Dynamic Belief Hierarchies John Bell and Zhisheng Huang Applied Logic Group Computer Science Department Queen Mary and West eld College University of London, London E1 4NS, UK E-mail: fjb,
[email protected]
Abstract. In this paper we introduce and formalise dynamic belief hier-
archies. We give a formal de nition of beliefs, which requires that they are coherent; that is, that each belief is jointly consistent with every other belief which the agent considers at least as reliable. Thus an agent's beliefs cannot be de ned independently, but only with reference to the agent's existing belief hierarchy. We then show how preferential entailment can be used, in conjunction with the rationality constraints on beliefs, to formalise the rational revision of beliefs and belief hierarchies. We then discuss the relationship between our theory and the AGM theory of belief revision. Finally we discuss resource-bounded reasoning.
1 Introduction Consider the following true story. On January 14, 1997, Zhisheng was travelling by train from Rome to Siena in order to participate in a workshop on agent modelling. He had to change trains at Chiusi, so on arrival at the station he hurried to discover the platform number of the next train departing for Siena. According to the published timetable the next train would depart from platform one at 19:34. So, he believed that this would be the case. He was very surprised when he looked at the electronic departures board in the station hall, as the board showed that the next train for Siena would depart from platform two. He considered that the information on the board was more reliable than that on the timetable, as it was more recent and more easily updated. So he dropped the belief that the train would depart from platform one in favour of the belief that it would depart from platform two. In order to be sure, he asked the man behind the information desk, who assured him that the next train for Siena would indeed depart from platform two. Zhisheng considered this to be the most reliable information so far. So he continued to believe that the train would depart from platform two despite the fact that at about 19:15 a train arrived at platform one labelled \Chiusi - Siena". However, by 19:28 there was still no sign of a train on platform two, so he started to have doubts. Fortunately there was a signalman on platform three,
so Zhisheng hurried over and asked him. The signalman told him that the next train for Siena was the one now on platform one. Zhisheng considered that the signalman was in a better position to know than the man at the information desk. So he revised his beliefs again and hurriedly boarded the train on platform one. At 19:34 the train on platform one pulled out and, happily, it arrived at Siena in due course. Despite the fact that Zhisheng was constantly changing his beliefs, he was clearly doing so in a rational way. In this paper we aim to model reasoning of this kind. In order to do so, we introduce the notion of a belief hierarchy. At any point in time an agent has a set of beliefs, a belief set, and considers some of these beliefs to be more reliable than, or preferable to, others. For example, Zhisheng considered the platform number given on the departures board to be more reliable than the one given in the published timetable. The agent's preferences de ne a preference ordering on the agent's belief set. Typically the preference ordering is partial. For example, Zhisheng believed that the timetable showed that the next train for Siena would depart from platform one, and he believed that the departures board showed that the next train for Siena would depart from platform two. Since both were observations, both were equally reliable, so he had no preference between them. Indeed, he regarded them with indi erence. At any point in time the agent's beliefs and preferences amongst them form the agent's belief hierarchy at that point in time. Typically the agent's belief hierarchy is dynamic; as time progresses the agent's beliefs and the preferences among them change. For example, Zhisheng initially believed that the next train for Siena would depart from platform one, however, after looking at the departures board, he believed instead that the train would depart from platform two. However, the belief hierarchies of rational agents tend to exhibit a certain stability. For example, Zhisheng did not reconsider his beliefs about what he had observed. The agent's beliefs and the preferences among them thus persist by default. Indeed, the belief hierarchies of rational agents tend to be upwardly stable; that is, the higher the belief in the hierarchy, the more it tends to remain in and maintain its relative position in it. For example, Zhisheng's beliefs about what he had observed were more stable than his beliefs about which platform the next train for Siena would depart from. This re ects the principle that rational agents should keep higher-level beliefs in preference to lower-level beliefs whenever possible. The beliefs in the hierarchy of a rational agent should also be coherent; that is, they should, in some sense, be jointly consistent. Roughly, an agent's belief hierarchy is coherent if every belief in the hierarchy is consistent with every belief which is not less preferred than it. If a rational agent realises that its beliefs are incoherent, the agent should revise them in order to restore coherence. In doing so the agent should retain more preferred beliefs in favour of less preferred ones wherever coherence permits. Moreover, the agent should only make those changes which are necessary in order to restore coherence. For example, Zhisheng's belief that the departures board was correct was inconsistent with his belief that the published timetable was
correct, so he restored consistency by dropping the latter, less preferred, belief. We believe that we are developing an interesting new de nition of beliefs and rational belief revision as part of the larger project of producing a formal theory of practical rationality [2]. Our de nition of belief hierarchies is motivated by Quine's \Web of Belief" metaphor [13, 14, 15], especially as explicated by Dummett [5]. There are also interesting similarities and di erences between our theory and the \AGM" theory of belief revision developed by Alchourron, Gardenfors and Makinson [7, 8]. Our approach can perhaps be seen as a generalisation of theirs. Moreover it is interesting in its own right as it aims to give a formal theory of beliefs, preferences, etc., which can be extended to include the interaction with goals, etc.; see e.g. the formalisation of goals in [3] and our commonsense theory of the interaction between perception and belief given the possibility of illusion [4]. Our theory is expressed in the language CA [1] which has been extended to include the preference operator of ALX [9, 10]. In order to make this paper self contained, we provide a concise account of the relevant features of this language in the next section. In Section 3 we then give the formal de nition of beliefs and belief hierarchies study their static properties. In the concluding section we consider the rational revision of beliefs and belief hierarchies and show how preferential entailment can be used to formalise this. By way of illustration, we show how our opening example can be formalised. We then discuss the relationship between our approach and the AGM theory of belief revision. Finally we discuss ongoing work on resource-bounded reasoning.
