.

Possibility theory is driven by the principle of minimal specificity. It states

that any hypothesis not known to be impossible cannot be ruled out. A
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possibility distribution is said to be at least as specific as another one if and

only if each state is at least as possible according to the latter as to the

former (Yager 1983). Then, the most specific one is the most restrictive and

informative.

In the possibilistic framework, extreme forms of partial knowledge can be

captured, namely:

Complete knowledge: for some state and for

other states s (only is possible)

Complete ignorance: , (all states are totally possible).

Given a simple query of the form does an event A occur?, where A is a

subset of states, or equivalently does the actual state lie in A, a response to

the query can be obtained by computing degrees of possibility andnecessity, respectively (if the possibility scale is ):

The possibility degree evaluates to what extent event A is consistent

with the knowledge , while evaluates to what extent is certainly

implied by the knowledge. The possibility-necessity duality is expressed by

where is the complement of Generally,

and . Possibility measures satisfy the basicmaxitivity property:

Necessity measures satisfy an axiom dual to that of possibility measures,

namely On infinite spaces, these axioms must

hold for infinite families of sets.

Human knowledge is often expressed in a declarative way using statements

to which some belief qualification is attached. Certainty-qualified pieces ofuncertain information of the form is certain to degree can then be

modelled by the constraint The least specific possibility

distribution reflecting this information assign possibility 1 to states where

is true and to states where A is false.


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Apart from , which represents the idea of potential possibility, another

measure ofguaranteed possibility can be defined (Dubois, Hajek and

Prade, 2000) : It estimates to what extent all states in are

actually possible according to evidence.

Notions ofconditioning and independence were studied for possibility

measures. Conditional possibility is defined similarly to probability theory

using a Bayesian like equation of the form (Dubois and Prade, 1988):

However, in the ordinal setting the operation cannot be a product and is

changed into the minimum. In the numerical setting, there are several ways

to define conditioning, not all of which have this form (Walley, 1996). There

are several variants of possibilistic independence (De Cooman, 1997;Dubois et al. 1997; De Campos and Huete, 1999). Generally, independence

in ordinal possibility theory is neither symmetric, nor insensitive to negation.

For non Boolean variables, independence between events is not equivalent

to independence between variables. Joint possibility distributions on

Cartesian products of domains can be represented by means of graphical

structures similar to Bayesian networks for joint probabilities (see Borgelt et

al. 2000; Benferhat Dubois Garcia and Prade 2002). Such graphical

structures can be taken advantage of for evidence propagation (Ben Amor et

al, 2003) orlearning (Borgelt and Kruse, 2003).

Historical Background

Zadeh was not the first scientist to speak about formalising notions of

possibility. The modalitiespossible and necessary have been used inphilosophy at least since the Middle-Ages in Europe, based on Aristotle's

works. More recently these concepts became the building blocks of modal

logic that emerged at the beginning of the XXth century. In this approach,

possibility and necessity are all-or-nothing notions, and handled at the

syntactic level. Independently from Zadeh's view, the notion of possibility asopposed to probability was central in the works of one economist, Shackle,

and is present in those of two philosophers, Cohen and Lewis:

The English economist G. L. S. Shackle (1962) introduced a full-

fledged approach to uncertainty and decision in the 1940-1970's based

on degrees of potential surprise of events. They are degrees of
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impossibility, that is, degrees of necessity of the opposite events. The

degree of surprise of an event made of a set of elementary hypotheses

is the degree of surprise of its least surprising realisation. Potential

surprise is understood as disbelief. The disbelief notion introduced

later in (Spohn 1990) employs the same type of convention as

potential surprise, using the set of ordinals as a disbelief scale.

Shackle also introduces a notion of conditional possibility. A

framework very similar to the one of Shackle was proposed by the

philosopher L. J. Cohen who considered the problem of legal

reasoning (Cohen, 1977).

The philosopher David Lewis introduced a graded notion of

possibility in the form of a weak order between possible worlds he

calls comparative possibility (Lewis, 1973). He relates this concept ofpossibility to a notion of similarity between possible worlds and a

most plausible one. Comparative possibility relations are instrumentalin defining the truth conditions of counterfactual statements.

Comparative possibility relations obey the key axiom, for all

events A, B, C: This axiom was later

independently proposed in (Dubois 1986) in an attempt to derive a

possibilistic counterpart to De Finetti and Savage comparative

probabilities.

Zadeh (1978) proposed an interpretation of membership functions of

fuzzy sets as possibility distributions encoding flexible constraintsinduced by natural language statements. Zadeh articulated the

relationship between possibility and probability, noticing that what is

probable must preliminarily be possible. However, the view of

possibility degrees developed in his paper refers to the idea of graded

feasibility (degrees of ease, as in the example ofhow many eggs can

Hans eat for his breakfast) rather than to the epistemic notion of

plausibility laid bare by Shackle. Nevertheless, the key axiom of

maxitivity for possibility measures is highlighted. Later on, Zadeh

(1979) acknowledged the connection between possibility theory,

belief functions and upper/lower probabilities, and proposed theirextensions to fuzzy events and fuzzy information granules.

