Desire is quite different from belief. While the accumulation of beliefs tend to reduce the remaining possible worlds they point at, the accumulation of desires tend to increase the set of states of affairs tentatively considered as satisfactory. Indeed beliefs are expected to be closed under conjunctions, while one can argue that endorsing ' _ as a desire means to desire both ' and . Still desiring ' and :' at the same time is not usually regarded as rational, since it does not make much sense to desire one thing and its contrary at the same time. Thus when a new desire is added to the set of desires of an agent, a revision process may be necessary. Just as belief revision relies on an epistemic entrenchment relation, desire relation is based on a hedonic entrenchment relation satisfying other properties, due to the different natures of belief and desire. Epistemic entrenchment relations are known to be qualitative necessity relations. In this paper it is shown that a well-behaved desire revision operation obeying a set of reasonable postulates is underlied by a qualitative guaranteed possibility relation in the sense of possibility theory. Then the general framework of possibilistic logic provides a syntactic setting for encoding desire change.
Desires constitute the primitive form of motivational attitude that
drives an agent to plan her action aimed at satisfying them.
Specifically, taking into account her beliefs about the world, the agent
chooses what to do in the pursuit of her desires. The result of the
agent’s choice constitutes her intentions to which she is then
committed . Such a simplified schema is for instance advocated in [
Desires and intentions are sometimes used more or less
interchangeably in the literature. However, desires and intentions should
be carefully distinguished. For instance, let us reconsider an example
adapted from [
In this paper, we do not consider intentions, but only desires. More precisely, we consider positive desires only, namely those that it would be really satisfactory to concretize, as opposed to negative desires corresponding to situations to be avoided because they are unsatisfactory, unbearable for the agent.
We advocate that an agent cannot simply cumulate desires without never making any revision, since it does not make sense to desire everything (at least according to the wisdom of mankind). This means that sometimes an agent has to revise her desires, not on the basis of some believed information about the state of the world which would trigger a change of intention, but just because the acceptance of a new desire altogether with her previous desires would lead her to desire everything and its contrary.
Such a situation is clearly similar to the revision of her beliefs by an agent receiving a new piece of information that she considers to be true, since she has to preserve the consistency of her beliefs. But desires and beliefs behave differently. Indeed, while believing ' and believing amounts to believing ' ^ , both desiring ' and desiring amounts to desiring ' _ , and conversely.
The difference of behavior between desire and belief has been
pointed out by several authors [
The paper is organized as follows. In Section 2, we highlight the main intuitions behind the concept of desire change in contrast with the concept of belief change, from a philosophical and AI perspective. Section 3 introduces the idea of a hedonic entrenchment relation that rank-order desires, and provide axioms for such a relation, whose unique numerical counterpart is a guaranteed possibility distribution, associated with a guaranteed possibility measure in the sense of possibility theory. In Section 4, desires are then represented in this setting. The guaranteed possibility distribution enables us to associate any set of desires with a level of unacceptability, which is the counterpart of the level of inconsistency for a set of beliefs represented in a possibilistic logic manner. Section 5 provides axioms for desire revision and Section 6 presents the revision of sets of prioritized desires axiomatically, semantically, and syntactically using a special type of possibilistic logic. Expansion and contraction of desires are also characterized and discussed.
An important and general distinction in philosophy of mind is
between epistemic attitudes and motivational attitudes. This distinction
is in terms of the direction of fit of mental attitudes to the world.
While epistemic attitudes aim at being true and their being true is
their fitting the world, motivational attitudes aim at realization and
their realization is the world fitting them [
Beliefs are mental representations aimed at representing how the physical, mental and social worlds are. In contrast, following the Humean conception, a desire can be viewed as an agent’s attitude consisting in an anticipatory mental representation of a pleasant state of affairs (representational dimension of desires) that motivates the agent to achieve it (motivational dimension of desires). The motivational dimension of an agent’s desire is realized through its representational dimension, in the sense that, a desire motivates an agent to achieve it because the agent’s representation of the desire’s content gives her anticipatory pleasure, following John Locke’s intuition. For example when an agent desires to eat sushi, she imagines herself eating sushi and this representation gives her pleasure. This pleasant representation motivates her to go to the “The Japoyaki” restaurant in order to eat sushi.