2 Time, Candidate Beliefs, Preferences, Indi erence CA is a rst-order many-sorted language with explicit reference to time. For
example, the sentence OnTable(B )(3) states that block B is on the table at time point 3. Thus time is taken to be composed of points and, for simplicity, we will assume that it is discrete and linear. The models of CA are fairly complex possible-worlds structures. Each model comes equipped with an interpretation function V which assigns an n-ary relation to each n-ary relation symbol at each time point at each possible world. Thus, for model M , world w in M and variable assignment g:1
M; w; g j= r(u1 ; : : : un )(t) i (u1 ; : : : ; un) 2 V (r; t; w) A sentence of the form CBel(a; )(t) states that proposition is one of agent a's candidate beliefs at time t; that is, that a is considering whether to adopt as a belief at t. The formal semantics for this operator is, for simplicity, the
standard possible-worlds semantics, but indexed by agent and time point. For example, for agent a, time t and world w, R(Bel;a;t;w) is a binary accessibility relation on worlds which represents a's candidate beliefs in w for at t. R(Bel;a;t;w) 1
For the sake of simplicity of presentation we will let the distinction between terms and their denotations in M given g take care of itself.
is required to be transitive and Euclidean, corresponding to positive and negative introspection, however it is not required to be serial, thus a's candidate beliefs at t need not be jointly consistent. The truth condition for the candidate belief operator is then as follows:
M; w; g j= CBel(a; )(t) i M; w ; g j= for all (w; w ) 2 R(Bel;a;t;w) : 0
0
A sentence of the form Pref (a; ; )(t) states that agent a prefers to at time t. The semantics of the preference operator begin with von Wright's conjunction expansion principle [18]. According to this principle, to say that you prefer an apple to an orange is to say that you prefer situations in which you have an apple and no orange to those in which you have an orange and no apple. In possible-worlds terms this principle might be stated as follows: agent a prefers to if a prefers ^ : -worlds to ^ : -worlds. However, this semantics is too simple, as it leads to paradoxes involving conjunction and disjunction. If is preferred to then _ is preferred to , and is preferred to ^ . For example, if a prefers co ee to tea, then a prefers co ee or poison to tea, and a prefers co ee to tea and a million dollars. Clearly we need to capture the ceteris paribus nature of preferences: we should compare ^ : worlds and ^ : -worlds which otherwise di er as little as possible from the actual world. In order to do so we introduce the selection function from the Stalnaker-Lewis analysis of conditionals [12, 17]. Thus the function cw is of type W P (W ) ! P (W ), and, intuitively, cw(w; [ ] M g ) is the set of closest worlds to w in which is true.2 Formally, cw is required to satisfy the conditions imposed by Lewis in [12]. The agent's preferences over time are represented by the function : A T ! P (P (W ) P (W )), which assigns a comparison relation over sets of worlds to each agent at each time point. Intuitively, for sets of worlds X and Y , X (a;t) Y means that agent a prefers the worlds in X to the worlds in Y at time t. Preferences are required to be irre exive and transitive, and should satisfy left and right disjunction. Accordingly, let X =(a;t;w) Y abbreviate cw(w; X \ Y ) (a;t) cw(w; Y \ X ). Then each =(a;t;w) is required satisfy the following properties: (irp) (trp) (orl) (orr)
X 6=(a;t;w) X:
If X =(a;t;w) Y and Y =(a;t;w) Z then X =(a;t;w) Z: If X =(a;t;w) Z and Y =(a;t;w) Z then X [ Y =(a;t;w) Z: If X =(a;t;w) Y and X =(a;t;w) Z then X =(a;t;w) Y [ Z:
The truth condition for preferences is then as follows: M M; w; g j= Pref (a; ; )(t) i [ ] M g =(a;t;w) [ ] g :
Given these semantics, we have the following axioms: 2 As usual, [ ] M g denotes the set of worlds in M in which is satis ed by g; i.e., [ ] M g = fw 2 W : M; w; g j= g.