This state of fact lays bare two branches of possibility theory: the qualitative

and the quantitative one (Dubois and Prade, 1998).

Qualitative possibility theory
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Qualitative possibility relations can be partially specified by a set of

constraints of the form , where A and B are events that may take the

form of logical formulae. If these constraints are consistent, this relation can

be completed through the principle of minimal specificity. A plausibility

ordering on S can be obtained by assigning to each state of affairs its highestpossibility level in agreement with the constraints (see Benferhat et al.1997,

1998). A general discussion on the relation between possibility relations and

partial orders on state of affairs is in (Halpern, 1997).

Qualitative possibility relations can be represented by (and only by)

possibility measures ranging on any totally ordered set (especially a finite

one, Dubois 1986). This absolute representation on an ordinal scale is

slightly more expressive than the purely relational one. When the finite set S

is large and generated by a propositional language, qualitative possibility

distributions can be efficiently encoded in possibilistic logic (Dubois, Lang,Prade, 1994). A possibilistic logic base K is a set of pairs , where is

a Boolean expression and takes on values on an ordinal scale. This pair

encodes the constraint where is the degree of necessity of the

set of models of . Each prioritized formula expresses a necessity-

qualified statement. It is interpreted as the least specific possibility

distribution on interpretations where this statement holds. Thus a prioritized

formula has a fuzzy set of models. The fuzzy intersection of the fuzzy sets of

models of all prioritized formulas in K yields an associated plausibility

ordering on S.

Syntactic deduction from a set of prioritized clauses is achieved by

refutation using an extension of the standard resolution rule, whereby

can be derived from and This rule,

which evaluates the validity of an inferred proposition by the validity of the

weakest premiss, goes back to Theophrastus, a disciple of Aristotle.

Possibilistic logic is an inconsistency-tolerant extension of propositional

logic that provides a natural semantic setting for mechanizing non-

monotonic reasoning (Benferhat, Dubois, Prade, 1998), with a

computational complexity close to that of propositional logic.

The main idea behind qualitative possibility theory is that the state of the

world is by default assumed to be as normal as possible, neglecting less

normal states. In particular, the events accepted as true are those true in all

the most plausible states, namely the ones with positive degrees of necessity

These assumptions lead us to interpret the plausible inference of a
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proposition B from another A, under knowledge as follows: B should be

true in all the most plausible states were A is true. It also means that the

agent considers B as an accepted belief in the context A. This kind of

inference is nonmonotonic in the sense that in the presence of additional

information C, B may fail to remain an accepted belief. This is similar to the

fact that a conditional probability may be low even if

is high. The properties of this consequence relation are now well-understood

(Benferhat, Dubois and Prade, 1997).

Decision-theoretic foundations of qualitative possibility were established by

Dubois et al. (2001) in the act-based setting of Savage. Two qualitative

decision rules under uncertainty can be justified: a pessimistic one and an

optimistic one, respectively generalizing Wald's maximin and maximax

criteria under ignorance, refined by accounting for a plausibility ordering on

the state space.

Quantitative possibility theory

The phrase "quantitative possibility" refers to the case when possibility

degrees range in the unit interval. In that case, a precise articulation between

possibility and probability theories is useful to provide an interpretation to

possibility and necessity degrees. Several such interpretations can be

consistently devised (see (Dubois, 2006) for a detailed survey):

a degree of possibility can be viewed as an upper probability bound(Walley, 1996; De Cooman and Aeyels, 1999), Especially,

probabilistic inequalities such as Chebychev one, can be interpreted as

defining possibility distributions (Dubois et al., 2004).

a possibility measure is also a special case of a plausibility function in

the theory of evidence (Shafer, 1987). It is then equivalent to a

consonant random set.

a possibility distribution can be viewed as a likelihood function

(Dubois, Moral and Prade, 1997).

Confidence or dispersion intervals are often extracted from statisticalinformation and are attached a confidence level like 0.95 per cent.

Varying this confidence level yields a family of nested intervals that

can be represented as a possibility distribution (Dubois et al. 2004).

Following a very different approach, possibility theory can represent

probability distributions with extreme, infinitesimal values (Spohn,

1990).
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The theory of large deviations in probability theory also handles set-

functions that look like possibility measures (Nguyen and Bouchon-

Meunier, 2003).

There are finally close connections between possibility theory and

idempotent analysis (Kolokoltsov and Maslov, 1997).

Applications

Possibility theory has not been the main framework for engineering

applications of fuzzy sets in the past. However, two directions where it

proved useful can be highlighted:

Interval analysis has been generalized in the setting of possibility

theory. This is the calculus of fuzzy numbers; see (Dubois, Kerre,

Mesiar and Prade, 2000) for a survey. It is then analogous to random

variable calculations. Finding the potential of possibilistic

representations in computing conservative bounds for probabilistic

calculations is certainly a major challenge.

Possibility theory also offers a framework for preference modeling in

constraint-directed reasoning. Both prioritized and soft constraints can

be captured by possibility distributions expressing degrees of

feasibility rather than plausibility (Dubois, Fargier and Prade, 1996).

