Hidden Uncertainty in the Logical Representation of Desires

1 Introduction

Autonomous agents reason frequently about preferences, de­ sires and goals. For example, Cohen and Levesque [1990] explore principles governing the rational balance among an agent's beliefs, goals, actions and intentions, Rao and Georgeff [ 1991] show how different types of rational agents can be modeled by imposing certain conditions on the per­ sistence of an agent's beliefs, desires or intentions (the BDI model) and work in qualitative decision theory [Pearl, 1993; Boutilier, 1994; Bacchus and Grove, 1996; Doyle and Thomason, 1999; Thomason, 2000] illustrates how planning agents are provided with goals - defined as desires together with commitments- and charged with the task of discovering and performing a sequence of actions to achieve these goals.

In logical formalizations of preferences, desires and goals serve as a computationally useful partial specification or heuristic approximation of the relative preference over the possible results of a plan [Doyle, 1991]. In this paper, we focus on desires although our observations may be relevant for goals as well. There are three different interpretations of sentences like "I desire B" or "in context A, I desire B".

First, desires may be formalized using only utility or pref­ erence. However, this leads to the problem that even if a util­ ity function or a preference relation over worlds is fixed, there is no unique way to extend it to formulas or, equivalently, sets of worlds. This distinction between utilities and probabilities is the main reason why formalizing desires is more problem­ atic than formalizing beliefs. Moreover, it does not seem to correspond to the meaning of desire in natural language. For

Emil Weydert

University of Applied Sciences Luxembourg Luxembourg emil. weydert@ ist.lu

instance, suppose that I exceptionally get sunburned when the weather is hot and sunny. Then, expressing "I desire a hot and sunny weather" intuitively means that the most normal or typ­ ical worlds satisfying hot 1\ sunny are preferred to the most normal worlds verifying-, (hot 1\ sunny), but it does not mean that I like exceptional effects of hot 1\ sunny such as sunburn.

Secondly, desires may express a combination of utility and probability in a classical decision-theoretic context, assuming a probability distribution over worlds. "In context A, I desire B" can then be interpreted as a raise of expected utility: my expected utility given A 1\ B is higher than my expected util­ ity given A 1\ ·B. A further development in terms of gain of expected utility is proposed in [Brafman and Friedman, 2001]. However, it is well-known that autonomous agents do not always have appropriate probabilistic information on the possible worlds. The probabilistic approach asks for costly, specific information and in practice often leads to an arbitrary choice of probability values.

Thirdly, desires may implicitly refer to the relative plausi­ bility or normality of worlds. When an agent states "I desire B", i.e., "I desire B to be satisfied", the agent often focuses on typical, normal worlds. The standard choice made in qual­ itative decision theory, see [Doyle and Thomason, 1999], is that uncertainty is described by a (total) pre-order expressing, e.g., plausibility, normality, or typicality. The main justifica­ tion is that ordinal uncertainty is Jess committing, cognitively speaking, than numerical uncertainty.

The research question of this paper is how we can formal­ ize desires to provide a realistic interaction between prefer­ ence and normality, and such that realistic examples like ex­ amples 2.4, 3.3, and 3.5 can be formalized. E.g., in the latter an airline company desires for an overbooked plane that an individual passenger shows up, but at the same time it desires that not all passengers show up. We start with Boutilier's no­ tion of so-called ideal goals [1994], but then we define condi­ tional desires that refer to both preference and normality, and that tum out to be more satisfactory.

The layout of this paper is as follows. Section 2 recalls the basics of Boutilier's semantics and discusses the choices upon which it is based; we then explain how to extend it to represent desires in a more appropriate way. The details of this modified interpretation of desires are given in Section 3. We study several examples and in particular, we illustrate how our semantics interprets sets of desires that are usually con­ sidered contradictory.

2 Boutilier's qualitative decision theory logic

A first class of solutions to combine preference and normal­ ity makes use of a trade-off between qualitative utilities and probabilities as in [Pearl, 1993], but this is not well-suited to situations with non-extreme utilities. A second approach is given in [Boutilier, 1994], which makes use of two ordering relations, one for preference and one for normality, represent­ ing the ordinal counterparts of utility and probability. How­ ever, here we see- informally speaking- a Jack of interaction between preference and normality.

