Dialogue Modelling in M-KRYPTON, a Hybrid Language for Multiple Bel ievers

Alessandro Saffiotti and Fabrizio Sebastiani

Department of Linguistics - University of Pisa Via S. Maria, 36 - 1-56 100 PISA

ABSTRACT We describe an attempt to combine insights from knowledge representation and epistemic logic. two research fields that, although closely related in principle, have not shown to date a high degree of cross-fertilization. On one side, hybrid knowledge representation systems embody a powerful representational paradigm, accounting for multiple, essentially different kinds of belief (or, popularly, "knowledge") in an integrated way. On the other side, the possible worlds semantics typical of epistemic logic is easily tailored to model various interesting notions, such as the existence of multiple believers and introspective belief. By recasting the KRYPTON hybrid KR system in terms of possible worlds semantics we have obtained a semantic account that, besides being "functionally" equivalent to the original one, is easily extensible to deal with operators (such as those for "propositional attitudes") that have traditionally been the subject of epistemic and related logics. In particular, we have concentrated on adding to KRYPTON the possibility of representing beliefs about the beliefs of multiple agents, where such agents may believe in propositions either of an "assertional" or of a "terminological" nature. The result is a knowledge representation language that, though absolutely general, is particularly geared to applications (such as user or dialogue modelling) where the interaction among multiple agents is involved.

BELIEF REPRESENTATION IN DIALOGUE MODELLING

In any situation involving a linguistic exchange, the way participants process information is influenced by a multifaceted body of knowledge. Endowing a computer system with the knowledge which is necessary to exhibit human-like dialogic behaviour requires designing a language that accommodates, in a principled way, the representation of the various types of knowledge involved.

The impulse for the work described in this paper has come from the necessity of defining such a knowledge representation language (KRL) to be used in a model of dialogue that also incorporates techniques for the detection and repair of miscommunications. We have focussed our attention on providing the representational primitives that are necessary for handling those instances of communication failure that can be traced back to undetected discrepancies between the knowledge bases (KBs) of participants 1171. The reason for this bias is that this kind of miscommunication, besides being pervasive in real life, is the one that is more deeply concerned with issues in the representation of knowledge and in the expressiveness of the representation language.

We will exemplify the importance of such issues in a context of dialogue modelling by hinting to the role of belief1 in dialogic

In the philosophical tradition the word "belief" is used instead of "knowledge" whenever there is no requirement of truth

behaviour. For example. if we were to formalize the INFORM speech act we might state that, as a result of agent a1 informing agent a2 that proposition p is true, either 1) a2 believes that p. or 2) a2 believes that al believes that p. or 3) a1 believes that a2 believes that p, or all of these things together. In typical accounts of speech acts (see for instance [15]). preconditions and effects do in fact involve instances of nested belief (i.e. beliefs about one's or other agents' beliefs); therefore, it seems unavoidable that a KRL adequate for dialogue modelling be capable of dealing, among other things, with the existence of multiple believers and with beliefs about (one's and other agents') beliefs.

We can hint to another, more general point concerning the role of belief in the use of natural language terms: typically, when using term t agent al believes that for agent a2 this term denotes the same concept as for her. and that, symmetrically, agent a2 believes the same thing of her. When these beliefs prove false, a miscommunication may result because the meaning of the term (hence, possibly of the whole utterance) that has been conveyed to a2 is different from the meaning al actually intended to convey. All this suggests that our representation language should also accommodate belief on the nature of concepts denoted by natural language terms (e.g. on their inner structure and on their relationships with other such concepts).

From a knowledge representation viewpoint, this latter problem can be recast in terms of the "hybrid knowledge representation systems" (hybrid KRSs - [l, 2 , 3, 201) paradigm. The notion of a hybrid KRS is that of a system which tightly integrates multiple representation languages to account for multiple, essentially different types of knowledge. The cited works have concentrated on accommodating both knowledge of facts ("assertional knowledge" - e.g. "Fabrizio lives in an igloo") and knowledge of the nature of entities these facts are about ("terminological knowledge" - e.g. "an igloo is made of ice"). Thus, assertional languages allow the creation of descriptive theories of the domain of discourse while terminological languages allow the creation of "conceptual dictionaries" (e.g. those semantic-network- and frame-based taxonomies well-known to all KL-ONE aficionados) describing the concepts involved in the domain of discourse and the conceptual relationships among them.