Other applications of possibility theory can be found in fields such as data

analysis (Wolkenhauer, 1998; Tanaka and Guo, 1999), structural learning

(Borgelt et al., 2000), database querying (Bosc and Prade, 1997), diagnosis

(Cayrac et al., 1996), belief revision (Benferhat, Dubois, Prade and

Williams, 2002), argumentation (Amgoud and Prade, 2004), case-based

reasoning (Huellermeier, 2007).

Lastly, possibility theory is also being studied from the point of view of its

relevance in cognitive psychology. Experimental results in cognitivepsychology (Raufaste et al., 2003) suggest that there are situations where

people reason about uncertainty using the rules or possibility theory, rather

than with those of probability theory.
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Exploring Extensions of Possibilistic Logic overGodel LogicPilar Dellunde1,2, Llus Godo2, and EnricoMarchioni2

1 Universitat Aut`onoma de Barcelona,08193 Bellaterra, [email protected] IIIA CSIC,08193 Bellaterra, Spain{enrico,pilar,godo}@iiia.csic.esAbstract. In this paper we present completenessresults of several fuzzy

logics trying to capture different notions of necessity(in the sense ofPossibility theory) for Godel logic formulas. In a firstattempt, based ondifferent characterizations of necessity measures onfuzzy sets, a group oflogics, with Kripke style semantics, are built over arestricted language,

indeed a two level language composed of non-modaland modal formulas,the latter moreover not allowing for nestedapplications of the modaloperator N. Besides, a full fuzzy modal logic forgraded necessity overGodel logic is also introduced together with analgebraic semantics, the

class of NG-algebras.1 IntroductionThe most general notion of uncertainty is capturedby monotone set functionswith two natural boundary conditions. In theliterature, these functions have


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received several names, like Sugeno measures [24]or plausibility measures [20].Many popular uncertainty measures, likeprobabilities, upper and lower probabilities,

Dempster-Shafer plausibility and belief functions, orpossibility andnecessity measures, can be therefore seen asparticular classes of Sugeno measures.In this paper, we specially focus on possibilisticmodels of uncertainty. A possibilitymeasure on a complete Boolean algebra of events U= (U, !, ", , 0U, 1U)

is a Sugeno measure ! satisfiying the following "-decomposition property forany countable set of indices I!("i"I ui) = supi"I!(ui),while a necessity measure is a Sugeno measure !satisfying the !-decomposition

property!(!i"I ui) = infi"I!(ui).Possibility and necessity are dual classes ofmeasures, in the sense that if !is a possibility measure, then the mapping !(u) = 1 !(u) is a necessity

C. Sossai and G. Chemello (Eds.): ECSQARU 2009,LNAI 5590, pp. 923934, 2009.#c Springer-Verlag Berlin Heidelberg 2009924 P. Dellunde, L. Godo, and E. Marchionimeasure, and vice versa. If U is the power set of aset X, then any dual pair of


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to fuzzy events, see e.g. [7,9,16,3,2,4]. All thenotions of necessity for fuzzy setsconsidered in the literature turn out to be of the formN(A) = infx"U !(x) ( A(x) (*)

where A is a fuzzy set in some domain U, ! : U $ [0,1] is a possibility distributionon U and ( is some suitable many-valued implicationfunction. Inparticular, the following notions of necessity havebeen discussed:(1) x (KD y = max(1 x, y) (Kleene-Dienes);(2) x (RG y = 1 if x ) y, and x (RG y = 1x otherwise

(reciprocal of Godel);(3) x (!L y = min(1, 1 x + y) ("Lukasiewicz).All these definitions actually extend the abovedefinition over classical sets orevents.In the literature different logical formalizations toreason about such extensionsof the necessity of fuzzy events can be found. In [19],

and later in [17], a fullmany-valued modal approach is developed over thefinitely-valued "Lukasiewiczlogic in order to capture the notion of necessitydefined using (KD. A logicprogramming approach over Godel logic isinvestigated in [3] and in [2] by relyingon (KD and (RG, respectively. More recently,

following the approach of[12], modal-like logics to reason about the necessityof fuzzy events in the frameworkof MV-algebras have been defined in [13], in order tocapture the notion ofnecessity defined by (KD and (!L.


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The purpose of this paper is to explore differentlogical approaches to reasonabout the necessity of fuzzy events over Godelalgebras. In more concrete terms,

our ultimate aim is to study a full modal expansion ofthe [0, 1]-valued Godel logicwith a modality N such that the truth-value of aformula N" (in [0, 1]) can beExploring Extensions of Possibilistic Logic overGodel Logic 925interpreted as the degree of necessity of ", accordingto some suitable semantics.