2.1 Definitions

Boutilier [1994] interprets a desire in terms of ideal worlds: if A then ideally B, denoted by I(BIAJ, is a conditional desire expressing on the semantic level that among all A-worlds, the most preferred ones also satisfy B . Quoting Boutilier, "from a practical point of view, I( BIAJ means that if the agent (only) knows A, and the truth value of A is fixed (beyond its control), then the agent ought to ensure B ( ... ) The statement can be roughly interpreted as if A, do B."

This definition enables the formalization of conditional de­ sires in the conditional logic QDT, which is based on his logic CO. Roughly, the semantics ofODT is the following: a COT­ model isM = (W, ~p, ~N, val) where W is a set of possi­ ble worlds, val a valuation function and ~P and ~N are two total pre-orders over W, i.e., transitive connected relations, where a relation is connected iff w ~ v or v ~ w for all w, v E W. They are also called total weak orders. ~P is a preference order and ~N a normality order: w ~P w' (resp. w ~N w') means that w is at least as preferred (resp. as normal) as w'. The conditional connectives 1(-1-J and N(-1-J have the following truth conditions. The two modalities do not mix preference and normality; in particular, conditional desires I( -I-) do not use the normality relation at all.

• M~I(BIAJifandonlyifeitherV'w E W, M,w ~·A, or there exists a w E W such that M, w ~ A and for all w' E W with w' ~P w, M,w' ~A-+ B. I(BIAJ ex­ presses the conditional preference "if A then ideally B".
• M~ N(BIAJ: as I(BIAJ , except that ~Pis replaced by ~N· N(BIAJ expresses the conditional default "if A then normally B".

In the rest of the paper we write I( B) for I(BITJ and N(B) for N( BIT), where T denotes a tautology.

Boutilier also introduces ideal goals that combine prefer­ ence and normality. However, they use only the upper cluster of the normality relation, nothing else. On the semantic level, if we assume that there are no infinite ascending chains, then the truth conditions for ideal goals are as follows. IG(BIAJ is true if the best of the most normal A worlds satisfy B.

Therefore, Boutilier's interpretation of conditional desires re­ lies on two strong assumptions, which we discuss in more detail below. The first assumption is the optimistic interpre­ tation of desires due to the ideality assumption. The second assumption is the weak interaction between preference and normality due to the focus on on the most normal worlds in a context.

2.2 Assumptions

Ideality

The ideality semantics consists in comparing sets of worlds by looking only at the most preferred worlds of these sets. It corresponds to an optimistic point of view in the sense that less preferred worlds are ignored in this process. What would be possible choices for comparing sets of worlds? If we do not want to bring in probabilistic information or as­ sumptions such as equiprobability, then we are left with the following four basic alternatives, and variations or combina­ tions thereof. Let A and B be two formulas. For the sake of simplicity we only consider the non-degenerate case where Mod( A A B) and Mod( A A ·B) are both non-empty.

• M M (ideality semantics): IM M (BIA) if and only if the best A A B-worlds are preferred to the best A A ·B-worlds.

mm: lmm(BIA) if and only if the worst A A B-worlds are preferred to the worst A A ·B-worlds.

mM: lmM (BIA) if and only if the worst A A B-worlds are preferred to the best A A ·B-worlds.

Mm: IMm(BIA) if and only if the best A A B-worlds are preferred to the worst A A ·B-worlds.

We have that lmM (BIA) implies both lmm(BIA) and IMM (BIA), and all of them imply IMm(BIA) . Note that only M M and mm are consistent with the semantics of con­ ditionallogics; for mm, just take the reverse preference rela­ tion. The M m variant is extremely weak and therefore tells us next to nothing. It may be useful only when it is extended with a non-monotonic reasoning mechanism. The mM vari­ ant is extremely strong and hard to satisfy; it certainly does not reflect the usual intuitive understanding of desires. It may be useful only when paired with normality, e.g., by focusing on the most normal worlds.

The mm variant has an underlying pessimistic semantics. It makes sense if "I desire B in context A" is interpreted as "in context A, given that I expect the worst outcomes to occur, I am happy to see B true". However, this semantics also does not fit well the intuitions behind the specification of desires. Consider the following example of two desires.

Example 2.1 (Game) Assume that an agent plays a game where two coins are tossed; he wins if both coins are heads (h) and loses otherwise. His preference ordering can be: (h1,h2) >p (h1,·h2) ""P (·h1,h2) ""P (·h1,·h2); the normality ordering ~N is the one where all worlds are equally normal. But, surprisingly, neither 1mm(h1) nor lmm ( h2) are satisfied.