(IRP ) :Pref (a; ; )(t) (TRP ) Pref (a; ; )(t) ^ Pref (a; ; )(t) ! Pref (a; ; )(t) (ORL) Pref (a; ; )(t) ^ Pref (a; ; )(t) ! Pref (a; _ ; )(t) (ORR) Pref (a; ; )(t) ^ Pref (a; ; )(t) ! Pref (a; ; _ )(t) (CEP ) Pref (a; ; )(t) $ Pref (a; ( ^ : ); (: ^ ))(t) (IRP ) and (TRP ) state the irre exivity and transitivity of preferences respectively, while (ORL) and (ORR) respectively state left and right disjunction of preferences.3 Finally, (CEP ) states the conjunction expansion principle. The following are theorems: (AS ) Pref ( ; )(t) ! :Pref ( ; )(t) (CP ) Pref (a; ; )(t) ! Pref (a; : ; : )(t) Thus preferences are asymmetric (AS ) and contraposable (CP ). Note that Pref (a; ; )(t) implies neither Pref (a; _ ; )(t) nor Pref (a; ; ^ )(t), so the paradoxes of conjunction and disjunction of preferences are avoided. We also require indi erence and weak preference operators. Informally, Ind(a; ; )(t) states that agent a is indi erent between and at time t, while PrefInd (a; ; )(t) states that a weakly prefers to at time t; that is, either a strongly prefers to at t, or a is indi erent between and at t. In order to do so, we require a stronger notion of (strong) preference. Each =(a;t;w) should now also be almost connected: (acp) If X =(a;t;w) Y then for any Z 2 P (W ) either X =(a;t;w) Z or
Z =(a;t;w) Y:
Then the indi erence relation, (a;t;w), can be de ned as follows:
X (a;t) Y i X 6=(a;t;w) Y and Y = 6 (a;t;w) X: We thus have the following additional axioms for preference and indi erence: (ACP ) Pref (a; ; )(t) ^ :Pref (a; ; )(t) ! Pref (a; ; )(t) (IND) Ind(a; ; )(t) $ :Pref (a; ; )(t) ^ :Pref (a; ; )(t) (TRI ) Ind(a; ; )(t) ^ Ind(a; ; )(t) ! Ind(a; ; )(t) (ACP ) states that preferences are almost connected and (TRI ) states that indi erence is transitive. Obviously it follows from (IND) that indi erence is also re exive (REI ), and symmetric (SY I ): (REI ) Ind(a; ; )(t) (SY I ) Ind(a; ; )(t) ! Ind(a; ; )(t) Finally, the weak preference operator is introduced by de nition: (WP ) PrefInd (a; ; )(t) $ Pref (a; ; )(t) _ Ind(a; ; )(t) 3
The disjunctive properties of preferences were suggested by Pierre-Yves Schobbens.
Proposition 1. Properties of (strong) preference, weak preference and indi er-
ence.
1. Consistency of preference and indi erence: Pref (a; ; )(t) ^ Ind(a; ; )(t) ! Pref (a; ; )(t) Pref (a; ; )(t) ^ Ind(a; ; )(t) ! Pref (a; ; )(t) 2. Weak preference is re exive, transitive, and comparable: PrefInd (a; ; )(t) PrefInd (a; ; )(t) ^ PrefInd (a; ; )(t) ! PrefInd (a; ; )(t) PrefInd (a; ; )(t) _ PrefInd (a; ; )(t) 3. Consistency of indi erence and weak preference:
Ind(a; ; )(t) $ PrefInd (a; ; )(t) ^ PrefInd (a; ; )(t) 4. Consistency of (strong) preference and weak preference: Pref (a; ; )(t) $ :PrefInd (a; ; )(t) 5. Exactly one of the following holds: Pref (a; ; )(t); Ind(a; ; )(t); Pref (a; ; )(t) Proof. (1) For the rst part, suppose that Pref (a; ; )(t) and Ind(a; ; )(t) but that :Pref (a; ; )(t). If Pref (a; ; )(t) then, by transitivity of preference, we have Pref (a; ; )(t), contradicting the supposition that Ind(a; ; )(t). So it must be the case that :Pref (a; ; )(t). Hence, by de nition, Ind(a; ; )(t). But then, as indi erence is transitive, we have Ind(a; ; )(t), contradicting the supposition that Pref (a; ; )(t). So it must be the case that Pref (a; ; )(t). For the second part, suppose that Pref (a; ; )(t) and Ind(a; ; )(t) but that :Pref (a; ; )(t). If Pref (a; ; )(t) then, by transitivity of preference, Pref (a; ; )(t), contradicting the supposition that Ind(a; ; )(t). So it must be the case that :Pref (a; ; )(t). Hence, by de nition, Ind(a; ; )(t). But then, as indi erence is transitive, we have Ind(a; ; )(t), contradicting the supposition that Pref (a; ; )(t). So it must be the case that Pref (a; ; )(t). (2) Re exivity. Since Pref is irre exive, we have :Pref (a; ; )(t). By the definition of indi erence, this means that Ind(a; ; )(t). Thus PrefInd (a; ; )(t). Transitivity. Suppose that PrefInd (a; ; )(t) ^ PrefInd (a; ; )(t), then there are four cases to consider. Case 1. Pref (a; ; )(t) ^ Pref (a; ; )(t). Since preference is transitive, we have Pref (a; ; )(t). So, by de nition, PrefInd (a; ; )(t). Case 2. Pref (a; ; )(t) ^ Ind(a; ; )(t). By part (1), Pref (a; ; )(t) holds. So, by de nition, PrefInd (a; ; )(t). Case 3. Ind(a; ; )(t) ^ Pref (a; ; )(t). Similarly, by part (1), we have PrefInd (a; ; )(t).
Case 4. Ind(a; ; )(t) ^ Ind(a; ; )(t). By the transitivity of indi erence we have Ind(a; ; )(t), so, by de nition, PrefInd (a; ; )(t). Comparability. Suppose that :PrefInd (a; ; )(t). Then, by de nition, :Pref (a; ; )(t) and :Ind(a; ; ). So it follows from the de nition of indi erence that Pref (a; ; )(t). So it follows from the de nition of weak preference that PrefInd (a; ; )(t). For (3), suppose that Ind(a; ; )(t). By the de nition of weak preference we have PrefInd (a; ; )(t). And, by the symmetry of indi erence and the de nition of weak preference, we have PrefInd (a; ; )(t). Conversely, suppose that PrefInd (a; ; )(t) ^ PrefInd (a; ; )(t). If :Ind(a; ; )(t) holds, then by the symmetry of indi erence, we also have :Ind(a; ; )(t). Furthermore, from the de nition of weak preference, we have Pref (a; ; )(t) ^ Pref (a; ; )(t). So, by the transitivity of (strong) preference, we have Pref (a; ; )(t). But this contradicts the irre exivity of preference. Thus, we conclude that Ind(a; ; )(t). For (4), suppose that Pref (a; ; )(t). If PrefInd (a; ; )(t), it follows by de nition that either Pref (a; ; )(t) or Ind(a; ; )(t). But, the former contradicts the asymmetry of preference, and the latter contradicts the irre exivity of preference by part (1). Conversely, suppose that :PrefInd (a; ; )(t). Then, by de nition of weak preference, :Pref (a; ; )(t) and :Ind(a; ; )(t). As :Ind(a; ; )(t), it follows that either Pref (a; ; )(t) or Pref (a; ; )(t). The former contradicts :Pref (a; ; )(t). So we conclude the latter. (5) is straightforward from (4). 2 In complex sentences the same agent and temporal terms are often repeated. When abbreviating such sentences we will usually adopt the convention that a missing agent term is the same as the closest agent term to its left, and that a missing temporal term is the same as the closest temporal term to its right. For example CBel(a; Pref (a; ; )(t))(t) can be abbreviated to CBel(a; Pref ( ; ))(t).