However, hybrid KRSs are only a partial solution to our belief representation problems, in that none of them is capable of dealing with belief (both in terminology and assertionality) in the case of multiple believers and nested belief.

This article describes our attempt to meet the representational needs of our dialogue model by extending the KRYPTON hybrid KRS [3] to handle these complex types of belief. This attempt has resulted in what we believe to be a promising paradigm of integration between hybrid knowledge representation and modal logic. Along this paradigm we have designed a

of what is known. As we do not make any assumption on the relationships between our representational structures and the outside world, any occurrence of the word "knowledge" in this paper will be best understood as actually referring to belief.

CH2552-8188/000010056$01 .OO 0 1988 IEEE 56

knowledge representation language (we have called it M-KRYPTON) that, though designed with our application in mind, has a character of generality. and is particularly geued to modelling situations where multiple reasoning agents uc involved (such as user modelling 1211 or communication protocols analysis [7]).

After a brief overview of KRyprON (roction 2) we give an alternative semantics for the KRYPTON assertional and terminological languages which seems to provide a more natural envinmment for developing the desired extensions; we also state its "functional" equivalence to the semantics originally given by B r a c h " and collugucs (roction 3). We then go on to describe our own extensions (section 4); these include developing a more unified approach towards asscrtionality and terminology. plus dealing with the representation of beliefs about the beliefs of multiple agents. Section 5 di.cusses some implications of our approach on dialogue modelling.

AN OVERVIEW OF KRYPTON

KRYPTON is a hybrid KRS that comprises two separate representation languages: 1) a terminological language. by meam of which concepts and

functional roles of these concepts can be descriW, for instance, the concept of a Logician can be expressed in KRYPTON by means of the term (VRGencric Scientist Subject Logic), where Scientist and Logic are themselves names of concepts and Subject is a name of a functional role;

2) an assertional language, by means of which facts about the domain of discourse may bc described that possibly make reference to such concepts and roles; for instance, the fact that some logicians are also poets can be expruscd in KRYPTON by the formula (EXISTS x (AND (Logician x) (Poet x))). where Logician and Poet arc names of concepts.

Both the differences and the relationships between the two languages arc accounted for by means of what the authors call a "hybrid S C ~ M ~ ~ C S " .

The other key feature of the KRYPTON system is that it embodier what Leverque [13] calls a "functional" approach to knowledge representation. This means that a user interacts with a KB (seen as an abstract data typc) only via an interface language consisting of a small set of precisely defined operations; thus, he/she may not be (and should not be) concerned with the actual data structures and mechanisms involved.

In this section we f m t give a complete description of the two representation languages together with an informal sketch of the underlying semantics; subsequently, we go on to briefly describe the interface-level operations that embody the "functional" specification of KRYPTON KBs. Space precludes us from giving a detailed formal description of the semantic account originally given to KRYPTON by its authors; for this we refer the reader to [3].

The assertional language of KRYPTON is a pure (i.e. with no individual constants and function symbols) first-order language. Hence, an assertional well-formed formula is defined according to the following syntax:

c w f b ::= (ck-ary predicate symbob < v w l ... e v e n ) I n 2 0 (NOT cwfb) I (OR <wfB cwfb) I (EXISTS evw <wfD)

In addition. the metalinguistic abbreviations AND, IMPLIES and FORALLm defmcd in the obvious way.

The terminological language permits the formation of complex terms ("gterms") according to the following syntax:

<gttnn> :F ck-ary prcdicate symbob I < c m c e p ~ I a010 cconcepb :F <unary predicate symbol> I

(ConGeneric <concepol ... cconccpb,,) I

57

(VRoencric <oareept> dole. < c o n e )

(Rolachain QolWl ... CrOlWJ e o l e ::= cbiiary prcdiicate symbol> I

We can thus express concepts obtained by role value restriction. such as the above mentioned Logician concept. by conceptual conjunction, such as (ConOeneric Logician Woman) ("a woman logician"). as well as roles obtained by role composition. such as (Rolcchain Parent Si*) ("a sister of a parent". i.e. an aunt).

Predicate symbols constitute the link between the two languages: each of them may be associated to a gterm which becomes its definition. as well used in an asscrtional wff.

As a starting point towards an integrated semantics for KRYPTON. the two languages are first given a semantics independently from each other: the truth of wffs is defied by mapping them into truth values, while the extension of gterms is defined by mapping them into sets of individuals.

