Persistence and Minimality in Epistemic Logic

Annals of Mathematics and Arti cial Intelligence 0 (1999) ?{?

Persistence and Minimality in Epistemic Logic

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Wiebe van der Hoek a Jan Jaspars b Elias Thijsse c Computer Science, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, the Netherlands E-mail: [email protected] b Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, the Netherlands E-mail: [email protected] c Faculty of Arts, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, the Netherlands E-mail: [email protected]

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Received 15 December 1998; revised 21 August 1999 We give a general approach to characterizing minimal information in a modal context. Our modal treatment can be used for many applications, but is especially relevant under epistemic interpretations of the operator 2. Relative to an arbitrary modal system, we give three characterizations of minimal information and provide conditions under which these characterizations are equivalent. We then study information orders based on bisimulations and Ehrenfeucht-Frasse games. Moving to the area of epistemic logics, we show that for one of these orders almost all systems trivialize the notion of minimal information. An other order which we present is much more promising as it permits to minimize with respect to positive knowledge. The resulting notion of minimality coincides with well-established accounts of minimal knowledge in S5. For S4 we compare the two orders. Keywords: epistemic and modal logic, minimal knowledge, Ehrenfeucht-Frasse orders, bisimulation, only knowing, disjunction property. AMS Subject classi cation: Primary 03B45; Secondary 03B46

1. Introduction This paper delivers a general report on the issue of the minimal informational content of modal assertions. This issue is most prominent in the area of modal epistemic logic [10] where the question has been addressed what it means

The authors would like to thank the referees for their detailed and constructive comments.

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van der Hoek, Jaspars, Thijsse / Persistence and Minimality in Epistemic Logic

that one agent `knows more than ' another. But also in the setting of a single agent, we want to compare his dierent states of knowledge. A central issue here is to obtain a formal description of the agent's knowledge containing not more than the information conveyed by some formula ', that is, the case in which ' is the agent's only knowledge. A satisfactory logical analysis of ònly knowing' is of essential importance for knowledge representation and inference, in analogy with closed world assumptions in the eld of database theory and logic programming. Formulas which can be ònly known', so-called honest formulas, can be used as a precise description of an agent's state of knowledge. Moreover, drawing `common sense' inferences from a knowledge base, describing the agent's knowledge, often relies on the assumption that this knowledge base represents the agent's only knowledge.1 A simple fact p should be rendered honest, and can therefore be used as a description of an agent's knowledge. The modal translation of this knowledge of p is written as 2p. From the additional assumption that p is only known, we can infer, apart from logical consequences like 2(p _ r), that the agent does not know any other atomic fact q: :2q. However, not every formula of single agent epistemic logic is honest. Consider the following proposition: \I know whether it is raining". Suppose that p represents \it is raining". Then the sentence above can be written in epistemic logic as 2p _ 2:p, and the fact that the sentence describes the agent's knowledge, as 2(2p _ 2:p). If we would accept that 2p _ 2:p is the only knowledge of the agent, then we could infer that the agent does not know that p: :2p, and in the same fashion, we would infer that he does not know that :p: :2:p. However, these two simple inferences together contradict the initial knowledge. This demonstrates the dishonesty of the formula 2p _ 2:p. The analysis of honesty generally depends on the epistemic background logic. What is especially important here, is which introspective capacities we are ready to attribute to the agent. For example, if the background logic contains the axiom of positive introspection 2 ! 22 we can infer 22p if only p is known. 1

Compare, for example Konolige's autoepistemic analysis of non-monotonic inferences in default logic [17]. Another important example, is the quantitative maxim in the logic of conversation [8]. This principle of cooperation says that the information conveyed by utterances should be the only information that a speaker has about the subject.


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This seems innocent since the inferred knowledge is still related to the initial description p. On the other hand, if we accept the axiom of negative introspection :2 ! 2:2 , then we can infer knowledge concerning q, for example 2:2q, from only knowing p. This knowledge cannot be derived from only knowing p ^ q, which intuitively represents more knowledge than only knowing p. As we will see, this kind of inferences eects the treatment of honesty for dierent modal systems. Halpern and Moses [10] proposed three dierent ways of formalizing the concept of honesty, and showed their equivalence for the strong, fully introspective epistemic logic S5. The system S5 contains veridicality 2 ! , saying that what is known is also true, in addition to positive and negative introspection. From studies of other, weaker, epistemic systems [24,26], it becomes clear that the three formalizations of [10] highly depend on the strength of S5. The question arises whether we need to adapt such de nitions according to the axiomatization of knowledge, or that it is possible to generalize these de nitions such that they can be used for any epistemic system. Of course, the latter situation would be highly preferable, and it is the aim of this paper to prove that such a generalization is possible. Before we introduce our generalization, we shortly give our explanation of the three formalizations of honesty, inspired by [10]. Stable sets The rst characterization of honesty is based on the set representing the knowledge which can be inferred from a formula ' representing the agent's only knowledge. The criterion of stability ([22,25]) for such a set is then the same as honesty for the initial formula '. This set is then called the minimal stable set for '. Dierent analyses ([10,26]) suggest that minimality and stability depend on the background logic, which will be addressed below. Information orders Whereas the existence of a minimal stable set can be seen as a syntactic characterization of honesty, the second characterization yields a semantic de nition. The aim is to capture what ònly knowing' means in terms of the possible world semantics of modal logic. Firstly, `knowing more' is de ned using a structural information order over possible worlds. Then, an honest formula ' can be characterized as having a minimal possible world with respect to this information order, that is, a world which does not assign more knowledge to the agent than '. In this paper, we emphasize that the notion of information order induces the de nition of minimality, whereas [10] uses minimal models as a primitive notion. Nevertheless, the superset relation between universal S5-models can

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be conceived as an information order underlying their de nition. Disjunction property As may be clear from the dishonesty example given above, disjunction plays an important role. This idea is captured by the third characterization of honest formulas as obeying a disjunction property. The intuition behind this property is that what an agent derives from knowing an honest formula ' may not lead to uncertainty about his knowledge. In terms of disjunctions, if 2'1 _ : : : _ 2'n can be derived from 2', then at least one disjunct 2'i itself should be derivable from 2'. Sometimes, as we will see, a restriction on the disjuncts 2'i is needed or wanted to obtain the equivalence with the other characterizations of honesty.