3 Belief Hierarchies In this section we will de ne a belief to be a coherent candidate belief. Informally, we assume that at any point in time the agent has a (perhaps empty) belief hierarchy, and is considering whether or not to adopt one or more candidate beliefs as beliefs. In order to do so, the agent needs to ensure that the candidate belief coheres (in a sense to be de ned) with all other beliefs in the hierarchy. The coherence requirement means that beliefs cannot be de ned independently, but only with reference to the agent's existing belief hierarchy. We will therefore give a recursive de nition of beliefs. By introducing an ordering on the agent's belief set at time t we obtain the agent's belief hierarchy at t. The ordering is represented by means of the preference operator; thus, for example, Pref (a; Bel( ); Bel( ))(t) states that a considers that is a more reliable belief than at t; that is, if, other things being equal, a could only believe one of the two beliefs, then a would believe rather than . Similarly Ind(a; Bel( ); Bel( ))(t) states that a regards the
beliefs and to be equally reliable, for brevity we will say that and are peer beliefs (of a's at t), and PrefInd (a; ; )(t) states that a regards the belief that to be at least as reliable as the belief that . In order to de ne beliefs, we will, for technical reasons, suppose that the agent also has a hierarchy of candidate beliefs which is required to satisfy the following conditions: (RPCB ) Pref (a; CBel( ); CBel( ))(t) ! CBel(a; )(t) ^ CBel(a; )(t) (PRB ) Pref (a; Bel( ); Bel( ))(t) ! Bel(a; )(t) ^ Bel(a; )(t) (PBCB ) Pref (a; Bel( ); Bel( ))(t) ! Pref (a; CBel( ); CBel( ))(t) (PCBB ) Pref (a; CBel( ); CBel( ))(t) ^ Bel(a; )(t) ^ Bel(a; )(t) ! Pref (a; Bel( ); Bel( ))(t) (RPCB ) is a realism condition on preferences between candidate beliefs. To say that at t, a prefers candidate belief to candidate belief should imply that and are candidate beliefs for a at t. Similarly (PRB ) is a realism condition on preferences between beliefs. (PBCB ) and (PCBB ) together ensure the agent's preferences on candidate beliefs are consistent with its preferences on beliefs. The last three conditions are, of course, equivalent to the following one: Pref (a; Bel( ); Bel( ))(t) $ Pref (a; CBel( ); CBel( ))(t) ^ Bel(a; )(t) ^ Bel(a; )(t) The axioms for the irre exivity and transitivity of preferences, (IR) and (TR), ensure that the preference orderings on beliefs and candidate beliefs are strict partial orderings. The corresponding weak preference orderings on (candidate) beliefs are, of course, pre-orderings. We also need to de ne what it means for a belief to be preferred to a candidate belief: Pref (a; Bel( ); CBel( ))(t) $ Pref (a; CBel( ); CBel( ))(t)^ Bel(a; )(t): We are now in a position to give our formal de nition of coherence, and thus beliefs. De nition 2. A candidate belief is P-coherent if the agent believes that it is jointly consistent with every belief that the agent prefers to it:4 V PCoherent(a; )(t) $ :CBel(a; ^ f : Pref (a; Bel( ); CBel( ))(t)g ! ?)(t)): 4
In keeping with the assumption of resource boundedness we are assuming that at any time point the agent's candidate beliefs can be represented by nite set of candidate beliefs. Intuitively, the agent is explicitly aware of the candidate beliefs in this set, and implicitly aware of those candidate beliefs which are consequences of them; this is discussed further in Section 5. In [9] section 11.6.1 (pp 179-184), there is a theorem which ensures that higher-order quanti cation over preferred formulas can be reduced to a nite set (and thus to a conjunction) of preferred formulas. Furthermore, this nite set always exists if we start with a niteVset of formulas and use the preference axioms. We use the set-theoretical notation f : Pref (a; Bel( ); CBel( ))(t)g to denote the conjunction on the nite formula set in which V every formula satis es the property Pref (a; Bel( ); CBel( ))(t). As usual, ; $ >.
De nition 3. A candidate belief is PI-coherent if it is P-coherent, and it coheres
with all peer candidate beliefs which are P-coherent: PICoherent(a; )(t) $ PCoherent )(t) ^ PCoherent(a; ^ Vf : Ind(a;(a;CBel ( ); CBel( )) ^ PCoherent( )g)(t):
De nition 4. A belief is a PI-coherent candidate belief:
Bel(a; )(t) $ CBel(a; )(t) ^ PICoherent(a; )(t): Proposition 5. Static properties of candidate beliefs and beliefs.