Desire and belief have also different origins. Belief revision is triggered either via direct sensing from the external environment (e.g., I believe that there is a fire in the house since I can see it) or via communication (e.g., I believe that there is a fire in the house since you told me this and I trust what you say). Desire change is triggered under other conditions. In the case of human agents, these conditions might be physiological or epistemic. For example, the desire of drinking a glass of water could be activated by the feeling of thirst (physiological condition) and the desire of going outside for a walk might be activated by the belief that it is a sunny day (epistemic condition). In the case of artificial agents, conditions of desire activation should be specified by the system’s designer. For example, a robotic assistant who has to take care of an old person could be designed in such a way that every day at 4 pm the desire of giving a medicine to the old person is activated in its mind. This highlights that belief change and desire change have different interpretations and meanings.
From the AI perspective, having a formal theory of desire change — and desire revision, as a kind of desire change operation — is important for at least two reasons: (i) desire change is a the heart of the concept of autonomous agent, (ii) a theory of desire change is required to design artificial systems who are expected to interact with humans in the appropriate way. Indeed, one important aspect of the concept of autonomy is the fact of being endowed with a mechanism responsible for the generation of internal motivations. From this perspective, an intelligent system (e.g., a robot, a virtual agent) is autonomous insofar it can generate its own desires on the basis of such a mechanism.
Moreover, an artificial agent interacting with a human should be capable of both ascribing desires to the human and understanding how the desires of the human evolve over time. 3
In this paper, desires are represented by means of a finite set D of sentences, denoted in the following by ', , or that belong to a Boolean algebra B. Hence :', ' ^ , ' _ belong to B as well. As usual > and ? are the top and bottom elements of B and denote the tautology and the contradiction respectively; ' ! =def :' _ ; ' =def(' ! ) ^ ( ! '). ` denotes the entailment defined as usual by ' ` if and only if ' ! >.
In this section we first present the notion of hedonic state. We
introduce this terminology, since volition is the name of the cognitive
process by which an agent decides on and commits to a particular
course of action, and since ultimately this process that takes into
account the agent’s beliefs about the world, relies on the desires of the
agent. Then, hedonic states will be described by means of an
ordering relation acknowledging the fact that desire is a matter of relative
strength. This relation should obey particular axioms, and has
guaranteed possibility measures [
If an agent is satisfied by having a cup of coffee or a cup of tea, she should be satisfied by having a cup of coffee. This simple example clearly suggests that desires behave in a reverse way with respect to logical entailment.
Due to this reverse behavior, we should note that desiring ' for an agent entails that she desires as well, as soon as ` '. If we prefer, desiring ' means desiring any situation where ' is true. For example, since having a cup of coffee ( ) logically entails having a cup of coffee or a cup of tea ( _ ), having a desire for a cup of coffee or a cup of tea ( _ ) means that the agent would enjoy a cup of coffee ( ), as well as she would enjoy a cup of tea ( ). We have to keep in mind that desires are considered as tentative in nature, and have not reached the step to be adopted as goals to pursue. Being pleased to have at least tea or coffee served does not mean that one has the goal to drink both in the case where both would be available. Here ‘desiring’ just means ‘finding satisfactory’, ‘finding enjoyable’, ‘having a taste for’ and so on. Note that the behavior of desires with respect to logical entailment is in full contrast with respect to beliefs, where believing ' for an agent entails that she believes as well as soon as ' ` .
This suggests that desires obey a reversed entailment, namely a desire for entails a desire for ', i.e., `des ' if and only if ' ` : This agrees with the fact that if all the models of are found satisfactory, then any interpretation taken in the subset made of the models of ' is a satisfactory state. As a consequence, in the same way as ' ` > trivially holds for any belief ', we have `des ? for any desire . In other words, desiring nothing is fully satisfactory, as believing tautologies is compulsory.