At rst sight, the three characterizations depend on the background logic. For example, minimality of stable sets in the context of a negative introspective logic cannot be de ned with respect to all possible knowledge. A stable set containing p ^ q does not contain :2q, whereas a minimal stable set for p does contain :2q. Hence, minimality has to be de ned with respect to an appropriate sublanguage of epistemic formulas, for example, excluding knowledge about ignorance. In the case of S5, Halpern and Moses even exploit a smaller sublanguage, viz. that of factual formulas : the formulas without epistemic operators. In order to make the disjunction property work in an equivalent manner, similar restrictions for disjunctive conclusions have to be imposed. In other epistemic logics, like the positive introspective system of knowledge S4, such a restriction is not necessary to obtain a sensible de nition of minimal stable sets and the disjunction property ([26]). It seems that the distinction between the relevant sublanguages for the systems S4 and S5 obstructs a general approach to honesty. How can we overcome this problem, and if we manage, can we lift our solution to a general modal level? Generalizing the notion of stability by using maximal consistent sets is a rst step ([15]). It turns out that for full introspective systems, the knowledge contained by a maximal consistent set is the same as a stable set. The advantage of maximal consistency is that it applies to any modal system. Moreover, it immediately settles an equivalence between the existence of minimal stable sets and the disjunction property. But still, a general semantic de nition of `knowing more' and `knowing only' requires an appropriate de nition of an information order. To arrive at such a general view, we need to implement general persistence or preservation results


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which can be derived from results in modal logic. As we will show, these results help to establish the equivalence between the three dierent notions of honesty in a general fashion. A sublanguage is persistent over an information order if its formulas that are true in one world are still true when passing to a larger (with respect to the order) world. We will show that in general, a persistence result for a sublanguage over an information order immediately settles a connection between minimal possible worlds and maximally consistent sets which are minimal with respect to that sublanguage. The existence of a minimal world implies the existence of such a minimal set. In order to establish a full equivalence in line with [10], the chosen sublanguage also has to characterize the information order. This re ects the converse of persistence: if a possible world contains more information of the sublanguage than some other world, then it must also be recognized as a larger world by means of the information order. The most well-known description of persistence of àll' modal information over possible worlds is bisimulation equivalence [2,11]. By a straightforward modi cation of the bisimulation de nition we arrive at describing persistence of knowledge. A weaker version of this de nition, based on the well-known EhrenfeuchtFrasse games for showing equivalence of rst-order models [4], yields a persistence and characterization result for this sublanguage and thus the desired equivalence for the generalized de nitions of honesty. As we mentioned above, the sublanguage of formulas of the form 2 is too large for a satisfactory analysis of honesty in the case of negative introspective systems. In fact, its successful application to the milder logic S4 seems to be sheer luck considering its failure in cases of other epistemic logics than only the negative introspective ones. For S4.2, an intermediate system which has also been advocated as an appropriate logic of knowledge [19], it would entail dishonesty for all formulas. We will show that this failure is much more general. The question is whether there is a dierent sublanguage which would give a satisfactory analysis for epistemic systems. Our answer is positive! If we take the sublanguage of positive formulas, leaving ignorance out by not accepting 2operators to appear in the scope of a negation, the derived notion of honesty entails a satisfactory de nition. It turns out that all knowledge of positive formulas is equivalent to some disjunction of factual knowledge for the fully introspective systems. On the basis of this insight, the de nitions in [10] are equivalent with

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our reformulation on the basisof this larger language. For systems like S4 this generalization does not t the de nitions as given in [15,26], but we will argue that our positive sublanguage yields a more intuitive notion of `knowing more' and `knowing only', also in the case of S4.

2. Preliminaries Language

Our language L is just that of modal logic, with the modal operator 2 (necessity) and the connectives : (negation), ^ (conjunction) and _ (disjunction). We assume that formulas ' and in L are composed from a nite set of propositional atoms P = fp; q; r : : :g, using these operators. Other operators are introduced as abbreviations: 3 (possibility) is de ned by 3' = :2:', ! (implication) by ' ! = :' _ . A special atom ? is de ned as (p ^ :p), whereas > = :?. The function d : L ! N calculates the modal depth of formulas as follows: d(p) = d(>) = d(?) = 0 (p 2 P ), d(:') = d('), d(' ? ) = maxfd('); d( )g for ? = ^; _; !; $, and, nally, d(2') = d(3') = 1 + d('). Some properties to be presented are relative to a given subset L0 L. An example of such a sublanguage of L is L(n) = f' 2 L j d(') ng. For a unary operator 4 = :; 3; 2 and language L0 L, 4L0 = f4' j ' 2 L0 g. We also use an ìnverse': 4? L0 denotes f j 4 2 L0 g. Semantics

We use Kripke models M = hW; R; V i as a standard interpretation of the modal language, where W is the set of worlds, accessibility R is a relation on W and V is a mapping from W to propositional valuations (i.e., 8w 2 W; p 2 P : V (w)(p) 2 f0; 1g). Instead of Rwv, we also write v 2 R[w], where w and v denote worlds in M . Here, the key notion is the pair hM; wi (often written as M; w), called a state, in which each modal formula ' receives its standard interpretation with the typical modal case: M; w j= 2' i for all v 2 R[w], one has M; v j= '. For any state hM; wi, its theory is Th(M; w) = f' j M; w j= 'g. For ? L, M; w j= ? means that for all 2 ? : M; w j= . Relative to a given set of models S , consequence is de ned by ? j=S ' i for all M 2 S ; w 2 W : M; w j= ? implies M; w j= '.