The example supports that Boutilier's ideality semantics best suits the commonsense intuitions concerning desired out­ comes of actions, and we therefore do not want to give it up.

At a first glance, it may seem paradoxical to favor an op­ timistic interpretation of desires while most papers in qual­ itative decision theory argue in favor of a pessimistic (max­ min) criterion for action selection [Brafman and Tennenholtz, 1997; Dubois et al., 1998]. The paradox is however only in appearence. When specifying I( Bl T ), for instance, an agent expresses that she has a preference for B, which is not at all the same as saying that she intends to take a decision making B true. This is illustrated by the following classical example.

Example 2.2 [Boutilier, 1994} Let DS = { I(· u), l(uir) } where u and r stand for umbrella and raining. I(· u) ex­ presses that the agent prefers not carrying an umbrella. This does not imply that if the agent has the choice between the ac­ tions take-umbrella and leave-umbrella (whose obvious out­ comes are u and ·U), she will choose leave-umbrella.

The example illustrates that the interpretation of desires and the action selection criterion are independent issues and we can consistently interpret desires in Boutilier's ideality se­ mantics while using a pessimistic criterion for action choice. This issue is discussed further in Section 4.

Weak interaction between normality and preference

This assumption is more problematic than the previous one. Boutilier's interpretation of ideal goals makes a very rough use of the normality relation, since it consists in focusing first, and once for all, on the most normal worlds, independently of the desires expressed, and then in interpreting desires by the ideality semantics.

This has unfortunate consequences. Ideal goals are differ­ ent from desires because we have the counterintuitive prop­ erty that if normally p then p is an ideal goal. Moreover, Boutilier's semantics makes intuitively coherent sets of de­ sires inconsistent, such as the dog and fence example of de­ feasible deontic logic [van der Torre and Tan, 1997].

Example 2.3 (Dog and fence)

DS = {1(-.j), l(f]d), I( d)}

1. John does not want a fence around his cottage;
2. If John owns a dog, then he wants a fence;
3. John wants to own a dog.

Example 2.3 is inconsistent, and would still be inconsistent if we would replace the I modality by IG.

Here is another example which is inconsistent in Boutilier's semantics. Suppose that I am going to my travel agent just one day before Christmas vacation, when normally all flights are fully booked.

Example 2.4 (Airplane ticket)

DS = {l(r), I( a),--, l(r 1\ a), N(· rl\--, a)}

1. 1 desire to have an airplane ticket to Rome;
2. I desire to have an airplane ticket to Amsterdam;
3. I do not desire to have both an airplane ticket to Rome and an airplane ticket to Amsterdam;
4. Normally, I will neither get an airplane ticket to Rome nor to Amsterdam.

Example 2.4 is inconsistent, and would still be so if we re­ placed the I modality by IG. However, this set of desires has an intuitive interpretation: when I think of having a ticket to Rome, I think of the most normal world where I have a ticket to Rome, in which I do not have a ticket to Amsterdam, and vice versa, and I prefer this world to the most normal world where I do not have a ticket to Rome, in which I do not have a ticket to Amsterdam either. See another interpretation of this set of desires in example 3.6.

3 Hidden uncertainty

Example 2.4 illustrates that, even if the intuitive expression of the desires does not mention normality or uncertainty issues, it implicitly refers to these. This is why we talk of hidden uncertainty in the specification of desires.

3.1 Definitions

We now introduce a notion of desires with hidden uncertainty that better fits the intuitive meaning of desires. To simplify the definition, we assume- in contrast to Boutilier- the exis­ tence of maxima. We may guarantee this by stipulating, e.g., that there are no infinite ascending chains. D( BIA) means that there are maximally normal A 1\ B-worlds which are strictly preferred to all the most normal A 1\ ·B-worlds.

Definition 3.1 Truth conditions for desires are as follows. M f=D(BIA) if and only if'tlw' E Max(?_N,Mod(A 1\ ·B)) :Jw E Max( ?.N, Mod(A 1\ B)) such that w > p w'.

In other words, "in context A, I desire B" is interpreted as "the best among the most normal A 1\ B worlds are preferred to the most normal AI\ ·B worlds". As usual, we write D( B) instead of D( Bl T ).

There exists also a probabilistic interpretation of desires which may help to clarify their meaning. The basic idea is to use the E-semantics for normality, a super-E-semantics - e.g. replacing E by E' -for utility, and the resulting concept of expected utility for interpreting desires.