Moreover desires are a matter of strength. Some situation may be more strongly desired than another one by an agent. Thus, a hedonic state of an agent will be described by an ordering relation on B, denoted by , called desirability relation. Such a relation compares sentences in terms of satisfaction they provide to the agent if made true. ' should be read “' is at least as desirable as ”; it means that concretizing ' should be at least as satisfactory as concretizing , or if we prefer that ' is desired more strongly than in the broad sense. As usual, ' > when ' but not '; and ' means ' and '.
This leads to suppose that should satisfy the following axioms: (A0) ? > (A1) ' (A2) ' (A3) ? (Pos) 8', if > '
Once recognized that desires behave in a reverse way with respect
to entailment, (A0) expresses non triviality, while axiom (A3) is a
limit condition; it states that having no desire (here represented by
?) cannot be unsatisfactory; in other words, having no desire for an
agent should lead her to be ever satisfied. The other axioms are
perfectly neutral with respect to a reverse, or a normal, behavior with
respect to entailment. Axioms (A1) and (A2) simply say that relation
is complete and transitive respectively. Axiom (A1) is a
working assumption; considering more generally a partial order is left for
further investigation. Axiom (Pos) states that if “ is at least as
desirable as ”, this preference in the broad sense cannot be altered
by enlarging the scope of the comparaison on both sides in the same
way by '. Indeed if you find more desirable (in the broad sense) to
drink tea than to drink coffee, then you should find at least as
desirable to drink tea or orange juice as to drink coffee or orange juice
(even if your actual preference is for orange juice). Axiom (Pos) that
makes sense for desires can be encountered in other modeling
problems such as conditional logics and comparative possibility theory
[
The previous set of axioms entail noticeable properties for desirability relations that agree with intuition. First, we can establish the following result.
Proposition 1 Under axioms (A0)-(A3), axiom (P os) is equivalent to ( ) if ' then ' _ .
( ) ) (Pos).
Assume . - If ' , ' _ ' _ ; - If ' , ' _ ' ' _ ; - If ', ' _ ' ' ' _ . (Pos) ) ( ).
Let = ' in (Pos). Then (Pos) ) 8'; if ' _ '. But (Pos) applied with = ? (and = to 8'; ' ' _ . Hence, if ' , then ' ' _ . ' then ) leads
Clearly, ( ) agrees with the reverse behavior of comparative desirability with respect to entailment, and expresses that desiring '_ has the same strength as desiring the least desired of ' and . Indeed if the agent desires coffee (') more strongly than tea ( ), it means that concretizing '_ has the same appeal as concretizing . Indeed, it seems intuitively satisfactory that the strength of desire of ' _ should be at most equal to the minimum of the desire strengths of ' and (when dealing with positive desires).
Moreover, under axioms (A0)-(A3) and (P os), it can be easily shown that the following properties hold.
[a] If ' ` [b] Either ' [c] 8'; ' Proof then ' > or :'
. >.
> or both.
[a] As already observed letting = ? in (Pos), the following holds 8'; ' ' _ . If ' ` , can be rewritten as ' _ . [b] It is an immediate consequence of (A2) and ( ), since letting = :' in ( ), we get > :' if ' :', and > ' if :' '.
[c] Since 8'; 8 ; ' ' _ , letting = :' yields the result.