1. Any maximal candidate belief is a belief: CBel(a; )(t) ^ :9 6= (PrefInd (a; CBel( ); CBel( ))(t)) ! Bel(a; )(t): 2. All beliefs are candidate beliefs: Bel(a; )(t) ! CBel(a; )(t): 3. Beliefs are consistent: Bel(a; )(t) ! :Bel(a; : )(t): 4. Beliefs are well behaved under conjunction and closed under implication: Bel(a; ^ )(t) ! Bel(a; )(t) ^ Bel(a; )(t); Bel(a; )(t) ^ Bel(a; )(t) ! Bel(a; ^ )(t); Bel(a; )(t) ^ Bel(a; ! )(t) ! Bel(a; )(t): 5. Consistency and maximality principles for peer beliefs: PrefInd (a; CBel( ); CBel( ))(t) ^ :PICoherent(a; ^ )(t) ! :(Bel( )(t) ^ Bel( )(t)); Ind(a; CB ( ); CB ( ))(t) ^ PCoherent(a; )(t)^ PCoherent(a; )(t) ! Bel(a; _ )(t): Proof. For (1), if is a candidate belief, for an agent a at time t,5 and there is no more reliable candidate belief than , then is a maximal CBel. As is a maximal CBel, it is coherent, and hence it is also a Belief. (2) and (3) follow from the de nition of beliefs. For (4), suppose that ^ is a belief for a at t. If :Bel(a; )(t) holds, then by the de nition of belief, either :PCoherent(a; )(t) holds, or PCoherent(a; )(t)^ 9 (ICBel(a; ; )(t)^PCoherent(a; )(t)^:PCoherent(a; ^ )(t) holds.6 The former contradicts Bel(a; ^ )(t). While from the latter it follows, by PCoherent(a; )(t) and Bel(a; ^ )(t), that PCoherent(a; ^ ^ )(t). We thus have PCoherent(a; ^ )(t), contradicting :PCoherent(a; ^ )(t). Thus, we conclude that Bel(a; )(t) holds. The proof for Bel(a; )(t) is similar. For (5), suppose that Bel(a; )(t) ^ Bel(a; )(t) holds. We know that either Pref (a; CBel( ); CBel( ))(t) or Pref (a; CBel( ); CBel( ))(t) or Ind(a; CBel( ); 5 In the sequel, we will often omit the agent name a and the time point t in proofs when it does not cause any ambiguity. 6 Where we use ICBel(a; ; )(t) to denote is a conjunction of some peers of . Namely, Ind(a; CBel( ); CBel( ))(t) ! CBel(a; ; )(t) and Ind(a; CBel( ); CBel( 1 ))(t) ^ ICBel(a; ; 2 )(t) ! ICBel(a; ; 1 ^ 2 )(t).
CBel( ))(t) holds. Suppose that Pref (a; CBel( ); CBel( ))(t) holds. Then, from Bel(a; )(t), we know that PCoherent(a; ^ )(t) holds. If :Bel(a; ^ )(t) holds, then this means that there exists a such that ICBel(a; ^ ; )(t) ^ PCoherent(a; )(t) ^:PCoherent(a; ^ ^ )(t) holds. However, from PCoher ent(a; )(t) and Pref (a; CBel( ); CBel( ))(t) and Pref (a; CBel( ); CBel( )), we have PCoherent(a; ^ ^ )(t), contradicting :PCoherent(a; ^ ^ )(t). The proof for the case where Pref (a; CBel( ); CBel( ))(t) holds is similar. For the case where Ind(a; CBel( ); CBel( ))(t) holds we know by the de nition of beliefs that PCoherent(a; ^ )(t) and there exists no other peer such that :PCoherent(a; ^ ^ )(t). Therefore, we conclude that Bel(a; ^ )(t). For (6), suppose that Bel(a; )(t) and Bel(a; ! )(t). Then it follows from the semantics for candidate beliefs and the de nition of belief that CBel(a; )(t). It also follows from the de nition of belief that PICoherent(a; )(t) and PICoher ent(a; ! )(t). From property (5) of Proposition 1 it follows that either PrefInd (a; CBel( ); CBel( ! ))(t) or PrefInd (a; CBel( ! ); CBel( ))(t). In either case it follows that Coherent(a; )(t). (7) is straightforward from the de nition of belief and the consistency of belief hierarchies. For (8), suppose that PrefInd (a; CBel( ); CBel( ))(t)^PCoherent(a; )(t)^ PCohrent(a; )(t). If :Bel(a; _ )(t) holds, then by the de nition of belief, we have either :PCoherent(a; _ )(t) or there exists a such that ICBel(a; ; _ )(t)^PCoherent(a; )(t)^:PCoherent(a; ^ ( _ ))(t). In the former case it follows that either :PCoherent(a; )(t) or :PCoherent(a; )(t), giving a contradiction in each case. While from the latter it follows that is inconsistent with _ , which contradicts the supposition that is P-Coherent.