In this context, although this does not extend theclassical possibilistic logic,it seems also interesting to investigate the notion ofnecessity definable fromGodel implication, which is the standard fuzzyinterpretation of the implicationconnective in Godel logic:(4) x (G y = 1 if x ) y, and x (G y = y otherwise

(Godel); This work is structured as follows. After thisintroduction, in Section 2 werecall a characterization of necessity measures onfuzzy sets defined by implications(KD and (RG and provide a (new) characterization ofthose defined by(G. These characterizations are the basis for the

completeness results of severallogics introduced in Section 3 capturing thecorresponding notions of necessityfor Godel logic formulas. These logics, with Kripkestyle semantics, are built over


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a two-level language composed of modal and non-modal formulas, moreover thelatter not allowing nested applications of the modaloperator. In Section 4 a

full fuzzy modal logic for graded necessity overGodel logic is introduced togetherwith an algebraic semantics. Finally, in Section 5 wemention some openproblems and new research goals we plan to addressin the near future.Due to lack of space, we cannot include preliminarieson basic notions regarding

Godel logic and its expansions with truth-constants,with Monteiro-Baazsoperator # and with an involutive negation, that willbe used throughout thepaper. Instead, the reader is referred to [17,10,11]for the necessary background.2 Some Necessity Measures over Godel Algebras ofFuzzy Sets and Their Characterizations

Let X be a (finite) set and let F(X) = [0, 1]X be the setof fuzzy sets overX, i.e. the set of functions f : X $ [0, 1]. F(X) can beregarded as a Godelalgebra equipped with the pointwise extensions ofthe operations of the standardGodel algebra [0, 1]G. In the following, for each r &[0, 1], we will denote by r

the constant function r(x) = r for all x & X.Definition 1. A mapping N : F(X) $ [0, 1] satisfying(N1) N(!i"Ifi) = infi"I N(fi)(N2) N(r) = r, for all r & [0, 1]is called a basic necessity.


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If N : F(X) $ [0, 1] is a basic necessity then it is easyto check that it alsosatisfies the following properties:(i) min(N(f),N(Gf)) = 0

(ii) N(f (G g) ) N(f) (G N(g) The classes of necessity measures based on theKleene-Dienes implicationand the reciprocal of Godel implication have beenalready characterized in theliterature. We do not consider here the one based on"Lukasiewicz implication.926 P. Dellunde, L. Godo, and E. Marchioni

Lemma 2 ([3,2]). Let N : F(X) $ [0, 1] be a basicnecessity. Consider thefollowing properties:(NKD) N(r (KD f) = r (KD N(f), for all r & [0, 1](NRG) N(r (RG f) = r (RG N(f), for all r & [0, 1]

Then, we have:(1) N satisfies (NKD) iff N(f) = infx"X !(x) (KD f(x)(2) N satisfies (NRG) iff N(f) = infx"X !(x) (RG f(x)

for some possibility distribution ! : X $ [0, 1] suchthat supx"X !(x) = 1. The characterization of the necessity measuresbased on Godel implication issomewhat more complex since it needs to consideralso an associated class ofpossibility measures which are not dual in the usualstrong sense.

Definition 3. A mapping $ : F(X) $ [0, 1] satisfying($1) $("i"Ifi) = supi"I $(fi)($2) $(r) = r, for all r & [0, 1]is called a basic possibility.Note that if $ : F(X) $ [0, 1] is a basic possibility thenit also satisfies


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max($(f),$(f)) = 1.For each x & X, let us denote by x its characteristicfunction, i.e. the functionfrom F(X) such that x(y) = 1 if y = x and x(y) = 0

otherwise. Observe thateach f & F(X) can be written asf =!x"Xx (G f(x) ="x"X

x ! f(x).Therefore, if N and $ are a pair of basic necessity andpossibility on F(X)respectively, by (N1) and ($1) we haveN(f) = infx"XN(x (G f(x)) and $(f) = supx"X

$(x ! f(x)).Then we obtain the following characterizations.Proposition 4. Let $ : F(X) $ [0, 1] be a basicpossibility. $ further satisfies($3) $(f ! r) = min($(f), r), for all r & [0, 1]iff there exists ! : X $ [0, 1] such that supx"X !(x) = 1and, for all f & F(X),$(f) = supx"X min(!(x), f(x)).

Proof. One direction is easy. Conversely, assume that$ : F(X) $ [0, 1] satisfies($1) and ($3). Then, taking into account the aboveobservations, we have$(f) = supx"X


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$(x ! f(x)) = supx"Xmin($(x), f(x)).Hence, the claim easily follows by defining !(x) = $(x)

. !Exploring Extensions of Possibilistic Logic overGodel Logic 927Proposition 5. Let N : F(X) $ [0, 1] be a basicnecessity and $ : F(X) $ [0, 1] be a basic possibilitysatisfying ($3). N and $ further satisfy($N) N(f (G r) = $(f) (G r, for all r & [0, 1]iff there exists ! : X $ [0, 1] such that supx"X !(x) = 1

andN(f) = infx"X!(x) (G f(x) and $(f) = supx"Xmin(!(x), f(x)).Proof. As for the possibility $, this is already shownabove in Proposition 4. Let