Property [a] expresses the reverse behavior of with respect to logical entailment (i.e., decreasingness with respect to entailment). Note that in particular, ? ; 8 , as requested by axiom (A3). Property [b] states that one cannot desire ' and :' at the same time, at least one the two options should not be desired more than what is the least desired, which is the tautology, whose non-desirability is stated by [c]. This expresses nothing but the fact that one cannot desire everything at the same time. This contrasts with the fact that desiring nothing (?) is no problem at all. Indeed there is no harm to desire ?, since you are then eversatisfied. Indeed it is not at all compulsory to desire something. 3.3
It is worth noticing that the set of axioms (A0)-(A3) and (P os)
depart from the ones characterizing epistemic entrenchment relations
epis that underly any well-behaved belief revision process [
N then ' ^
N ^ ; which under axioms (A’0)-(A1)-(A2)-(A’3) is equivalent to if '
N then ' ^
N ; where ' N means that ' is at least as certain as .
By duality, comparative necessity relations N are associated
with comparative possibility relations through the equivalence
' , : N :'. Comparative possibility relations [
We have seen that an ordering relation obeying axioms (A0)-(A3) and (Pos) may be appropriate for modeling (positive) desirability in a relative way. A natural question is then to wonder what are the absolute scale-valued functions, if any that agree with a desirability relation.
A numerical function F from B to [0; 1] is said to agree with a relation R if 8'; ; ' R , F (') F ( ). In the following we assume that the set of literals ', , ... of the considered language is finite. Thus the set W of associated interpretations is finite. We denote by jwj the proposition whose unique model is w 2 W .
The only numerical functions compatible with the desirability
relation ordering are guaranteed possibility measures [
We first establish the following proposition, before proving the announced result.
Proposition 3 8' 6= ?; 9w '; jwj '.
Proof If ' = jwj, this is obvious. If ' 6= jwj, let '1 6= ? be a strict implicant of ', such that ' ^ :'1 '1. It is always possible to find such a '1 thanks to axiom (A1). Using ( ), since (' ^ :'1) _ '1 = ', we conclude ' '1. If '1 has not a unique model, we define '2 6= ? as a strict implicant of '1 such that '1 ^ :'2 '2, and so on. The sequence ', '1, '2, ..., 'i, ... is a chain of implicants which is strictly decreasing (in terms of number of models). Since we assume a finite setting, 9n; 9w; 'n = jwj. Then from axiom ( ), ' '1 '2 'n. Proposition 4 Any numerical function F , from B to [0; 1], agreeing with an ordering relation obeying axioms (A0)-(A3) and (Pos) is a guaranteed possibility measure. Conversely any guaranteed possibility measure from a Boolean algebra B to [0; 1] satisfying (?) > (>) induces a qualitative relation satisfying (A0)-(A3) and (Pos).
Proof
()) From Proposition 3 and its proof, we know that 8' 6= ?; 9w ', such that ' '1 '2 'n = jwj, where ' is decomposed in a chain of implicants '1, '2, ... and ' ^ :'1 '1, '1 ^ :'2 '2; ; 'n 1 ^ :'n 'n. Taking ' = >, ' ^ :'1, '1 ^ :'2, ..., 'n 1 ^ :'n, jwj make a partition of W . Starting from 'n 1 ^ :jwj jwj, any model w0 of 'n 1 ^ :jwj is either such that jw0j jwj or jw0j > jwj. Let '0 be the proposition whose set of models is exactly W n fw0 s.t. jw0j jwjg. Let us apply Proposition 3 to '0 and find w00 such that '0 jw00j. We can iterate this process until we reach k with '0k 1 = ?. As a result, we can organize W into a set of layers of strictly increasing desirability (two interpretations in the same layer having the same desirability), and associate the value of a numerical function to each layer. Then it is possible to build a function (') = minw ' (w). Due to axiom ( ), is an agreeing function, and it is clear that is satisfy (1). Moreover, (w) can be taken to be equal to 0. (>) = minw2W (() Conversely, a guaranteed possibility measure, in a finite setting, is based on a distribution such that (') = minw ' (w), and it is easy to check that it induces an ordering relation that satisfies axioms (A0)-(A3) and (Pos).
Thus, Property 4 states that the only numerical functions agreeing with a qualitative ordering are those obeying decomposability property (1). Note that the range [0; 1] may be replaced by any linearly ordered, possibly finite, scale.