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A major issue in this paper is how truth is preserved by moving from one state to another. Let L L and let S be a set of states. We say that an order on S preserves the sublanguage L, or that L is persistent over i for all states hM; wi and hM 0 ; w0 i:

M; w M 0 ; w0 ) for all ' 2 L : (M; w j= ' ) M 0; w0 j= '): In other words, preserves L if moving to a -greater state preserves the truth of L formulas. If the overall converse holds, we say that L characterizes on S . Note that, in the case of persistence of all formulas, the information order is an equivalence relation. Inference and modal systems

We will discuss several logical systems S on top of the minimal modal system K, assuming familiarity with the notion of derivability in a modal system S. In particular, for a set of premises ?, we write ? `S ' if there is a derivation of ' (without applications of necessitation to the premises) from ? in S. When S is clear from context, or when the particular normal system is not relevant, we will simply drop it. For a single premise we omit parentheses, i.e., ' ` means f'g ` ; for derivability without premises we write ` ' instead of ; ` '. The set CnS (') = f j ' `S g contains all formulas that can be derived from ' in S. The formulas ' and are equivalent in S, or S-equivalent , if both ' `S and `S '. The logic S is called nitary for a sublanguage L if it induces nitely many S-equivalence classes in L . Notice that such an L is not in general nite. As an immediate consequence of our assumption that P is nite, we have that every S we will consider is nitary layered , in the sense that for each n 2 N, S is nitary for L(n) .2 The generalisation that we use to lift `S from a subset of 2L L to a subset of 2L 2L is more analogous to the one used in sequent calculi (cf. [16]) than to our Hilbert-style presentation of a logical system: ? `S , 91 : : : n 2 : ? `S (1 _ _ n ) if 6= ;, and ? `S ; , ? `S ?. The minimal system K contains the rule of necessitation (` ' ) ` 2') and the modal axiom K : 2(' ! ) ! (2' ! 2 ) on top of any Hilbert2

This can be shown by use of normal forms, see [6].

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style axiomatization of propositional logic. Here, we are interested in normal systems S, i.e., systems that are obtained by adding modal axioms to K. One such extension, T, is obtained by adding the axiom T : 2' ! ' to K. The system KD is named after the axiom that distinguishes it from K, which is D : 3> or, equivalently, 2' ! 3'. In epistemic logic, systems that include axiom 4: 2' ! 22' are called positively introspective, those that have axiom 5: :2' ! 2:2' are called negatively introspective. Two other axioms worth mentioning are B : ' ! 23' and G: 32' ! 23'. Any combination of the axioms mentioned is called an epistemic logic, here. Typical examples of such systems are S4 (T + 4), S5 (S4 + 5) and S4.2 (S4 + G). When 2 is interpreted as belief, the axiom T is often replaced by D, which gives rise to systems that obtain their name directly from the constituting axioms: KD, KD4, KD45, etc. Given a logic S, a set of formulas ? is S-consistent if ? 6`S ?. We say that ? L satis es the S-disjunction property 3 (S-DP) over a sublanguage L , if ? is S-consistent and for every 1 ; 2 ; : : : ; k 2 L: ? `S ( 1 _ _ k ) ) for some i k : ? `S i : A state veri es a logic S if it veri es all the theorems of S, i.e., all that is derivable without premises. The set of states verifying S (S-states for short) is called stateS . For a given formula ' we de ne stateS (') = fhM; wi 2 stateS j M; w j= 'g. An S-model is a model of which all states verify the logic S. Completeness

A set of formulas ? is maximal S-consistent (S-m.c.) if it is S-consistent and moreover contains all the formulas ' for which ? [ f'g is consistent. We use a strong form of the Lindenbaum Lemma (see, e.g., [3]): if ? 6`S then there is an S-m.c. ? such that \ = ;. In other words, such a consistent set ? can be expanded to an S-m.c. set which is disjoint from . 3

The disjunction property originates from intuitionistic logic, where it holds in the case ? = ; ([7]). In the context of modal logic a similar notion called the 'rule of disjunction' has been introduced by Lemmon ([18]) as a property of modal systems, with ? = ; and L = 2L; see also [14]. The seminal paper [10] considered the disjunction property for single knowledge formulas, ? = f2g, in the setting of S5, restricted to disjunctive conclusions of factual knowledge: L = 2L(0) .


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Linking up deduction and semantics, a crucial role is played by the maximal consistent sets ?. All S-m.c. sets together constitute the set of possible worlds WS in the canonical model MS = hWS ; RS ; VS i for S. The de nition of VS , saying that VS (?)(p) = 1 , p 2 ?, together with that of RS (RS ? , 2? ? ), guarantees the following truth lemma to hold:

MS ; ? j= ' , ' 2 ? Combining this truth lemma with Lindenbaum's Lemma then immediately proves (strong) completeness of S with respect to any class of models S that contains MS : ? j=S ' ) ? `S '. The converse of this implication is called (strong) soundness of S with respect to S . Soundness trivially holds for the class of S-models. Thus, for any class of S-models S that contains MS , indeed ? j=S ' , ? `S '. In particular, since MS is an S-model,4 S is strongly sound and complete for the class of all S-models. Notice, however, that the class of S-models and most of its subclasses are `chaotic' in the sense that the models in it are not characterized by their accessibility relation alone. Many classes of models S have been identi ed that are sound and complete with respect to the systems S that were mentioned above (see [3] or the more recent [23]). Most signi cant are those classes of models that are determined by a property of their accessibility relation.5 For instance, for KD one takes the serial (i.e., 8x9yRxy) Kripke models, for T the accessibility relation has to be re exive, in KD4 it is serial and transitive, and in S5 it is an equivalence relation.

3. Minimal information in modal logic Let S be an arbitrary modal system and the information order a preorder6 on stateS . A formula ' is called honest with respect to S and , if there exists a least S-state verifying 2'. More precisely,

De nition 1. A formula ' is S-honest (for ) i there is an S-state M; w such that:

For, if ? is a world in MS and `S ', then ' 2 ?, and so, by the truth lemma MS ; ? j= '. If this is possible, the system is frame complete wrt the (frame) class. Not all systems can be characterized in such a way, i.e., there are (frame) incomplete systems, see [14]. All the speci c systems discussed in this paper are frame complete. 6 A relation is a pre-order if it is re exive and transitive.

4

5

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M; w j= 2', M 0 ; w0 j= 2' ) M; w M 0; w0 for all hM 0 ; w0 i 2 stateS. In other words, ' is honest (for S and ) i there is a -least element in stateS (2'). Although our initial motivation for the notion of honesty is strictly semantic in nature, we would like to relate it to syntactic and deductive criteria. This will be the topic of investigation in Section 3.1. 3.1. General characterizations of minimality

Let L be a sublanguage of L, such that L is persistent over . We consider the following approaches to minimality: (1) Formula ' has a -least verifying S-state (2) ' has an L -smallest S-m.c. expansion (3) ' has S-DP with respect to L The next result, which is visualized in Figure 1, relates the three approaches.