N be defined as N(f) = infx"X !(x) (G f(x) for thepossibility distribution! : F(X) $ [0, 1] determined by $. We have N(f (G r) =infx"X(!(x) (G(f(x) (G r)) = infx"X((!(x) ! f(x)) (G r) = (supx"X !(x) !f(x)) (G r =$(f) (G r. Hence, $ and N satisfy ($N).Conversely, suppose that N and $ satisfy ($N). Then,

using the fact that$(x) = !(x) for each x & X, we have N(f) = infx"X N(x(G f(x)) =infx"X $(x) (G f(x) = infx"X !(x) (G f(x). !3 Four Complete Logics: The Two-Level LanguageApproach


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The language of the logics we are going to considerin this section consists oftwo classes of formulas:(i) The set Fm(V ) of non-modal formulas ", % . . .,

which are formulas of G!(Q)(Godel logic G expanded with Baazs projectionconnective # and truthconstants r for each rational r & [0, 1]) built from theset of propositionalvariables V = {p1, p2, . . .};(ii) And the set MFm(V ) of modal formulas &, ' . . .,built from atomic modal

formulas N", with " & Fm(V ), where N denotes themodality necessity,using the connectives from G! and truth constants rfor each rational r & [0, 1]. Notice that nestedmodalities are not allowed.

The axioms of the logic NG0 of basic necessity arethe axioms of G!(Q) fornon-modal and modal formulas plus the following

necessity related modalaxioms:(N1) N(" $ %) $ (N" $ N%)(N2) N(r) * r, for each r & [0, 1] + Q.

The rules of inference of NG0 are modus ponens (formodal and non-modalformulas) and necessitation: from , " infer , N".It is worth noting that NG0 proves the formula N(" !

%) * (N" ! N%),which encodes a characteristic property of necessitymeasures.928 P. Dellunde, L. Godo, and E. MarchioniAs for the semantics we consider several classes ofpossibilistic Kripke models.


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A basic necessity Kripke model is a system M= -W, e,I. where: W is a non-empty set whose elements are callednodes or worlds,

e : W V $ [0, 1] is such that, for each w & W, e(w,) : V $ [0, 1]is an evaluation of propositional variables which isextended to a G!(Q)-evaluation of (non-modal) formulas of Fm(V ) in theusual way. For each " & Fm(V ) we define its associatedfunction "W : W $ [0, 1],

where "W(w) = e(w, "). Let #Fm = { " | " & Fm(V )} I : #Fm $ [0, 1] is a basic necessity over #Fm (as aG-algebra), i.e. it satisfies(i) I(rW) = r, for all r & [0, 1] + Q(ii) I(!i"I"iW) = infi"I I( "iW).Now, given a modal formula &, the truth value of & inM = -W, e, I., denoted0&0M, is inductively defined as follows:

If & is an atomic modal formula N", then 0N"0M = I("W) If & is a non-atomic modal formula, then its truthvalue is computed by evaluatingits atomic modal subformulas, and then by using thetruth functionsassociated to the G!(Q) connectives occurring in &.We will denote by N the class of basic necessity

Kripke models. Then, takinginto account that G!(Q)-algebras are locally finite,following the same approachof [13] with the necessary modifications, one canprove the following result.


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Theorem 6. NG0 is sound and complete for modaltheories w.r.t. the class N of basic necessitystructures.Now our aim is to consider extensions of NG0 which

faithfully capture thethree different notions of necessity measureconsidered in the previous section.We start by considering the following additionalaxiom:(NKD) N(r " ") * (r "N"), for each r & [0, 1] + Q.Let NGKD be the axiomatic extension of NG0 with(NKD). Then, using Lemma

2, it is easy to prove that indeed NGKD captures thereasoning about KDnecessitymeasures.

Theorem 7. NKD is sound and complete for modaltheories w.r.t. the subclassNKD of necessity structures M = -W, e, I. such thatthe necessity measure I isdefined as, for every " & Fm(V ), I( "W) = infw"W !

(w) (KD "W(w) for somepossibility distribution ! : W $ [0, 1] on the set ofpossible worlds W.

To capture RG-necessities, we need to expand thebase logic G!(Q) with aninvolutive negation 1. This corresponds to the logicG$(Q), as defined in [10].So we define NGRG as the axiomatic extension of

NG0 over G$(Q) (instead ofover G!(Q)) with the following axiom:(NRG) N(1" $ 1 r) * (1N" $ 1 r), for each r & [0, 1]+ Q.Exploring Extensions of Possibilistic Logic overGodel Logic 929


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Then, using again Lemma 2 and the fact that also G$(Q)-algebras are locallyfinite, one can also prove the following result.