Thus, we can interpret ('), where is a guaranteed possibility
measure, as the extent to which the agent desires ' to be true. As
suggested in [
N (' ^
) = min(N ('); N ( )) i.e., believing ' and amounts to believing ' and to believing . As a consequence of property (1) we have min( ('); (:')) = (>) = 0 which is the numerical counterpart of property [b] in Proposition 2. Moreover (?) = 1 by convention, since is monotonically decreasing with respect to entailment. Besides, (' ^ )
max( ('); ( )): This is the consequence that is decreasing with respect to entailment (i.e. property [a] in Proposition 2). This makes perfect sense for motivational attitudes like desires, as suggested by the following example.
Example 1 Suppose Paul has a taste for cheese with strength (i.e., (eat cheese) = ) and, at the same time, he likes to drink wine with strength (i.e., (drink wine) = ). Then, according to the preceding property, Paul likes to eat cheese and drink wine with strength at least max( ; ) (i.e., (eat cheese ^ drink wine) max( ; )). This is a reasonable conclusion because the situation in which Paul achieves his two desires is (for Paul) at least as pleasant as the situation in which he achieves only one desire.
One might object that if it is generally the case that satisfying
simultaneously two desires is at least as good as satisfying one of them,
there may exist exceptional situations where it is not the case. Just
imagine, in the above example, the case where the wine is corked,
and so Paul would not like to drink it with his cheese. This is a
situation of nonmonotonic desires that could be also coped with in this
setting; see [
As in Ga¨rdenfors [
min wi2jj'jj (wi) (2) where (wi) = (!'i) with jj!'ijj = fwig. The function is called a guaranteed possibility distribution; its domain W and its range is [0; 1], or more generally any linearly ordered scale S. Thus, (w) represents the degree of desirability of a given world w 2 W . We assume that satisfies the following normality constraint: there exists w 2 W such that (w) = 0 (i.e., at least one state of the world is not desired at all). This ensures that (>) = 0 since jj>jj = W . Thus the normality constraint of ensuring that not everything is desired, entails that if (') > 0 then (:') = 0. This means that if an agent desires ' to be true – i.e., with some strength > 0 – then she does not desire at all ' to be false. This is a form of consistency requirement. Clearly, the distribution is just the numerical, or more generally the graded counterpart, of the qualitative ordering .
A desire ' with strength is expressed by a constraint of the form (p) . It will be denoted [p; ]. A set D of desires ['i; i] (for i = 1; : : : ; m) is semantically associated to a guaranteed possibility distribution
D(w) =
max i=1;:::;m
min(jj'ijj(w); i): (3) where jj'ijj(w) = 1 if w is a model of ', and jj'ijj(w) = 0 otherwise. D is the smallest possibility distribution (maximum specificity principle) such that ('i) i for i = 1; : : : ; m. This maximum specificity principle may be understood here as a minimal desire principle: there is no more desire that those expressed in the desire set D. The distribution D rank-orders the interpretations of the language induced by the 'i’s according to their satisfaction level on the basis of the strength of the desires in D. A hedonic state can then be viewed as a fuzzy (or graded) subset of worlds.
Because we should have (>) = 0, minw D(w) = 0 should hold. More generally, una(D) = min D(w) w may be viewed as a level of unacceptability of D. The larger una(D), the more unacceptable the set of desires D. 4.2
Expression (2) can be contrasted with the expression of a necessity measure in terms of a possibility distribution
N (') = 1
max w2jj:'jj (w) which estimates the extent to which the agent believes ' to be true, all the more as :' is found impossible in the sense of . Indeed, the necessity measure of N is the dual of the possibility measure , namely (') = 1 N (:') (where 1 ( ) denotes the orderreversing map of S).