2
4 Belief Revision Thus far our analysis has been concerned with the static properties of beliefs and belief hierarchies, with the properties of agents' beliefs and belief hierarchies at particular points in time. In this section we consider the dynamic properties of beliefs and belief hierarchies; that is, how they should be revised over time. As suggested in the introduction we follow what Gardenfors calls the principle of informational economy. Thus the rational agent should only revise its beliefs when they become incoherent; for example, when a new candidate belief is preferred to but not coherent with an existing belief, or when the preferences between beliefs change. When this is the case the agent should keep higher-level beliefs in preference to lower-level beliefs wherever coherence permits, and the agent should only make those changes which are necessary in order to restore coherence. Otherwise the agent's beliefs and preferences among them should persist by default. In order to represent the persistence of beliefs and preferences, we use the a ected operator, A , of CA. This modal operator is analogous to the Ab predicate of the Situation Calculus. Let be a meta-variable which ranges over the
non-temporal component of atomic modal formulas.7 Then a formula (t) is a ected at t if its truth value at t di ers from its truth value at t + 1:
M; w; g j= A ()(t) i M; w; g j= :((t) $ (t + 1)): We thus have the following persistence rule:
(t) ^ :A ()(t) ! (t + 1): Intuitively we are interested in models in which this schema is used from left-to-right only in order to reason \forwards in time" from instances of its antecedent to instances of its consequent. Typically also we want to be able to infer the second conjunct of each instance nonmonotonically whenever it is consistent to do so. For example, if we have Bel(a; )(t) then we want to be able to use the rule to infer Bel(a; )(t +1) if A (Bel(a; ))(t) cannot be inferred. In order to enforce this interpretation, we de ne a prioritised form of preferential entailment [16]. De nition 6. Let A1 ; : : : ; An be a partition of the atomic modal sentences of
n di erent types according to their type.8 For each Ai , model M and time point t, let MA =t = f i (t ) 2 Ai j t t; M j= i (t )g. Then a model M is chronologically less de ned than a model M on the basis of the priorities hA1 ; : : : ; An i, written M A1;:::;A M i M and M di er at most on the interpretation of A1 ; : : : ; An and there is a time point t such that: { for some i such that 1 i n; MA =t MA =t, and { for all j such that 1 j i; MA =t MA =t: 0
i
0
0
0
h
0
ni
0
0
i
i
0
j
j
De nition 7. A model M is an hA1 ; : : : ; An i-preferred model of a sentence
if M j= and there is no model M such that M j= and M A1 ;:::;A M . Similarly, M is an hA1 ; : : : ; An i-preferred model of a set of sentences if M j= and there is no model M such that M j= and M A1 ;:::;A M . 0
0
0
0
0
0
h
h
ni
ni
De nition 8. A set of sentences preferentially entails a sentence given
the priorities hA1 ; : : : ; An i (written j A1 ;:::;A ) if, for any hA1 ; : : : ; An ipreferred model M of , M j= . h
ni
In the case of belief hierarchies, we are interested in hCBel; Pref ; A i-preferred models. Candidate beliefs, preferences and a ected atoms should be minimised chronologically while, at any time point, candidate beliefs should be minimised before preferences, and preferences should be minimised before a ected atoms. In the sequel we will abbreviate j CBel;Pref;Aff to j . As a result of the de nitions we have: 7 Atomic modal formulas are formulas of the form op(a; 1 ; : : : ; n )(t), where n 1 and op is a modal operator other than A . 8 For example, A1 might be the set Bel of all belief atoms Bel(a; )(t), A2 might be the set Pref of all preference atoms Pref (a; ; )(t), etc. h
i
Proposition 9. Dynamic properties of beliefs and belief hierarchies.
1. 2. 3. 4.
Beliefs persist by default. Preferences on beliefs persist by default. Belief hierarchies persist by default. Belief hierarchies are upwardly stable.
Proof. For (1), let be an appropriate theory of beliefs such that j Bel(a; )(t) and j :A (Bel(a; ))(t). Then it follows from the persistence rule that j Bel(a; )(t + 1). The proof for (2) is similar. Part (3) follows from (1) and (2). Part (4) follows from the maximality and default persistence of belief hierarchies. 2
By way of illustration, we show how the opening example can be formalised. Example 1. Let 1; 2; ::; denote time points, and one and two denote the two platforms. Then the given facts about the agent beliefs, preferences and candidate beliefs are expressed by the following set, , of sentences: (A) Pref (a; CBel(Timetable(one));CBel(8x(Timetable(x) ! Platform(x))))(1); (B ) Pref (a; CBel(Board(two)); CBel(8x(Board(x) ! Platform(x))))(2); (C ) Pref (a; CBel(8x(Board(x) ! Platform(x)); CBel(8x(Timetable(x) ! Platform(x)))(2); (D) Pref (a; CBel(Infoman(two)); CBel(8x(Infoman(x) ! Platform(x))))(3); (E ) Pref (a; CBel(8x(Infoman(x) ! Platform(x));CBel(8x(Board(x) ! Platform(x)))(3); (F ) Pref (a; CBel(Train(one)); CBel(8x(Train(x) ! Platform(x))))(4); (G) Pref (a; CBel(8x(Board(x) ! Platform(x)); CBel(8x(Train(x) ! Platform(x)))(4); (H ) Pref (a; CBel(Signman(one)); CBel(8x(Signman(x) ! Platform(x))))(5); (I ) Pref (a; CBel(8x(Signman(x) ! Platform(x)); CBel(8x(Infoman(x) ! Platform(x)))(5); (J ) 8 Pref (a; CBel((Platform(one) _ Platform(two)) ^ :(Platform(one)^ Platform(two)); CBel(a; ))(1):