Theorem 8. NGRG is sound and complete for modal

theories w.r.t. the subclassNRG of necessity structures1 M= -W, e, I. such thatthe necessity measure I isdefined as, for every " & Fm(V ), I( "W) = infw"W !(w) (RG "W(w) for somepossibility distribution ! : W $ [0, 1] on the set ofpossible worlds W.It is worth pointing out that if we added the Boolean

axiom " " " to thelogics NKD and NRG, both extensions would basicallycollapse into the classicalpossibilistic logic.Finally, to define a logic capturing NG-necessities, weneed to expand thelanguage of NG0 with an additional operator $ tocapture the associated

possibility measures according to Proposition 5.Therefore we consider theextended set MFm(V )+ of modal formulas &, ' . . . asthose built from atomicmodal formulas N" and $", with " & Fm(V ), truth-constants r for eachr & [0, 1] + Q and G! connectives. Then the axiomsof the logic N$G are

those of G!(Q) for non-modal and modal formulas,plus the following necessityrelated modal axioms:(N1) N(" $ %) $ (N" $ N%)(N2) N(r) * r,($1) $(" " %) * ($" " $%)


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($2) $(r) * r,($3) $(" ! r) * ($" ! r)(N$) N(" $ r) * ($" $ r)where (N2), ($2), ($3) and (N$) hold for each r & [0,

1] + Q. Inference rulesof N$G are those of G!(Q) and necessitation for N and$.Now, we also need to consider expanded Kripkestructures of the form M=-W, e, I, P., where W and e are as above and themappings I, P :$ [0, 1] are suchthat, for every " & Fm(V ), I( "W) = infw"W !(w) (G

"W(w) and P( "W) =supw"W min(!(w), "W(w)), for some possibilitydistribution ! : W $ [0, 1]. CallNPG the class for such structures. Then, usingProposition 5 we get the followingresult.

Theorem 9. N$G is sound and complete for modaltheories w.r.t. the class

NPG of structures.4 Possibilistic Necessity Godel Logic and ItsAlgebraicSemantics: The Full Modal Approach

The logics defined in the previous section are notproper modal logics sincethe notion of well-formed formula excludes thoseformulas with occurrences of

1 With the proviso that the evaluations e ofpropositional variables extend to G!(Q)-evaluations for non-modal formulas and not simply toG!(Q)-evaluations.930 P. Dellunde, L. Godo, and E. Marchioni


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nested modalities. Our aim in this section is then toexplore a full (fuzzy) modalapproach.We start as simple as possible by defining a fuzzy

modal logic over Godelpropositional logic G to reason about the necessitydegree of G-propositions.

The language of Possibilistic Necessity Godel logic,PNG, is defined as follows:formulas of PNG are built from the set of G-formulasusing G-connectives andthe operator N. Axioms of PNG are those of Godel

logic plus the followingmodal axioms:1. N(" $ %) $ (N" $ N%).2. N% * NN%.3. N0.Deduction rules for PNG are Modus Ponens andNecessitation for N (from %derive N%). These axioms and rules define a notion

of proof ,PNG in the usualway.Notice that in PNG the Congruence Rule from " * %derive N" * N%as well as the theorems N1 and N(" ! %) * (N" ! N%)are derivable. Alsoobserve that, if we had restricted the NecessitationRule only to theorems, we

would have obtained a local consequence relation(instead of the global onewe have introduced here). For this weaker version ofthe logic, the Deduction

Theorem in its usual form would holds, neverthelessthis logic turns out not to


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be algebraizable.Theorem 10. [Deduction Theorem] If T 2{", (} is anyset of PNG-formulas,then T 2 {"} ,PNG ( iff T ,PNG (" !N") $ (.

Kripke style semantics based on possibilisticstructures (W, e, I) could be alsodefined as in Section 3, but now the situation is morecomplex due to the factthat we are dealing with a full modal language.Moreover, it seems even morecomplex to try to get some completeness resultswith respect to this semantics

so this is left for future research. This is the reasonwhy in the rest of the paperwe will turn our attention to the study of an algebraicsemantics, following theideas developed in [15,14] for the case of aprobabilistic logic over "Lukasiewiczlogic, and see how far we can go.We start by defining a suitable class of algebras

which are expansions of Godelalgebras with a new unary operator trying to capturethe notion of necessity.Definition 11. An NG-algebra is a structure (A,N)where A is a G-algebraand N : A $ A is a monadic operator such that:1. N(x ( y) ( (Nx ( Ny) = 12. Nx = NNx

3. N1 = 1The function N is called an internal possibilistic stateon the G-algebra A.Observe that, so defined, the class of NG-algebras isa variety. Examples of


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internal possibilistic states are the identity functionId, the # operator and theExploring Extensions of Possibilistic Logic overGodel Logic 931

operator. The variety of G-algebras can beconsidered as a subvariety of NGalgebras,namely the subvariety obtained by adding theequation N(x) = x. Itis easy to check, using the definition of NG-algebrathat, for every NG-algebra(A,N) such that N(A) = A we have N = Id, and that,given a, b & A, a ) b

implies Na ) Nb.Definition 12. An NG-filter F on an NG-algebra (A,N) isa filter on the GalgebraA with the following property: if a & F, then Na & F.By an argument analogous to the one in Lemma2.3.14 of [17], if 1F is therelation defined by: for every a, b & A, a 1F b iff (a (b) & F and (b ( a) & F,

then 1F is a congruence on (A,N) and the quotientalgebra (A,N)/ 1F is anNG-algebra.Lemma 13. Let F be an NG-filter on an NG-algebra(A,N). Then, the leastNG-filter containing F as a subset and a given a & AisF% = {u & A : 3v & F such that u 4 v 5 a 5 Na}

By Corollary 4.8 of [5], it is easy to check that PNG isfinitely algebraizable andthat the equivalent algebraic semantics of PNG is thevariety of NG-algebras.As a corollary we obtain the following generalcompleteness result.