Formula (3) can be contrasted with the possibilistic
representation of a belief set B expressed by a set of possibilistic logic
formulas ( j ; j ) (for j = 1; : : : ; n) encoding constraints of the form
N ( j ) j . B is semantically associated with a possibility
distribution [
B (w) =
min i=1;:::;n max(jj j jj(w); 1 j ):
B is the largest possibility distribution (minimum specificity
principle) such that N ( j ) j for j = 1; : : : ; n. The distribution
B rank-orders the interpretations of the language induced by the
j ’s according to their plausibility on the basis of the strength of
the beliefs in B. If the set of beliefs B = f j ; j = 1; : : : ; ng is
consistent then the distribution B is normalized in the sense that
9w; B (w) = 1. More generally the level of inconsistency of B is
defined by inc(B) = 1 maxw B (w). Thus una(D) should play
the same role in desire revision as inc(B) in belief revision [
There are two slightly different views of a belief set. In the dominant
one initiated by Ga¨rdenfors [
A more practical approach [
These two points of view similarly exist when revising a desire set. A desire set D is a collection of propositional sentences, closed under reversed entailment, i.e., D = f j D `des g = f j ` Dg, or may be a set of propositions associated with desire strengths, similarly closed.
It is clear that the two views are of interest. They lead to state the axioms governing revision in two different ways. We start by the view without explicit desire strengths. 5.1
As already said, one cannot desire ' and desire :' at the same time, without being led to a meaningless plethora. This parallels the fact that one cannot believe and believe : at the same time, without being led to inconsistency. Revising beliefs copes with this constraint. Similarly, revising desires should cope with the previous constraint.
Having in mind the reverse behavior of desires with respect to
beliefs, one is naturally led to state axioms that parallel the AGM
axioms [
D'+ = f j ` D [ f'gg (D*1) is a closure property. (D*2) is a success postulate: the new desire should enter in the desire set. (D*3) and (D*4) guarantee that the revision is an expansion that amounts to add the new desire ' to the desire set when :' is not already in the closure of the desire set. (D*5) states that the revision cannot result into desiring everything except if the new desire would be to desire everything. (D*6) is the independence with respect to syntax. (D*7) and (D*8) clearly parallel (D*3) and (D*4) when revision is decomposed in two steps.
As for guaranteeing the existence of an epistemic entrenchment in
belief revision where the last two AGM axioms are necessary, (D*7)
and (D*8) are required for ensuring the existence of a hedonic
entrenchment relation in the sense of the postulates of subsection 2.1.
This can be established following a route very similar to the one of
Grove [
DA = (D [ A if it is not W: some C 6= W; C
A otherwise.
Besides, DA+ = D [ A.
The two last axioms come from the choice function area, and
specify that C is selected with respect to an ordering on W (the most
desired states outside A) [
One could think of applying here the maximum specificity principle for desires, counterpart of the minimum specificity principle in belief representation. Namely, unless desire is explicit, one assume states are not desirable. Under this assumption, C should be A in the above set revision rule (since there is no desire strength for discriminating the states outside A). This is clearly too drastic, and in the next section we investigate desire revision with explicit desire strengths. 6
We now turn towards the case where the hedonic entrenchment can
be computed from the desires given with their explicit strength. We
first briefly recall how belief revision works in the possibility
theory setting. Indeed it has been recognized early that the epistemic
entrenchment relations underlying any well-behaved belief revision
process obeying AGM postulates [
We first need to recall the possibilistic expression of conditioning underlying belief revision and its counterpart for guaranteed possibility measure, before considering the revision of beliefs, and then the revision of desires. 6.1
In qualitative possibility theory [
) = min( ( j'); (')): The quantitative version would use the product instead of min, but here we prefer a qualitative setting which agrees with the nature of the hedonic entrenchment. Applying the minimum specificity principle which leaves the possibility degrees as high as possible given the constraints (for avoiding arbitrary restrictions of the possible states), we get the possibility distribution ( j') associated with the possibility measure ( j'): (wj') = 8 <
As can be seen, what is no longer reachable (conditioning by ' means that, for some reason, the possible states are restricted to be those where ' is true) is fully desirable by default ( (:'j') = 1), while what we have is no longer desired since ('j') = 0, but still preserving what is strictly above ('). where B is the possibility distribution associated with the belief base B, as recalled in Section 3.2. Conditioning by ', acknowledges the fact that according to the input of the new piece of belief ', states where ' is false have become impossible.