In a sense, our definition of desires is still reminiscent of the ideality semantics, because for a desireD( BIA) we only consider the most normal A 1\ B and the most normal A 1\ · B worlds. However, the fact that we do not only consider the most normal A worlds makes a crucial difference. This is illustrated by the properties and examples below.

3.2 Properties

The definition guarantees that the conditional desire always holds if the implication is strict, i.e., if AI\ ·B is inconsistent. When all worlds are equally normal, desires represent pure preference in the sense that both modalities D() and I() coin­ cide. However, in general none of the following implications D(AIB) =?I( AlB), I( AlB)=? D(AIB), D(AIB) =? IG(A IB) and IG(AIB) =? D(AIB) holds. Furthermore it avoids- as opposed to some of its variants- the validation of conditional desires of the form D( ·AlA) for consistent A.

The following example illustrates that we have no longer right weakening (strictly speaking, left weakening, given our notation) for the desires, i.e. D( A) does not imply D( A VB).

Example 3.1 Let M be the following model in which ·p is always more plausible than p, whereas pis always preferred to ·p.

Then M satisfies D(p) but not D(p V q).

This absence of right weakening, a controversial property when reasoning with obligations in deontic logics, is also very natural for desires: take p = "the woman of my dreams falls in love with me" and q = "I receive my electricity bill" (needless to say where the most normal worlds are).

The violation of right weakening also explains the distinc­ tion between the two conflicts in the following example. The example also illustrates that the conjunction (or AND) rule is not valid for D.

Example 3.2 {D(p) , D(· p)} is inconsistent, but {D(p ), D(..., p 1\ q)} is consistent when q is exceptional, as is witnessed by the following model:

Interestingly, many rules valid for I( BIAJ do no longer hold forD( BIAJ, as illustrated by the two examples above. Or, more precisely, they hold only conditionally. The reason is a kind of context-dependence or "higher-order nonmonotonic­ ity". For instance, if M od(A 1\ B) increases or decreases, it does not follow that max(?.N, Mod(A 1\ B)) increases or decreases as well. One way to design proof rules is to en­ sure with additional clauses that this implication holds, which has been done for decision-theoretic defaults [Brafman and Friedman, 2001]. However, our logic of desires also vali­ dates inference rules which do not directly depend on such conditions.

The first formula of the following proposition illustrates that the conjunction rule holds under normality conditions N(A IB 1\ C) and N(BIA 1\ C), and the latter two formulas hold without any conditions.

Proposition 3.1 The following formulas are theorems of our logic:

1. D(AICJ 1\ D(BICJ 1\ N(AiB 1\ C) 1\ N(BIA 1\ C) =? D(A 1\ BICJ;
2. D(AIB 1\ C) 1\ D(BICJ =} D(A 1\ BICJ;
3. D(AIA v B) 1\ D(· AI· A v B) =} D(· BITJ;

Properties expressed in the dyadic modal logic are often hard to read. We therefore propose an alternative strategy for gen­ erating proof rules. The idea is to express desires with a suit­ able order modality <nd where A <nd B means that, taking into account normality and adopting an optimistic perspec­ tive, A is less desirable than B, or inconsistent. Formally speaking, we get the following truth condition for <nd (as­ suming finitely many worlds).

Definition 3.2 Truth condition of preference is as follows. M f=A < nd B if and only ifVw' E Max(?.N,Mod(A)) 3w E Max(?.N,Mod(B)) such thatw >p w'.

We have the following relationship between the conditional desires and preferences.

Proposition 3.2 The following translation rules are valid be­ tween <nd and D(.! .).

• M f=A < n d B iff M f=D(·AIAV B)
• M f=D(BIAJ iff M f=(A 1\ ·B) <nd (A 1\ B)

Similarly, we can introduce <n for expressing propositional normality comparisons. Taken together, these modalities al­ low us to formulate a number of features which are concep­ tually easier to grasp and to handle than the corresponding conditional notions. The following proposition lists several properties using this alternative representation.

Proposition 3.3 The following formulas are theorems of the logic. They are called left weakening ( Ll ), right weakening ( L2 ), left strengthening ( L3 ), left impossibility ( IA ), right pos­ sibility ( L5 ), asymmetry ( L6 ), left disjunction ( L7), right dis­ junction (L8), transitivity (L9), left cautious transitivity ( LJO), right cautious transitivity ( Lll ).

Due to lack of space we must omit proofs and discussion.

3.3 Examples

Let us consider Example 2.3 and 2.4 taking hidden uncer­ tainty into account.