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Theorem 14. The logic PNG is strongly complete withrespect to the variety ofNG-algebras. This means that for any set of formulas) 2 {&}, ) ,PNG & iff,

for all NG-algebra A and for all evaluation e on A, ife(') = 1A for all ' & ),then e(&) = 1A.Observe that it is not possible to prove completenesswith respect to linearlyordered NG-algebras. Otherwise N(& " ') * (N& "N')would be a theorem.Now we prove some satisfiability results of formulas

of the logic PNG.Formulas of the language of PNG can be seen also asterms of the language ofNG-algebras. Therefore for the sake of clarity, in thefollowing proofs we workwith first-order formulas of the language of NG-algebras proving that they aresatisfiable, if the corresponding formulas of the

language of PNG are satisfiable.Proposition 15. Let ((x1, . . . , xn) be a PNG-formula. If( is satisfiable, then( = 1 is satisfiable in an NG-algebra (B,*), by asequence (b1, . . . , bn) of elementsof B such that, for every 0 < i ) n, we have either bi =1 or *(bi) = 0.Proof. Let (A,N) be an NG-algebra such that ( = 1 is

satisfiable in (A,N) by(a1, . . . , an). Without loss of generality we assumethat there is k ) n such thatfor every 0 < i ) k, N(ai) '= 0 and for i > k, N(ai) = 0.Now we build a finite sequence of NG-algebras(B1, . . . ,Bk) and homomorphisms


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(h1, . . . , hk) such that for every 0 < i ) k, ( issatisfied in Bi by(c1, . . . , ci1, hi 6 hi1 6 6 h1(ai), . . . , hi 6hi1 6 6 h1(an))

where each ci & {0, 1}. We define only the firsthomomorphism, the others canbe introduced analogously. Let F = {x & A : Nx 4Na1}. So defined, it is easy932 P. Dellunde, L. Godo, and E. Marchionito check that F is an NG-filter. And since, by aprevious assumption, Na1 '= 0,the filter F is proper. Thus, (A,N)/ 1F is an NG-

algebra. Now let h1 be thecanonical homomorphism from (A,N) to (A,N)/1F ,and let B1 = (A,N)/1F .It is easy to check that ( = 1 is satisfied in B1 by(h1(a1), . . . , h1(an)), thath1(a1) = 1 and that for i > k, N(h1(ai)) = 0. Finally,take (B,*) = (Bk, hk 6 6 h1 6 N). !

Definition 16. An unnested atomic formula of thelanguage of NG-algebras,is an atomic formula of one of the following fourforms: x = y, c = y (where cis a constant c & {0, 1}), Nx = y or F(x) = y (forsome function symbol F ofthe language of the Godel algebras).Lemma 17. Let ( be a term of the language of NG-

algebras. Then there is aset )" of unnested atomic formulas such that, forevery NG-algebra (A,N):( = 1 is satisfiable in (A,N) iff )" is satisfiable in (A,N).Proof. It is a direct consequence of Theorem 2.6.1 of[21]. !


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Example: Let ( be the term x1 " N(x2 ( N(x3 ( 0)),take )" to be thefollowing set of unnested atomic formulas:{x1 " y = z,1 = z,Nw = y, (x2 ( v) = w,Nq = v, (x3 (

p) = q,0 = p}Theorem 18. Let ((x1, . . . , xn) be a PNG-formula. If (is satisfiable, then ( =1 is satisfiable in the NG-algebra ([0, 1]G,#) by asequence of rational numbers.Proof. Let (A,*) be an NG-algebra in which ((x1, . . . ,xn) = 1 is satisfiable byan n-tuple (a1, . . . , an). Without loss of generality

we may assume that: ( is a conjunction of unnested atomic formulas (byusing Lemma 17); for every 0 < i ) n, ai '= 0 and ai '= 1 (otherwise wecan work with theformulas ((xi/1) or ((xi/0), by substituting thecorresponding variables bythe constants 0 or 1);

for every i, we have *(ai) = 0 (by Proposition 15).Now we consider the unnested conjuncts of (. For thesake of simplicity, assumethat there is k ) n such that only in case that 0 < i ) k,the variable xi has anoccurrence in an unnested atomic formula of theform Nxi = y. We work nowwith the formula + = ((Nxi/0), obtained by

substituting in ( all the occurrencesof Nxi by the constant 0, for every 0 < i ) k.Observe that, so defined, + is a conjunction ofunnested atomic formulasin the language of the G-algebras which is satisfied in(A,*) by (a1, . . . , an).