' has a -least

verifying S-state (1)

/

' has an L -smallest

S-m.c. expansion (2)

@ @

-

@

@@ R

' has S-DP

with respect to L (3)

Figure 1. Relating least states, expansions and disjunction properties

Theorem 2. Let L be persistent over . Then the minimality approaches (2) and (3) are equivalent, while (1) implies both (2) and (3).

Proof. The proof of this and the next theorem invokes the canonical S-model and exploits the usual properties of m.c. sets. The equivalence of (2) and (3) is mirrored in the fact that is an S-m.c. set i both


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is deductively closed, i.e., `S ) 2 , and has S-DP, i.e., is S-consistent and for all ; : : : ; k 2 L: `S _ _ k ) `S i for some i k. (1) ) (2) Assume that there is a -least element hM; wi in stateS ('). Let ' = Th(M; w). Clearly ' is S-m.c. and ' 2 ' . Now suppose is an S-m.c. set such that ' 2 . Let MS be the canonical S-model and consider the canonical state hMS ; i. Since ' 2 we have that, by the truth lemma, MS ; j= ', thus hM; wi hMS ; i. Therefore, if 2 ' \ L , then M; w j= , hence (by persistence) MS ; j= , and therefore (by the truth lemma) 2 . Consequently, ' \ L . (2) ) (3) Assume (2) holds, that is, that there is an S-maximal consistent set ' with the least L -part among the maximal consistent expansions of a given '. To show that ' has the S-disjunction property over L , suppose ; : : : k 2 L and ' `S _ _ k . Then, since ' is an S-maximal consistent set, we have ( _ _ k ) 2 ' and, so there is an i k for which i 2 ' . In order to show ' `S i we will use the completeness theorem. Let hM; wi be any state in stateS ('). Since Th(M; w) is an S-maximal consistent set containing ', we have, by (2), ' \ L Th(M; w), hence M; w j= i . In short, ' j=S i , and therefore (using strong completeness) ' `S i . (3) ) (2) Assume ' has the S-disjunction property over L (3). We are 1

1

1

1

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looking for a maximal consistent expansion ' of ' with the smallest intersection with L among maximal consistent expansions of '. This set ' is de ned by considering rst = L n CnS ('). Suppose that would satisfy ' `S (). Then there are 1 ; : : : ; k 2 such that ' `S 1 _ _ k and so, by (3), there is a i 2 with ' `S i , i.e., i 2 CnS ('). Since this contradicts i 2 , we conclude that our assumption () cannot be true, and hence we have ' 6`S . Applying the strong Lindenbaum Lemma now gives us an S-m.c. set ' f'g with ' \ = ' \ L n CnS (') = ;, implying ' \ L CnS (') (*). To show that ' is the L -smallest S-m.c. expansion of ', let 2 ' \ L . By (*), we nd 2 CnS (') and so, the S-consequence of ' must be contained in every S-m.c. expansion of '.

Notice that L in Theorem 2 need not be the largest persistent sublanguage. Moreover, it may be too weak to characterize . Consequently, persistence does not guarantee the implications (3) ) (1) and (2) ) (1) to hold in general. They can be established by adding the converse of persistence. Let us say that in

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that case the minimal information equivalences hold for L and . This is the situation displayed in Figure 2.

' has a -least

/

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' has an L -smallest

S-m.c. expansion (2)

verifying S-state (1)

I @ @

-

@

@@ R

' has S-DP

with respect to L (3)

Figure 2. Minimal information equivalences for S, L and

Theorem 3. Let L be a persistent sublanguage of L which also characterizes . Then the minimal information equivalences hold for L and . Proof. In the light of Theorem 2, we only have to verify (2) ) (1). Assume there is an S-m.c. set ' with smallest L -part among the S-m.c. expansions of '. Consider the canonical S-model MS . Since ' 2 ' we have that, by the truth lemma, MS ; ' j= '. Take any hM; wi 2 stateS (') and notice that ' 2 Th(M; w) (an m.c. set), thus ' \ L Th(M; w). Since L characterizes and ' = Th(MS ; '), it follows that hMS ; 'i hM; wi, and so hMS; ' i is a least S-state verifying '.

Corollary 4. Let L L be persistent and characterizing for . Then the following propositions are equivalent:

' is S-honest for 2' has an L-smallest S-m.c. expansion 2' has S-DP over L In the literature ([22,25]) there is a lot of emphasis on stable sets; in our terminology a stable set is simply the knowledge contained in an m.c. set, or more formally: is stable if = 2?? for some S-m.c. ?, as introduced in [15]. If


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' 2 and is stable, is called a stable expansion of '.7 The second condition for ' being honest can now be rephrased as:

' has a 2?L-smallest stable expansion Although Corollary 4, together with the condition above, solves the problem of alternative characterizations of honesty in an abstract sense, the solution is not entirely satisfactory. Notice that Theorem 3 and the resulting minimal information equivalences are based on the assumption that L is both persistent and characterizing for . However, it is unclear, up to this point, whether suitable orders exist that enable persistent sublanguages to characterize them. And, if so, we would like to specify them in an independent, insightful way. With this end in view we propose several speci c orders in the next subsections. 3.2. Bisimulation and minimality

Our de nitions of information orders over states are related to the notion of bisimulation, which constitutes a standard method in modal logic and process algebra to show that two states have the same informational content [2,11]. The generality of this technique enables us to de ne information orders that can be used for arbitrary modal systems. However, unrestricted bisimulation turns out to be too strong for a proper information order.

De nition 5. Let M = hW; R; V i and M 0 = hW 0 ; R0; V 0i be two Kripke models. A bisimulation is a non-empty relation B W W 0 such that for all w 2 W , w0 2 W 0 with Bww0 : V (w) = V (w0 ) if Rwu for some u 2 W , then there is a u0 2 W 0 with R0w0 u0 and Buu0 (forth) if R0w0 u0 for some u0 2 W 0, then there is a u 2 W with Rwu and Buu0 (back) If there exists a bisimulation B between the models M and M 0 such that Bww0 for worlds w in M and w0 in M 0 , we also say that the states hM; wi and hM 0 ; w0 i bisimulate and we write hM; wi $ hM 0 ; w0 i. 7

Our direct de nition of `stable expansion' generalizes the notion characterized by Moore's xpoint equation in [22].