Example 3.3 (Dog and fence, continued)

{D(d},D(·f),D(f ld)} is consistent. Here are four classes of models, which illustrate that for each normality ordering there are various preference orderings.

class 1: dogs are exceptional (and nothing else)

?.N: {(·d, f) , (·d, ·f)} >N {(d, f) , (d , ·f)}

?_p: any complete preordering satisfying constraints

(d , f) >p (d, ·f)

class 2: not having a fence is exceptional (and nothing else)

?_p: any complete preordering satisfying constraints (·d, ·f) >p (·d, f) , (d, f) >p (d , ·f) ( d, f) > p ( ·d, f)

class 3: dogs and not having a fence are both exceptional

?.N: ( ·d, f) > N { (d, f), ( ·d, ·f)} > N (d, · f)

?_p: any complete preordering satisfying constraints ( ·d, ·f) > p ( ·d,J), (d, f) > p ( ·d, f) (d , f) > p (d, ·f)

class 4: neither dogs nor not having a fence is exceptional

?_p: any complete preordering satisfying constraints (d,·f) > p (·d,f), (d,f) >p (·d, f)

Example 3.4 (Airplane ticket, continued)

DS = {D(r), D(a),..., D(r 1\ a), N(· r/\..., a)}. Here is a set of models satisfying DS:

?_p : any order where (r, ·a), ( ·r, a) are strictly more pre­ ferred than the other worlds.

The following example is a reformulation of the lottery para­ dox in terms of desires.

Example 3.5 (overbooking)

The agent is an airline company which has sold 301 tick­ ets for a flight on an airplane of 300 seats. For each seat occupied the company gains 100 utility units, but if all 301 persons show up, then the company looses 1000 utility units. The agent may consistently express {D( show-up( 1 )), .. . , D(show-up(301)), D(-. (show-up(l) /\ ... /\ show-up(301))}, because, individually, passenger #i showing up makes the ex­ pected utility of the company increase (slightly), due to the fact that it is very unlikely that all passengers show up.

3.4 Normality and update

There are several perspectives on normality. One consists in viewing normality as distance to the current situation. When an agent figures out a "normal A-world" he often figures out the closest A-world to the actual world. This is in accordance with the principle used for evaluating counterfactuals. Update could come very intuitively into this framework.

The normality ordering is then defined by the proximity to the current world, which is defined by a faithful proximity re­ lation in the sense of [Katsuno and Mendelzon, 1991], i.e., a collection of weak orders {~w. w E W}, where faithfulness is the condition: w <w w' for all w' =1- w; WI ~w w2 means WI is closer to w than w2. We simply have WI "?.N w2 iff WI ~w w2, and the set of most normal worlds (i.e., clos­ est worlds to w) satisfying A is the update of w, denoted by w <> A. The simplest and most frequent choices are: WI ~w W2 iff Diff(w, WI) ~ Diff(w, W2) and WI ~w W2 iff IDiff(w, wi)I ~ IDiff(w, w2)l, where Diff(w, w') is the set of variables assigned a different value by w and w'.

Let us consider example 2.4 but with a slightly different interpretation. We do not have to suppose here that flights are normally fully booked. We just assume that in the current situation, the agent does not have any airplane ticket.

Example 3.6 (Airplane ticket, continued)

DS = {l(r), I( a),-. l(r /\a),-. r A-. a}

1. 1 desire to have an airplane ticket to Rome;
2. 1 desire to have an airplane ticket to Amsterdam;
3. 1 do not desire to have both an airplane ticket to Rome and an airplane ticket to Amsterdam;
4. In the current situation I do not have any airplane ticket.

If the proximity relation is such that ( r, a) is closer to ( -.r, a) and (r,-.a) than to (-.r,-.a), and equally close to (-.r,a) and (r, -.a), then we get the following normality ordering: ( ·r, -.a) > N ( -.r, a) "'N (r, -.a) > N (r, a), because the ini­ tial situation is ( -.r, -.a). Normality reflects proximity to the initial situation.

Of course, this assumes that a preliminary step has been done so as to translate the proximity to the current world into a normality ordering; this issue has been considered several times in the literature. See for instance [ Grahne, 1991] for the intertranslation between update and conditional logics, and [Herzig, 1998] for a review of logics for belief update.