Bridging the gap between the two languages starts from the notion of symbol table, i.e. a function S that associates to each predicate symbol of the assertional language a gterm which becomes its definition.

Consideration is then restricted to those notions of truth and extension that are "consistent" with the contents of a given symbol table S: these arc called truth valuations wrt S and ex tens ion functions wrt S. respcctively.

All the extension functions consistent with a symbol table S are then considered together in the definition of a subsumpt ion relationship between gterms: a g tmn g, is said to subsume gterm g2 ( w n S) iff its extension is a superset of g2 according to every extension function wrt S. Abstracting away from particular extension functions results in the required relationship between terminology and assertionality. with the former constraining the latter but not vice-versa.

An ordered pair formed by a subsumption relationship =)

and a truth valuation 0 is called an outcome, and may be s c m as a complete specification of a state of affairs and of a "conceptual dictionary".

Next, we turn to briefly describe the interface level of KRYPTON. The fact that a KB is usually an incomplete specification of a state of affairs and of a Conceptual dictionary is captured by defining a KRYPTON KB as the set of all outcomes that arc consistent with this specification. The following clauses define the main interface-level operations that implement the "functional" vicw of a KR"

NEWKB [I = [-,e I -, CO> is anoutcome) ASK [s KB] = yes if for each <a, m> in KB, m(a) = true,

no otherwise TELL [a.KB] = (<*,CO> inKB I Ma) = m) DEFINE [t. g.KB] = (Ca. e i n K B I t a gsndga t) SUBSUMES [gi. g2. KB] = YCS if for -h <*, CO> in KB. 82 * g1;

no otherwise

where a ranges on assertiond wffs. g, g1. and g2 on gt". t on unary and binary predicate symbols and KB on knowledge bases. NEWKB crcatcs an empty KB. TELL ud DEFINE update a KB and ASK and SUBSUMES query it; roughly speaking. tapdate operations act by restricting the number of outcomes of the KB. while query operations by ins-ting them.

We close this section with some informal comments on the properties of these operations as deriving from the semantics given in [3] and sketched above.

F m t of all, it might be the case that ASK [a. KB] = no and. at thc same time, ASK [(NOT a), KB] = no.

Next, each valid formula of the first order predicate calculus

holds in all outcomes (i.e. in NEWKB [I). Analogously, in all outcomes the relation of subsumption comprises all the expected pairs of terms (e.g. gl subsumes (ConGeneric g l g2)). as resulting from the semantics of the above described term calculus.

Third, properties that seem reasonable to expect of the interaction between ASK and TELL and between SUBSUMES and DEFINE do in fact hold for any KB. such as

ASK [a, TELL [a. KB]] =yes SUBSUMES [t, g, DEFINE [t. g, KB]] =yes SUBSUMES [g. t. DEFINE [t, g. KB]] =yes

Finally, also the interaction between terminology- and assertionality-related operations works well: for instance. for every KB

ASK [(FORALL x (IMPLIES (p x) (q x))), DEFINE [p, (ConGeneric q r), KB]] =yes

while it is not true in general that

SUBSUMES [p. q. TELL [(FORALL x fJMPLIEs (p x) (q x))). KBll= yes

AN ALTERNATIVE SEMANTICS FOR KRYPTON

In this paragraph we propose an alternative semantic account of the KRYPTON system in the style of the "possible worlds semantics" (PWS) originally developed for modal logic. The reasons for undertaking such a task are manifold.

First of all, the way KRYPTON distinguishes between terminology and assertionality can be recast in terms of the distinction between necessary and contingent truths respectively, as enforced in classical modal systems.

Second, a PWS for KRYPTON appears more simple and intuitive than the "hybrid semantics" given in [3]. allowing, among other things, to highlight the difference between terminology and assertionality by relating in a clearer way terminological facts to truth in a multiplicity of states of affairs, and assertional facts to truth in a particular state of affairs only.

Third, the possible worlds approach our semantics relies on is somehow more orthodox; in particular, a significant body of work on logics for knowledge and belief ("epistemic logics") has been carried on in this framework. In the next chapter we will see how this fact allows us to extend KRYPTON in a natural way to deal with modal operators for representing beliefs about the beliefs of multiple agents; interestingly, our framework will make it possible to distinguish between beliefs in propositions of an assertional nature and in propositions of a terminological nature. Remarkably, the possible worlds framework would allow us. if desired, to add to the language other modal operators (e.g. in order to model other "propositional attitude" notions such as "wanting that") with the same naturalness.