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It is a well-known result (cf. [2]) that any sublanguage L is invariant (i.e., two-way persistent) over bisimulations, i.e., for all ' 2 L and all states hM; wi; hM 0 ; w0 i one has:

hM; wi $ hM 0 ; w0 i ) (M; w j= ' , M 0; w0 j= '): Nevertheless, the existence of a bisimulation is not a necessary condition for modal equivalence of two states.

Example 6. Consider the following pair of models M and M 0, with every world

having the same local valuation of propositional variables (see Figure 3). For every natural number n the model M has a branch of length n. M 0 is similar to M , except that it also has an in nitely long branch. In this example the accessibility relation R is irre exive and intransitive.

0 M

0 M’

Figure 3. Two models with the same theory that do not bisimulate

For M and M 0 thus de ned, we have that: for all ' 2 L: hM; 0i j= ' , hM 0 ; 0i j= ' hM; 0i 6$ hM 0; 0i, i.e., there exists no bisimulation between hM; 0i and hM 0 ; 0i Summing up, the models of Example 6 are such that world 0 has the same theory in both models, but a bisimulation between those two states cannot be


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given (subsequent steps in M 0 on the in nite path cannot be simulated in M ). Similar diculties arise when we implement straightforward modi cations of the bisimulation de nition to obtain appropriate information orders re ecting knowledge persistence. The most obvious is to de ne that a state M2 ; w2 extends a state M1 ; w1 if for every successor v2 of w2 in M2 , an M1 -successor v1 of w1 can be found such that M1 ; v1 and M2 ; v2 bisimulate, which intuitively represents that in the state M2 ; w2 we have 'less possibilities' or 'less uncertainties'. In this case the two models in Figure 3 can be used as an illustration that this de nition of extension is not a necessary condition for knowledge persistence: M; 0 is a proper extension of M 0 ; 0, while they contain the same knowledge. This illustrates that a modi cation of information orders is therefore needed. 3.3. Ehrenfeucht-Frasse orders

We will now inspect orders inspired by Ehrenfeucht-Frasse games.8 The idea behind this move is that the notion of bisimulation is too strong in general: it may involve an unlimited chain of accessible worlds in one model, whereas such an in nite branch is not directly relevant for modal formulas, for the simple reason that each formula has a nite modal depth, and there is no point to look for worlds in chains beyond that depth. We will illustrate this point on the two models displayed in Figure 3. Starting from the root 0 in both models, we look at `spheres' of worlds that are successively further from 0. Within these spheres the models should behave the same, and indeed they do. So: up to any nite depth the models are equivalent. This situation is depicted in Figure 4. The Ehrenfeucht-Frasse orders are de ned by means of underlying, `layered' pre-orders. To make the connection, we will present a general lemma that paves the way. Suppose n is a pre-order on stateS for each natural number n (`layer n'). Moreover, let be de ned by: (M = hW; R; V i and M 0 = hW 0 ; R0; V 0 i)

M; w M 0 ; w0 () 8n 2 N 8v0 2 R0 [w0 ] 9v 2 R[w] : M; v n M 0 ; v0 Finally, let L be a sublanguage and L(n) = L \ L(n) be its subset of formulas of modal depth up to n. 8

See [4] for the use of Ehrenfeucht-Frasse games in rst-order predicate logic, in which modal logic can be embedded [2].

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0

0

M

M’

Figure 4. Two models with the same theory that are equivalent up to each nite depth

Lemma 7 (Collecting). If Ln is persistent and characterizing for n, and L is closed under _, then 2L is persistent and characterizing for , i.e., M; w M 0 ; w0 () 8' 2 2L : (M; w j= ' ) M 0 ; w0 j= ') ( )

Proof. Let L(n) be persistent and characterizing for n and L closed under _. For the `)' direction, assume that M; w M 0 ; w0 and let ' = 2 2 2L be such that the depth of ' is d(') = d( ) + 1 = n + 1. Suppose M 0 ; w0 6j= ', then there is a v0 2 R0 [w0 ] with M 0 ; v0 6j= . Since M; w M 0 ; w0 , we nd a v 2 R[w] for which M; v n M 0 ; v0 . Persistence of L(n) over n then guarantees that M; v 6j= , hence M; w 6j= 2 , i.e., M; w 6j= '. For the proof of `(', suppose that M; w 6 M 0 ; w0 . This means that for some n 2 N and some v0 2 R0 [w0 ], we have that for all v 2 R[w], M; v 6n M 0; v0 . Since the language L(n) is characterizing for n , we then can nd, for every v 2 R[w], a formula 'v 2 L(n) for which M; v j= 'v whereas M 0 ; v0 6j= 'v . Now note that every logical system S that we consider is nitary layered, which implies that there are only nitely many S-equivalence classes for L(n) . Thus we can nd nitely many formulas 'v ; : : : ; 'vm 2 L(n) such that each of the 'v 's mentioned W above is S-equivalent to some 'vi i m. Now, consider = im 'vi . Then 2 L(n) and M; w j= 2 , but at the same time M 0; w0 6j= 2 . 1

With this tool we present two important Ehrenfeucht-Frasse orders.


17

3.3.1. General information order In the rst Ehrenfeucht-Frasse order the underlying, layered order is an equivalence relation. The relation 'n is de ned recursively by:

M; w ' M 0; w0 () V (8w) = V 0 (w0 ) > M; w 'n M 0 ; w0 & < M; w 'n M 0 ; w0 () 8v0 2 R0[w0 ] 9v 2 R[w] : M; v 'n M 0; v0 (back) & > : 8v 2 R[w] 9v0 2 R0[w0 ] : M; v 'n M 0; v0 (forth) Then it can be shown that two states are 'n -equivalent i they verify the same 0

+1

formulas up to depth n:

M; w 'n M 0 ; w0 () 8' 2 L(n) : (M; w j= ' () M 0; w0 j= '):

(1)

From left to right the external equivalence expresses persistence over 'n, from right to left it says that L(n) characterizes 'n. At rst blush it may seem that the latter involves an ordering rather than an equivalence relation between the verifying models. However, on second thought there is no problem: if 8' 2 L(n) : (M; w j= ' ) M 0 ; w0 j= '), then we also have the converse (take ' = : ). Therefore we can make use of Lemma 7. For the sake of illustration, reconsider the two states labelled 0 in Figure 4. Instead of judging the bisimulative behaviour on the basis of all accessibilities, the Ehrenfeucht-Frasse equivalence compares the two states on the basis on what information is accessible in a given nite number of steps. It can be seen quite easily that in this way the two states hM; 0i and hM 0 ; 0i in Figure 4 are Ehrenfeucht-Frasse-equivalent for each nite number, and therefore contain the same information. First we de ne the general information order v based on strati ed Ehrenfeucht-Frasse equivalence:

De nition 8. The general information order v between two states hM; wi and hM 0 ; w0 i is de ned by: M; w v M 0 ; w0 () 8n 2 N 8v0 2 R0 [w0 ] 9v 2 R[w] : M; v 'n M 0 ; v0 Since L(n) is persistent and characterizing for 'n, Lemma 7 shows that 2L is both persistent and characterizing for v.