For natural numbers n1 and n2 such that 1 n1 n2 7, we will use ([n1 : : : n2 ]) to denote the conjunction (n1 ) ^ (n1 + 1) ^ ::: ^ (n2 ). Then the
following sentences are true in all hCBel; Pref ; A i-preferred models of : (a) Pref (a; CBel(Timetable(one));CBel(8x(Timetable(x) ! Platform(x))))([1 : : : 5]); (b) Pref (a; CBel(Board(two)); CBel(8x(Board(x) ! Platform(x))))([2 : : : 5]); (c) Pref (a; CBel(8x(Board(x) ! Platform(x));CBel(8x(Timetable(x) ! Platform(x)))([2 : : : 5]); (d) Pref (a; CBel(Infoman(two)); CBel(8x(Infoman(x) ! Platform(x))))([3 : : : 5]); (e) Pref (a; CBel(8x(Infoman(x) ! Platform(x));CBel(8x(Board(x) ! Platform(x)))([3 : : : 5]); (f ) Pref (a; CBel(Train(one)); CBel(8x(Train(x) ! Platform(x))))([4 : : : 5]); (g) Pref (a; CBel(8x(Board(x) ! Platform(x));CBel(8x(Train(x) ! Platform(x)))([4 : : : 5]); (h) Pref (a; CBel(Signman(one)); CBel(8x(Signman(x) ! Platform(x))))(5); (i) Pref (a; CBel(8x(Signman(x) ! Platform(x)); CBel(8x(Infoman(x) ! Platform(x)))(5); (j ) 8 Pref (a; CBel((Platform(one) _ Platform(two)) ^ :(Platform(one)^ Platform(two)); CBel(a; ))([1 : : : 5]): So in all hCBel; Pref ; A i-preferred models of a's beliefs change as follows during the period: (0) Bel(a; (Platform(one) _ Platform(two))^ :(Platform(one) ^ Platform(two))([1 : : : 5]) (j ); (1) Bel(a; Timetable(one))(1) (a) (2) Bel(a; 8x(Timetable(x) ! Platform(x))(1) (1); (a) (3) Bel(a; Platform(one))(1) (1); (2) (4) :Bel(a; Platform(two))(1) (0); (3) (5) Bel(a; Board(two))(2) (b) (6) Bel(a; 8x(Board(x) ! Platform(x))(2) (b); (c) (7) :Bel(a; 8x(Timetable(x) ! Platform(x))(2) (c) (8) Bel(a; Platform(two))(2) (5); (6)) (9) :Bel(a; Platform(one))(2) (0); (8) (10) Bel(a; Infoman(two))(3) (d) (11) Bel(a; 8x(Infoman(x) ! Platform(x)))(3) (10) (12) Bel(a; Platform(two)(3) (10); (11) (13) :Bel(a; Platfrom(one))(3) (0); (12) (14) Bel(a; Train(one))(4) (f ) (15) :Bel(a; 8x(Train(x) ! Platform(x))(4) (g) (16) Bel(a; Platform(two))(4) (Persistence) (17) :Bel(a; Platform(one))(4) (0); (16) (18) Bel(a; Signman(one))(5) (h) (19) Bel(a; 8x(Signman(x) ! Platform(x))(5) (h) (20) :Bel(a; 8x(Infoman(x) ! Platform(x))(5) (i) (21) Bel(a; Platform(one))(5) (18); (19) (22) :Bel(a; Platform(two))(5) (0); (21)
2
A further illlustration of the use of the theory is its use in our commonsense theory of the interaction between perceptions and beliefs given the possibility of illusion [4].
It is interesting to compare our work with the \AGM" theory of belief revision developed by Alchourron, Gardenfors and Makinson, e.g. [7, 8]. On the AGMview, an agent's beliefs are represented by a knowledge set; a deductively closed set of sentences. At any stage, a knowledge set can be modi ed in one of three ways: Expansion : A proposition , which is consistent with a knowledge set K , is added to K . The result is denoted K + . Revision : A proposition , which may be inconsistent with a knowledge set K , is added to it. In order to maintain consistency some of the propositions which were in K may have to be removed. A revision of K by is denoted by K . Contraction : A proposition is removed from a knowledge set K . A contraction of K by is denoted by K ?_ .
Alchourron, Gardenfors and Makinson propose a number of plausible postulates which any de nition of these operations should satisfy. The postulates for expansion are straightforward. The postulates for revision are as follows:9 (a) (Closure) K is a closed theory. (b) (Inclusion) K K + : (c) (Vacuity) If : 62 K; then K + K : (d) (Success) 2 K : (e) (Consistency) If ? 2 K ; then : 2 Cn(;): (f ) (Extensionality) If Cn(K ) = Cn(K ); then K = K : The postulates for contraction need not concern us as the Harper identity shows that the contraction operation can be de ned in terms of the revision operation: K ?_ = (A : ) \ K: Our theory di ers from the AGM theory in many important respects. For example, in our theory beliefs are represented as hierarchies of propositions, rather than sets of sentences, and the preferences among beliefs must be considered when revision takes place. Secondly, revision of a hierarchy from one time point to the next may correspond to several AGM operations; several beliefs may have to be removed in order to incorporate new ones, while several others may simply be added or deleted. The revision of a hierarchy will always be unique; unlike the result of an AGM revision. Finally, an agent's preferences may change over time, unlike a given entrenchment ordering in the AGM approach. In order to make a comparison we consider the special case in which each revision of a belief hierarchy correponds to a single AGM operation. Given a theory and resulting belief hierachy at time t, we can de ne an agent a's belief set at t as follows: 0
9
Bel(a; t) = f : j Bel(a; )(t)g: Where, Cn(S ) is the deductive closure of S .