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Therefore, the conjunction +0 = + !$0


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formulas for both NGKD, NGRG and N$G. Given theresults in [18],we conjecture that the problem of checkingsatisfiability for those logics is in

PSPACE. As for PNG, notice that from the results inthe above section and thefact that satisfiability for G! is an NP-completeproblem (easily derivable from[17]), we immediately obtain that the set ofsatisfiable PNG-formulas is in NP.Acknowledgments. The authors are grateful to theanonymous referees for their

valuable comments for improving the final version ofthis paper. They also acknowledgepartial support from the Spanish projects MULOG2(TIN2007-68005-C04) and Agreement Technologies (CONSOLIDERCSD2007-0022, INGENIO2010), as well as the ESF Eurocores-LogICCC/MICINNproject (FFI2008-03126-

E/FILO). Marchioni also acknowledges partial supportof the Juan de la CiervaProgram of the Spanish MICINN.References1. Aguzzoli, S., Gerla, B., Marra, V.: De Finettis no-Dutch-book criterion for Godellogic. Studia Logica 90, 2541 (2008)2. Alsinet, T.: Logic Programming with Fuzzy

Unification and Imprecise Constants:Possibilistic Semantics and Automated Deduction.Monografies de lInstitutdInvestigacio en Intellig`encia Artificial, CSIC,Barcelona (2003)


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3. Alsinet, T., Godo, L., Sandri, S.: On the Semanticsand Automated Deduction forPLFC: a logic of possibilistic uncertainty andfuzziness. In: Proc. of 15th Conference

on Uncertainty in Artificial Intelligence ConferenceUAI 1999, Stokholm, Sweden,pp. 312 (1999)4. Alsinet, T., Godo, L., Sandri, S.: Two formalisms ofextended possibilistic logicprogramming with context-dependent fuzzyunification: a comparative description.Electr. Notes Theor. Comput. Sci. 66(5) (2002)

5. Blok, W.J., Pigozzi, D.: Algebraizable logics.Memoirs of the American MathematicalSociety A.M.S. 396 (1989)6. Burris, S., Sankappanavar, H.P.: A Course inUniversal Algebra. Graduate textsin mathematics, vol. 78. Springer, Heidelberg (1981)7. Dubois, D., Prade, H.: Possibility theory. PlenumPress, New York (1988)

8. Dubois, D., Prade, H.: Resolution principles inpossibilistic logic. InternationalJournal of Approximate Reasoning 4(1), 121 (1990)934 P. Dellunde, L. Godo, and E. Marchioni9. Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In:Gabbay, et al. (eds.) Handbookof Logic in Artificial Intelligence and LogicProgramming. Nonmonotonic

Reasoning and Uncertain Reasoning, vol. 3, pp. 439513. Oxford University Press,Oxford (1994)10. Esteva, F., Godo, L., Hajek, P., Navara, M.:Residuated fuzzy logics with an involutive


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negation. Archive for Mathematical Logic 39(2), 103124 (2000)11. Esteva, F., Gispert, J., Godo, L., Noguera, C.:Adding truth-constants to logics

of continuous t-norms: Axiomatization andcompleteness results. Fuzzy Sets andSystems 158, 597618 (2007)12. Flaminio, T., Godo, L.: A logic for reasoning aboutthe probability of fuzzy events.Fuzzy Sets and Systems 158, 625638 (2007)13. Flaminio, T., Godo, L., Marchioni, E.: On theLogical Formalization of Possibilistic

Counterparts of States over n-Valued "LukasiewiczEvents. Journal of Logic andComputation (in press), doi:10.1093/logcom/exp01214. Flaminio, T., Montagna, F.: MV-algebras withinternal states and probabilisticfuzzy logics. International Journal of ApproximateReasoning 50, 138152 (2009)15. Flaminio, T., Montagna, F.: An algebraic approach

to states on MV-algebras. In:Proc. of EUSFLAT 2007, Ostrava, Czech Republic, vol.II, pp. 201206 (2007)16. Godo, L., Vila, L.: Possibilistic temporal reasoningbased on fuzzy temporal constraints.In: Proceedings of IJCAI 1995, pp. 19161922. MorganKaufmann, SanFrancisco (1995)

17. Hajek, P.: Metamathematics of Fuzzy Logic.Trends in Logic, vol. 4. Kluwer AcademicPublishers, Dordrecht (1998)18. Hajek, P.: Complexity of fuzzy probability logicsII. Fuzzy Sets and Systems 158,26052611 (2007)


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19. Hajek, P., Harmancova, D., Esteva, F., Garcia,P., Godo, L.: On Modal Logicsfor Qualitative Possibility in a Fuzzy Setting. In: Proc.of the 94 Uncertainty in

Artificial Intelligence Conference (UAI 1994), pp. 278285. Morgan Kaufmann, SanFrancisco (1994)20. Halpern, J.Y.: Reasoning about uncertainty. MITPress, Cambridge (2003)21. Hodges, W.: Model Theory. Encyclopedia ofMathematics and its Applications,vol. 42. Cambridge University Press, Cambridge

(1993)22. Mundici, D.: Averaging the truth-value in"Lukasiewicz logic. Studia Logica 55(1),113127 (1995)23. Shafer, G.: A mathematical theory of evidence.Princeton University Press, Princeton(1976)24. Sugeno, M.: Theory of Fuzzy Integrals and its

Applications. PhD thesis, Tokyo

Institute of Technology, Tokio, Japan (1974)

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