18


Lemma 9 (Modal characterization of v). M; w v M 0 ; w0 () 8' 2 2L : (M; w j= ' ) M 0 ; w0 j= '). This lemma justi es calling v an information order, as we did in De nition 8. If 2 is the knowledge operator of epistemic logic, the lemma above says that a state is smaller than another state if and only if in the latter state more information is known than in the former. By Theorem 3, we obtain the minimal information equivalences for the general information order (see also Figure 2).

Theorem 10. For any S, the minimal information equivalences hold for v and 2L. From now on, let us call honesty with respect to v general honesty. Corollary 4 then implies that ' is generally honest i 2' has an 2L-smallest m.c. expansion, i.e., ' has a smallest stable expansion. And in terms of disjunction properties, ' is generally honest i 2' has S-DP over 2L. Despite the result in Lemma 9, we will show that v is not a proper order for many epistemic systems. In the next section, we therefore introduce yet another Ehrenfeucht-Frasse order to overcome such problems. 3.3.2. Positive information order The second order is based on a genuine strati ed pre-order. The relation n is de ned recursively by:

M; w M 0 ; w0 () V ((w) = V 0(w0 ) w n M 0 ; w 0 & M; w n M 0; w0 () M; 8v0 2 R0[w0 ] 9v 2 R[w] : M; v n M 0 ; v0 (back) 0

+1

Notice the `forth' direction is typically missing here. Next we de ne the positive information order based on the strati ed Ehrenfeucht-Frasse pre-orders:

De nition 11. The positive information order between two states hM; wi and hM 0; w0 i is de ned by: M; w M 0; w0 () 8n 2 N 8v0 2 R0 [w0 ] 9v 2 R[w] : M; v n M 0 ; v0


19

In terms of knowledge, the order preserves positive knowledge. Technically this amounts to persistence of formulas that do not contain boxes in the scope of a negation. So let us de ne the persistent sublanguage of `positive knowledge':

L = f' 2 L j ' contains no 2 in the scope of :g +

In other words, formulas in L+ contain no negative occurrence of 2; therefore 3 is also not allowed in a positive knowledge formula (since 3 = :2:). Thus the sublanguage L+ amounts to the closure under ^, _ and 2 of propositional formulas. So 2p _ 2q, 2:p and 2p ^:q are members of L+ , but :2p and 3p _ 2q are not. Recall that L+(n) = L+ \ L(n) . One can prove, by induction on n, the following persistence and characterization result for L+(n) :

M; w n M 0 ; w0 () 8' 2 L+(n) : (M; w j= ' ) M 0 ; w0 j= ')

(2)

Notice that on the right hand side of (2), we do not have an equivalence as in (1); this is due to the fact that L+(n) is not closed under : (if n > 0). Again, by the equivalence (2) and the Collecting Lemma 7, we can easily prove that 2L+ is both persistent and characterizing for .

Lemma 12 (Modal characterization of ). M; w M 0; w0 () 8' 2 2L : (M; w j= ' ) M 0; w0 j= '). +

The next theorem then follows immediately from Lemma 12 and Theorem 3.

Theorem 13. For any S, the minimal information equivalences hold for and 2L . +

Let us call honesty with respect to positive honesty. Using Corollary 4, the last theorem implies that ' is positively honest i 2' has a 2L+ -smallest m.c. expansion, i.e., i ' has an L+ -smallest stable expansion. And in terms of disjunction properties, ' is positively S-honest i 2' has S-DP over 2L+. A simple, yet important case of the positive information order is the submodel relation. hM 0 ; wi is a submodel of hM; wi (M; w M 0 ; w) i W 0 W , R0 = R \ W 0 W 0 and V 0 (u) = V (u) for all u 2 W 0 . As a consequence of the (downward) Los theorem in rst-order logic, a modal formula is preserved under

20


submodels i it is equivalent to a `positive knowledge' formula, i.e., a formula in L+.9 Since 2L+ L+ this implies, using Lemma 12, that

Corollary 14. 1. preserves L 2. If M; w M 0 ; w, then M; w M 0 ; w +

Since the submodel relation is easily established, this provides a convenient tool for proving that two models are related by the positive information order. In [10] the submodel relation has been used for the de nition of minimal models for the system S5.

4. Evaluating minimality in epistemic systems In this section we evaluate the two information orders introduced before in the light of our broad class of epistemic systems, as introduced in Section 2. In fact, we will rst prove a negative result for an even larger class. To this purpose we de ne the notion of a Geach logic , see [3]. This is a normal modal system which contains (in addition to K) axioms of the form:

3k 2l ' ! 2m3n' with k; l; m; n 2 N; where 3k is de ned recursively: 3 ' = ' and 3k ' = 33k'. 0

(3)

+1

The most usual systems of epistemic logic, and in fact all the systems we consider epistemic logics here (recall that these amount to a selection of D; T; 4; 5; B; G added to K) consist of a combination of axioms of this format. For example, KD45 has three Geach axioms: D (k = m = 0; l = n = 1), 4 (k = n = 0; l = 1; m = 2) and 5 (k = m = n = 1; l = 0). The regularity of Geach logics is caused by the general correspondence between an axiom of the form (3) and the class of models in which the accessibility relation is k; l; m; n-con uent: 9

See [21] and [1]; since negations can be pushed to the propositional atoms, our L+ amounts to the closure under ^, _ and 2 of literals (atoms and negated atoms). This shows that our result is completely equivalent to its dual formulation in [1, thm.2.10], stated for the ùpward' version of the Los theorem.