This expression covers the expansion B'+ of B by ' as a particular case:
B'+ (w) = min( (w); jj'jj(w)) provided that the consistency condition core( ) \ jj'jj 6= ; holds, where core( ) = fw j (w) = 1g.
Besides, the contraction B' of B by ' is semantically expressed
by [
Let D be a set of prioritized desires. Let D be the associated hedonic distribution (as defined by (3) in Section 3.1).
The contraction of D by ' amounts to no longer desire ' at all after contraction. It is semantically expressed by:
D' (w) = 0 (w) if D(w) = (') and w otherwise '
The expansion of a set of desires D by ' amounts to cumulating desire ' with the desires in D, providing that the result is not the desire of everything to some extent (due to the postulate (>) = 0). Thus, we have
D'+ (w) = max( D(w); jj'jj(w)) provided that support( D) [ jj'jj 6= W , where support( ) = fwj D(w) > 0g.
While the revision of a set of beliefs B by ' exactly corresponds to the conditioning of B by ', this is no longer the case with respect to D for the revision of a set of desires D by '. Indeed, while a belief input ('; 1), i.e., N (') = 1, really means that all the models of :' should be impossible, i.e., (:') = maxw :' B (w) = 0, a desire input ['; 1] means (') = minw ' D(w) = 1, which says that all the models of ' are satisfactory after revision.
Moreover, we have observed in Section 5.1 that ('j') = 0 and (:'j') = 1. But ('j') = 0 does not fit with the idea that ' is a new desire, nor (:'j') = 1. Indeed, conditioning by ' does not mean to get a new desire. It means that for some reason, the possible states are restricted to be those where ' is true (which indeed confirms ' is not a new desire). So the agent can only desire such states, which would favor D' (:') = 0.
Due to this change of focus from :' to ', when moving from beliefs to desires, desire revision is expressed by:
D' (w) =
D(wj:')
D' (w) = 8 < As can be seen we have D' (:') = 0 and D' (') = 1.
Having D' (') = 1 may be considered as too a strong expression of the success postulate when revising the desire set D by the new desire '. We may think that this interpretation of an input is too strong for revising a gradual desire profile. Introducing a new desire does not necessarily mean that the new desire should be desired with the highest strength. As revision is a merging of two entities of the same nature, we may prefer considering revision by (') (rather than (') = 1). This leads to
D('; ) (w) = 8 > > < > > :
It can be checked that we now have D' (') = . We may also
think of weakening the success postulate into D' (') > 0. It can be
defined by taking lesson of what is done in belief revision, where this
corresponds to the idea of natural revision in the sense of Boutilier
[
Let D = f[' ^ ; ]; [ ; ]g, be a desire base where > , where '; ; are literals. Applying (3), we get its semantical counterpart under the form of the distribution D. Namely we have D(' ) = D(' : ) = ; D(': ) = D(:' ) = D(:': ) = ; D(': : ) = D(:' : ) = D(:': : ) = 0.
Clearly, una(D) = 0.
Now, assume we want to add desire [:'; 1]. Let us compute D:' . We get: D:'(' ) = D:'(' : ) = ; D:'(': ) = ; D:' (': : ) = 0, which remain unchanged, while it gives D:'(:' ) = D:'(:': ) = D:'(:' : ) = D:'(:': : ) = 1.
Observe that una(D [ f[:'; 1]g) = 0, which means that after addition of the new desire, the set of desires remains acceptable. In fact, we have just performed an expansion here.
Now suppose we only add the desire [:'; ]. Then the modified part of D would be now D:'(:' ) = D:'(:': ) = max( ; ), D:'(:' : ) = D:'(:': : ) = .