0
Given an operation on a's belief set at t and the proposition , we are thus interested in a's belief set at t + 1. An AGM-type expansion operator can be partially de ned as follows:
Bel(a; t) + = f : j Bel(a; )(t)g [ f : j Bel(a; ! )(t + 1)g when j Bel(a; )(t + 1): The assumption that the expansion Bel(a; t) + is the only operation which occurs at t is captured by the following condition: (Uni+) If Bel(a; t) + is de ned, then Bel(a; t) + = Bel(a; t + 1): An AGM-type revision operator can be partially de ned in a similar way:
Bel(a; t) = f : j Bel(a; )(t) ^ :A (Bel(a; )(t))g[ f : j Bel(a; ! )(t + 1)g when j Bel(a; )(t + 1): The assumption that the revision Bel(a; t) is the only operation which occurs at t is captured by the following condition: (Uni*) If Bel(a; t) is de ned, then Bel(a; t) = Bel(a; t + 1): Proposition 10. The belief revision operator de ned above satis es AGM Pos-
tulates (a)-(f).
5 Resource-bounded Reasoning Thus far we have been considering agents who are ideal KD45 reasoners, and so have given an idealised formalisation of Quine's web of belief. In this section we outline some preliminary ideas for formalising resource-bounded reasoning. The one limitation was the assumption behind the de nition of coherence (De nition 2) that the agent's candidate beliefs can be represented by a nite set of candidate beliefs. This assumption can be made explicit by means of Fagin and Halpern's awareness operator [6]. A sentence of the form Aware(a; )(t) states that agent a is aware of at time t. The semantics of the awareness operator is given by an awareness function which assigns a set of formulas to each agent at each time point. The truth condition for the awareness operator is then as follows: M; w; g j= Aware(a; )(t)i 2 Aw(a; t): In keeping with resource-boundedness, we will additionally require that each awareness set, Aw(a; t), is nite, and will use it to represent the nite set of candidate beliefs that agent a is aware of at t. The agent then considers those candidate beliefs of which it is aware:
ACBel(a; )(t) $ Aware(a; CBel( ))(t) ^ CBel(a; )(t):
Then, for example, coherence can be de ned as follows:
PCoherent(a; )(t) $ :CBel(a; ^ Vf : Pref (a; Bel( ); ACBel( ))(t)g ! ?)(t)):
As a result, the belief hierarchies of agents are nite. However the problem with using an awareness operator is, as Fagin and Halpern point out, its syntactic nature. In order to overcome this it is necessary to add additional conditions in order to ensure for example that the agent is aware of CBel( ^ )(t) i it is aware of CBel( ^ )(t). In on-going work, we are considering a suitable set of such conditions. Morever, the use of the CBel operator in the above de nition means that, while the the agent may have a nite set of beleifs, its reasoning about them is still ideal. In order to capture resource-bounded reasoning, we are investigating the use of an additional awareness operator to represent the consequences of the agent's resource-bounded reasoning at t which could then be used to modify the use of the CBel operator in the above de nition. Once again, the problem is to give a plausible de nition of the awareness set; agent's should draw \interesting" or \relevant" consequences of other candidate beliefs, but not trivial or irrelevant consequences; for example those which follow from the idempotence of ^ and _, or those which result from the purposeless use of tautologies such as ! ( _ ). The overall e ect would be to combine our hierarchical approach to beliefs with the resource-bounded view suggested by Konolige [11]; where the agent has an initial belief set and forms the closure of this using rules which will typically be weaker than those of belief logics such as KD45.
Acknowledgements This research forms part of the Ratio Project and is supported by the United Kingdom Engineering and Physical Sciences Research Council under grant number GR/L34914.
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4. J. Bell and Z. Huang. Seeing is Believing. Proceedings of Common Sense-98. R. Miller and M. Shanahan (eds.), pp. 391-327. Available at: http://www.dcs.qmw.ac.uk/conferences/CS98/. 5. M. Dummett. The Signi cance of Quine's Indeterminacy Thesis. Synthese 27 1974, pp. 351-97. 6. R. Fagin and J. Halpern. Belief, Awareness, and Limited Reasoning, Arti cial Intelligence 34 (1988) 39-76. 7. P. Gardenfors. Knowledge in Flux; Modeling the Dynamics of Epistemic States. MIT Press, Cambridge, Massachusetts, 1988. 8. P. Gardenfors, and D. Makinson, Revisions of knowledge systems using epistemic entrenchment, in: M. Y. Vardi (ed.), Proceedings of TARK'88, Morgan Kaufmann Publishers, Inc., 1988. pp. 83-95. 9. Huang, Z., Logics for Agents with Bounded Rationality, ILLC Dissertation series 1994-10, University of Amsterdam, 1994. 10. Huang, Z., Masuch, M., and Polos, L., ALX: an action logic for agents with bounded rationality, Arti cial Intelligence 82 (1996), pp. 101-153. 11. K. Konolige. A Deduction Model of Belief, Morgan Kaufmann, Los Altos, California, 1986. 12. Lewis, D., Counterfactuals, Basil Blackwell, Oxford, 1973. 13. W.V.O. Quine. Two Dogmas of Empiricism. In: From a Logical Point of View. Harvard University Press, Cambridge, Massachusetts, 1953. 14. W.V.O. Quine. Word and Object. MIT Press, Cambridge, Massachusetts, 1960. 15. W.V.O. Quine, and J.S. Ullian, The Web of belief, Random house, New York, 1970. 16. Shoham, Y. Reasoning About Change. MIT Press, Cambridge, Massachusetts, 1988. 17. Stalnaker, R., A theory of conditionals, Studies in Logical Theory, American Philosophical Quarterly 2 (1968), pp. 9 8-122. 18. von Wright, G., The Logic of Preference, Edinburgh University Press, Edinburgh, 1963.
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