21

8x; y; z 2 W : (Rk xy & Rm xz) ) 9w 2 W : (Rl yw & Rnzw): (4) Here R is simply the identity relation, and Rk = R Rk , the relational com0

+1

position of R and Rk . Every Geach logic corresponds to a conjunction of relational restrictions as given in (4). One easily veri es that all our epistemic logics (and many others) are in fact Geach logics. 4.1. General minimality

The general information order speci es that one world is smaller than another world if and only if the rst world represents less knowledge than the second. It turns out that for most systems this order is not appropriate. In most Geach logics this order trivializes the notion of general honesty: either all formulas are honest, or (nearly) all formulas are dishonest. Trivial honesty. For weaker modal systems such as K, K4, KD and KD4, which are sometimes used for belief and in which neither T nor 5 holds, it can be proved by a simple model-theoretic technique that all formulas ' such that 2' is consistent are honest with respect to the general information order.10 This technique is called simple amalgamation.11 For two S-states a simple amalgamation is constructed by adding one world from which all worlds are accessible which are accessible from the original two states. The construction is depicted in Figure 5. For every formula ' we obtain:

(M; w j= 2' & M 0 ; w0 j= 2') , M ; w j= 2':

(5)

Let S be a class of S-models which are characterized by the Geachean relational restrictions corresponding to S, MS 2 S (and therefore S is sound and complete with respect to S), and S be closed under amalgamation |which is the case for each of these weak modal systems. This proves the disjunction property of any consistent 2' over 2L, by using contraposition: Suppose that 2' 6`S 2 1 , 2' 6`S 2 2 , then we can nd S-states with M; w j= 2' ^ :2 1 and

Since all 2-formula are K- and K4-consistent (then hf0g; ;; V i with V arbitrary is a proper counter-example to 2' ` ?) for all ', in these two systems all formulas are generally honest. 11 Simple amalgamation is a slight modi cation of what is called àmalgamation' in [14]. The latter notion is what we will call `re exive amalgamation' in the remainder of this section.

10

22


@IPiPP 3 M

QkQ

1 3 P I@ @ w PPP QQ PPPQ @ w0 M 0 w

>

M Figure 5. A simple amalgamation of the states hM; wi and hM 0 ; w0 i.

M 0 ; w0 j= 2' ^ :2 2 . By (5) we know that M ; w j= 2', but obviously, also M ; w j= :(2 1 _ 2 2 ). In other words, 2' 6`S 2 1 _ 2 2 .

This argument shows that in every system S with only axioms of the form ! 2m 3n' with l; m + n > 0 trivial honesty occurs, since the semantic conditions for such Geach logics are preserved under simple amalgamation.

2l '

Trivial dishonesty. On the other hand, for many Geach logics the notion of general honesty de ates seriously. Such systems often contain theorems of the form 2'1 _ 2'2 where 2'1 and 2'2 are not theorems (*). In particular, systems which incorporate Geach axioms with k; m > 0 have the property (*).12 The disjunction property is then easily violated. In fact, one can prove the following theorem, bearing in mind that epistemic logics are de ned here to be systems extending K with a selection of T; D; 4; 5; B and G.

Theorem 15. For any epistemic logic S except for T, S4, and the weak systems K, KD, K4, and KD4, no S-consistent formula is generally honest. Examples of systems without consistent generally honest formulas are K45, KD45 and S5, but also those with a milder form of negative introspection such as S4.2. The only generally honest formulas in these systems are those ' that are S-inconsistent whereas 2' is S-consistent. 12

Another set of Geach axioms satisfying (*) are B and the like (k; l = 0; m > 0).


23

Remnants of general honesty. From the previous paragraphs it follows that the only epistemic systems which do not suer from trivial honesty or trivial dishonesty with respect to the general information order are T and S4. Purely propositional (i.e., factual, non-modal) formulas such as p _ q are both T- and S4-honest, while 2p _ 2q is dishonest in these two systems. A semantic test for general honesty in such systems as T and S4 is provided by the notion of rootability [15], which is de ned on the basis of re exive amalgamation. A re exive amalgamation of two states is de ned in the same way as a simple amalgamation with the only exception that the `root world' w is taken to be re exive: R w w . A formula ' is called rootable if for every pair of 2'-states, a re exive amalgamation can be found such that 2' holds in the root world. Rootability implies honesty, but not the other way around. Nevertheless, for every system which is closed under re exive amalgamation, such as S4 and T, every positive knowledge formula is generally honest if and only if it is rootable. So, for example, 2p _ q is rootable, and therefore, generally S4-honest. 4.2. Positive minimality

As we have seen the systems T and S4 permit non-trivial generally honest and dishonest formulas. However, there are good reasons to question the feasibility of the notion of general honesty. To begin with, general honesty cannot serve as a paramount notion of honesty, since in many systems this notion trivializes, as we have seen in the previous section. Moreover, for epistemic purposes it seems intuitively more sound to exclude formulas which represent ignorance, i.e., formulas of the form :2', when it comes to minimizing knowledge.

M1

up

w1

-:p u

v1

M2

up

w2

Figure 6. Two S4-models.

To understand the problem, consider the two models above (in all gures, we omit re exive arrows). Intuitively, one would say that the agent knows more in state hM2 ; w2 i than in hM1 ; w1 i, since the agent considers less possibilities in

24


hM ; w i. However, hM ; w i is not smaller than hM ; w i in terms of the general 2

2

1

1

2

2

information order. In the rst con guration the agent knows that he does not know that p, while in the second he does not know that he does not know that p, since he knows that p. This shows that the general information order on possible worlds does not t in with our intuition that `more knowledge' corresponds to `less uncertainty'. In the case of the positive information order, however, we do obtain M1 ; w1 M2 ; w2 , i.e. M1 ; w1 M2 ; w2 but M2 ; w2 6 M1 ; w1 For the system S5, the restriction to positive minimality turns out to be equivalent with the original analysis of honesty in [10]. In fact a more restricted version of minimality is given in [10], viz. with respect to the language 2L(0) (factual knowledge). However, it can be shown that in the system S5 the disjunction property with respect to this restricted language is equivalent to the disjunction property with respect to the language of positive knowledge formulas, using the fact that every formula 2 is S5-equivalent to a formal of modal depth 1, see e.g., [3]. For some modal systems such as S4, in which neither general nor positive minimality trivializes, it is interesting to compare the two orders. First let us note that for arbitrary normal systems general honesty implies positive honesty (this easily follows from the disjunction properties, or from the fact that if M; w v M 0 ; w0 then M; w M 0 ; w0 ):