4 Related research

4.1 Two tasks in qualitative decision theory

Qualitative decision theory aims at developing mainly non­ numerical - and therefore non-probabilistic - normative frameworks for decision making under uncertainty, e.g. look­ ing for minimal sets of minimal behavioral properties or ax­ ioms of a rational agent that correspond to a given action se­ lection criterion. Most approaches, e.g., [Brafman and Ten­ nenholtz, 1996; Lehmann, 2001; Dubois et al., 2002], use ordinal structures for preference and uncertainty.

In a decision-theoretic context we can distinguish at least the task of interpreting desires, which aims at reasoning about the mental state of the agent (what he likes and what he be­ lieves) and the task of selecting an action that uses the pos­ sible mental states induced by the upstream task, like quali­ tative analogs of maximum expected utility. Both tasks are complementary. The logic developed in this paper does not investigate criteria for action selection and therefore it is not really a new approach to qualitative decision theory. Our logic aims at interpreting desires as they can be expressed by agents, for instance in an interactive elicitation process. Our logic can infer some information about the normality order­ ing or the preference ordering of an agent's mental state but cannot predict which action he will perform.

There are several ways in which our logic can be extended with action selection. In example 2.2, when a model con­ tains both rand -.r normal worlds, 1(-. u) merely expresses that preferred worlds satisfy -.u while the selection of ac­ tion take-umbrella may well reflect that the worst among the most normal effects of this action are preferred to the worst among the most normal effects of leave-umbrella. The latter criterion is used by [Brafman and Tennenholtz, 1997] who model agents as pessimistic decision makers. Noticeably, it is similar to our interpretation of desires, except that worst states have to be focused on instead of best states. It would be worth extending our logical framework with a modality PA(.I.J: the preference of taking action a over action (3 would be expressed by PA(do(a)ldo(a) V do((J)), expressing that the worst normal effects of action a are preferred to the worst normal effects of action (3. This modality would have the same properties as the D(.I.J modality, except that the prefer­ ence order has to be reversed: if M = (W, "?_p, "?.N, val) and M' = (W, reverse("?_p ), "?.N, val) then M f= PA(BIA) iff M' f= D(BIA). This extension is left for further research.

4.2 Formalisms of desires

There are several other formalisms that represent desires and pure preference, especially ceteris paribus constraints [Doyle and Wellman, 1991; Boutilier et al., 1999]. However, this framework deals with pure preference only and not with un­ certainty and normality, except maybe a preliminary attempt in [Tan and Pearl, 1994]. The combination of ceteris paribus and normality is an issue for further research.

Thomason's framework [Thomason, 2000] builds on Re­ iter's default logic and deals with both normality and prefer­ ence defaults, but with a procedural strategy which departs from our completely semantical interpretation: goals are de­ rived by first closing the facts under beliefs defaults, and thereafter under desire defaults. The same mechanism is used in BOlD architectures [Broersen et al., 2002]. The goals are thus restricted to the most normal state.

Finally, decision-theoretic defaults [Brafman and Fried­ man, 2001; Poole, 1992] also have a semantics based on nor­ mality and preference. For example, Brafman and Friedman give a detailed analysis of the use of expected utility, they ex­ plain a drawback of straightforward expected utility, and they introduce defaults based on gain of utility G(A). Moreover, they define weak and strong notions of defaults, where the first does not satisfy right weakening, because G(A) is not informative about the behavior on subsets of A. This seems related to what we here called higher-order nonmonotonicity.

5 Summary

The research question of this paper is how desires can be for­ malized with a realistic interaction between preference and normality. We start with Boutilier's logic for QDT, in which preference and normality have been combined in a notion of ideal goal. However, there is only weak interaction, such that the dog and fence example as well as the airplane ticket exam­ ple cannot be represented in a consistent way. We therefore interpret desires in a different way: "in context A, I desire B" is interpreted as "the best among the most normal A 1\ B worlds are preferred to the most normal A 1\ ·B worlds".

We study various formal properties of these desires. We show that our D(AIBJ does not imply Boutilier's I(AIBJ or IG(AIBJ, nor vice versa. We show that they do not satisfy weakening nor the the conjunction rule. We show that these properties and many others hold under conditions of normal­ ity, and we show that some properties like transitivity and cumulativity hold unconditionally. We illustrate the expres­ sive power on several classes of examples including the dog and fence and airline ticket examples, and we illustrate how update can be introduced in the framework. Finally we po­ sition the desires with respect to previous work in qualitative decision theory, where we mention as subjects of further re­ search an extension with an action selection criterion, and the introduction of normality in ceteris paribus preferences.