Suppose now D has to be modified by input [: ; ]. Then we have ) = ; ) = D[f[: ; ]g(:'
) = ; D[f[: ; ]g(' D[f[: ; ]g(': ) = D[f[: ; ]g(:': D[f[: ; ]g(': : ) = D[f[: ; ]g(:' : ) = D[f[: ; ]g(:': : ) = and D[f[: ; ]g(' : ) = max( ; ). Thus una(D [ f[: ; ]g) = min( ; ; ) = min( ; ) = assuming > (the new desire is not less strong than the desires in D). The result of the revision is D[: ; ] (': ) = D[: ; ] (:' ) = ) = 0, while for the other interpretations, we keep
D[f[: ; ]g(w). This preserves una(D[: ; ]) = 0.
D[: ; ] (:': D[: ; ] (w) = 6.4
It is easy to write the possibilistic counterpart of the axioms for desire revision presented in subsection 5.1. Namely
1)] For any sentence ' 2 B, D' represents a hedonic [( 2)] D' (') = 1. This a (strong) priority to the new desire. [( 3)] D'+ is not more specific than D' [( 4)] If D(:') = 0 then D' D'+ [( 5)] D' = > if and only if ' > [( 6)] Equivalent pieces of desires lead to equivalent revisions. We have it for free in the semantic view.
[(
( 2) may be weakened into D' (') > 0. It can be easily
checked from the definition of D' that they all hold in the possibility
theory setting. These axioms are the exact counterpart of the axioms
for gradual belief revision [
The possibilistic logic of desires does not obey the same rules as
the possibilistic logic of beliefs. Indeed now the resolution rule writes
[' ^ ; ] and [:' ^ ; ] entails [ ^ ; min( ; )], which echoes
the reverse rule (4) for defining expansion, and contrasts with the
more classical resolution rule for prioritized beliefs: (' _ ; ) and
(:' _ ; ) entails ( _ ; min( ; )) [
Analogously to the belief revision case, it can be checked that only the desires strictly above the level of unacceptability are saved:
D' = f['i; i] 2 D s.t. i > una(D [ f['; ]g)g [ f['; ]g: the others being drown, as it is the case for [ ; ] 2 following example D0 in the
Let D0 = f['; ]; [ ; ]g with > .
Then una(D0) = 0 since D0 does not entail > at any non zero degree. Now, let us add desire [:'; 1].
We have una(D0 [ f[:'; 1]g) = and then D:0' = f[:'; 1]g. If we rather consider D00 = f['; ]; [ ; ]g (always with > ), then we have una(D00 [ f[:'; 1]g) = , and D:00' = f[ ; ]; [:'; 1]g.
Similarly in Example 2, it can be checked, we have D:' = D:+', and the syntactic counterpart is D:' = f[' ^ ; ]; [ ; ]; [:'; 1]g. Moreover D[: ; ] = f['; ]; [: ; ]g assuming > .
Note that in all the above examples, we have assumed that none of the interpretations induced by the language used for specifying the desire set is impossible in the real world. In any case, if such an impossibility exists for some of them, this has to be taken into account when adopting goals, but not in the revision of desires. 7
The paper has presented a formal approach to the revision of desires. By desire, we mean potential desires and distinguishing them from goals. Goals are desires that have been actualized by the agent and to which she is committed. Their revision is not the same problem as the one of desire revision, and is in fact quite similar to belief revision (since having goal ' and having goal should be the same as having goal ' ^ ). The goal revision of a set of prioritized goals would be based on a volitive entrenchment, formally similar to an espistemic entrenchment. The particular nature of desires with respect to beliefs has been advocated and emphasized. Roughly speaking, desires behave in a reverse way. This is reflected in the different series of axioms characterizing the hedonic entrenchment and then desire revision that have been proposed. Several directions remain to investigate, such as studying iterated desire revision.
Besides, it is known that belief revision and nonmonotonic
reasoning are two sides of the same coin [