Theorem 16. For any normal system S, if ' is S-honest with respect to v then ' is also S-honest with respect to . However, a similar transfer between dierent modal systems (and one kind of honesty) is not easily obtained. It may therefore be illuminating to contrast general and positive honesty (based on v and , respectively), for S4 with posip tive honesty for S5. Table 1 displays formulas which are honest ( ) or dishonest (?) in the indicated sense. From Theorem 15, we know that there are no (consistent) formulas that are generally honest in S5. Also, Theorem 16 explains why there are no witnesses for the cases 3 and 4 in the table. Cases 5 and 6 show that Theorem 16 cannot be strengthened to an ìf and only if' statement. Finally, note that there is no relationship between positive honesty in S5 and either general or positive honesty in S4. For illustrative purposes, let us prove both a positive and a negative entry in

van der Hoek, Jaspars, Thijsse / Persistence and Minimality in Epistemic Logic case

formula

1

p_q 2p _ q none none 2p _ 23q 2p _ 232q (2(2p _ q) ^ :2q) _ 2(p _ r) 2p _ 2q

2 3 4 5 6 7 8

25

S4gen S4pos S5pos p p p p ? ? ? ?

p p

? ? p p ? ?

p

? p ? p ? p ?

Table 1 Several (dis-)honest formulas for S4 and S5

Table 1. To start with, let us consider ' = 2p _ 23q (case 5 of Table 1). In order to demonstrate that 2' has the S4-DP with respect to 2L+, let ; 2 L+. To prove that 2' `S4 2 _ 2 implies that either 2' `S4 2, or 2' `S4 2 , we argue by contraposition. Hence assume that 2' 6`S4 2, and 2' 6`S4 2 . Using completeness, we nd two S4-models M = hW; R; V i and M 0 = hW 0 ; R0 ; V 0 i, with M; w j= 2' ^ :2; and M 0 ; w0 j= 2' ^ :2 Thus, there must be v and v0 such that M; v j= ' ^ :, and M 0 ; v0 j= ' ^ : (see Figure 7). Now, we build a new model M out of M and M 0 as follows. To the re exive amalgamation of the two models, we add a `common ceiling' u in which q is true, i.e., let W = W [ W 0 [ fw ; ug; R = R [ R0 [ (W fug) [ (fw g W ); and let V equal V on W , V 0 on W 0 , V (u)(q) = 1, and V on w be arbitrary. M is an S4-model, since it is both re exive and transitive. Because M ; u j= q, we have that M ; w j= 2'. Finally M ; w 6j= 2 _ 2 , for suppose that M ; w j= 2, then in particular M ; v j= . Since M is carefully constructed to be a submodel of M , by the corollary of the Los theorem: M; v j= , contradictory to assumption. Thus M ; w 6j= 2. By a similar argument, using the fact that M 0 is also a submodel of M , M ; w 6j= 2 . In all, M ; w 6j= 2 _ 2 , hence by

26


u

M

q M

M0

' : v

w

' : v0

w0

w

Figure 7. Adding a common ceiling and root to two S4-models

soundness 2' 6`S4 2 _ 2 . Since 2' is obviously S4-consistent, this completes the proof that ' is S4-honest with respect to the positive information order. To illustrate the derivation of a negative result in Table 1, let us consider the rst `?' in case 7. Let ' = (2(2p _ q) ^ :2q) _ 2(p _ r). We will disprove the S4-DP for ' with respect to 2L. First of all, it is easily veri ed that 2' `S4 2(2p _ q) _ 2(p _ r). However, the two models M and M 0 of Figure 8 illustrate that 2' 6j=S4 2(2p _ q) and that 2' 6j=S4 2(p _ r), respectively. This example also shows that ' is positive S4-dishonest, since the disjuncts are elements of 2L+.

M0

M

:p; :q; r

:p; q; :r Figure 8. Two counter models

p; :q; r


27

Let us summarize the main results of this evaluation. For our class of epistemic logics, only the systems T and S4 are non-trivial with respect to general honesty, and all systems apart from K, KD,K4, and KD4 are non-trivial with respect to positive honesty. This formally justi es our preference for the positive information order and related notions of sublanguage and honesty.

5. Conclusion Until now research in the notion of minimal information was mainly devoted to particular modal logics such as S4 and S5. Most often authors also used nonstandard semantics and specialized techniques to model minimal information. In this paper we oered a general approach to the representation of minimal information on the basis of standard Kripke models for arbitrary normal modal logics. The key idea is the use of preservation results for modal logic, and can be sketched as follows. A formula expresses minimal information i there is a least verifying model; this presupposes an order with respect to which the model is minimal. Next determine which formulas are preserved by this order, i.e., which formulas remain true when moving to a `greater' (more informative) model. If this sublanguage also characterizes the order, this is a suitable information order. Such an order and the sublanguage it preserves then provides precise syntactic, deductive and semantic criteria for minimality which can be used as equivalent tests for representation of minimal information by single, so-called honest, formulas. In the context of epistemic logic this analysis led to the choice of the positive Ehrenfeucht-Frasse order on models for the semantic analysis of minimality and the sublanguage of positive knowledge formulas for the syntactic and the deductive description of minimality. The conclusion is that this description oers an adequate analysis of minimality for epistemic systems. One may argue that for this analysis trivial honesty (i.e., all consistent formulas are honest) still occurs on the level of weak modal logics such as K, KD, K4 and KD4, and that, therefore, our choice is not adequate for those systems. In epistemic logic, however, these systems are simply too weak since in these systems a set as f2(2p _ 2:p); :2p; :2:pg is consistent, which seems unacceptable when 2 represents some epistemic attitude (which is, incidentally, an argument against Hintikka's KD4 axiomatization of belief [12]). It is implausible that an

28


agent may have the information that he has the information whether p, while on the other hand, he doubts whether p is the case. Needless to say the generality of our approach may inspire further research in this area. One obvious topic is the application of our results to particular systems, such as extensions of S4 like S4.2 and S4F, vide [24]. Other lines of future research may be the adaptation of our techniques to multiple agent systems and the move to partial semantics.

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