Fourth, our semantics suggests a natural extension of the KRYPTON language to express terminological definitions as wffs. so yielding, in a certain sense, a truly unified hybrid language. This modification brings with it a uniform treatment of belief, because the object of belief will then be a proposition. regardless of its assertional or terminological nature.

In this section we will concentrate on our alternative semantics for the KRYPTON language, while in the next one we will discuss our extensions.

Before specifying our semantics in detail it is perhaps worthwhile to briefly recall the basic notions of possible worlds semantics. PWS was formalized by Kripke 112) in terms of what are now called Kripke structures to provide a semantic account of the modal notions of necessity and possibility (see [ll]). Roughly speaking, while standard Tarskian semantics works by mapping

syntactic objects on a particular state of affairs ("interpretation"). PWS interprets these objects on a Kripke structure, which incorporates a number of alternative states of affairs ("possible worlds"), together with a binary relation (the "accessibility" relation) on these worlds. Intuitively, the accessibility relation tells us which worlds are possible alternatives given a particular one; accordingly, proposition p is necessarily true in world w if it is true in all worlds that stand in the accessibility relation with w.

Our PWS for KRYPTON is then given in terms of a Kripke structure M=<S, D. V. T>, where S is a non-empty set of states, D is a domain of individuals, V is a function from individual variables to elements of D and from pairs <p. s> (with p an n-ary predicate symbol and s an element of S) to n-ary relations over D, and 7 is a binary reflexive relation over S. A possible world is then defined to be an ordered pair w=<M, s>; intuitively. a possible world represents a complete specification of a state of affairs.

The 7 relation allows us to capture the difference between sentences of an assertional nature and sentences of a terminological nature: intuitively, sentences of the first kind have a "contingent" import, and their truth is thus evaluated with respect to a specific world, whereas sentences of the second kind have a "necessary" import, and their truth is thus evaluated with respect to a set of worlds (the ones accessible through T)..

Formally, we define the truth with respect to a world <M. s> of an assertional wff a of KRYPTON by means of a support relation I=, where <M, s> I = a is to be read as "a is true in world <M, s>":

DEFINITION 1: Let <M, s> be a possible world, with M=<S. D. V. 7> and s an element of S. A support relntion I= is such that, for all predicate symbols p and for all wffs a and B: <M, s> I = (p x1 ... x,,) iff a, s> I= (NOT a) iff a, 0 I = (OR a p) iff <M, s> I = Q or <M. s> I= p <M, s> I= (EXISTS x a)

<M [x,d~. s> I= a for some dc D

<V(xl), ..., V(x,,)> E V(p. s) it is not the case that <M, s> I = a

iff

where M [x/d~ is defined as the Kripke structure that is exactly like M except that V(x)=d. A formula a is vnlid iff <M. s> I = a for all possible worlds <M. s>. Next, similarly to [3]. we have:

DEFINITION 2: Let <M, s> be a possible world, with M = <S. D, V, 7> and s an element of S. Then, for any gterm g. the extension of g in world <M, s> (written E,M, &)) is given by: 1. E,M, ,>@) is V(p. s) for any predicate symbol p. 2. E,M, ,>(ConCeneric c1 _.. ck) is the intersection of the E,M, s>(ci)

for i = 1, .... k, and D if k=O. 3. E,M, ,,(VRGeneric c l r cz) is those elements x belonging to E,M,

s>(cl) such that <x, y> is in E,M, &) only when y is in E,M, S>(CZ).

4. E,M, ,,(RoleChain r1 ... rk) is the relational composition of the E,M, &) fori = 1, ..., k.

Next, we define what it means, in terms of possible worlds, that gterms stand in a relation of subsumption:

DEFINITION 3: Let <M, s> be a possible world, with M = <S, D. V, 'T> and s an element of S. Let gl and g2 be gterms. Then gl subsumes g2 in <M, s> iff for every s' such that s 7 s ' . ECM, s.>(gl)

2 E<M, sdg2).

Hence, in a given possible world <M. s>. the support relation tells us which (assertional) formulae are true, while the subsumption relation tells us which (terminological) relationships between

58

gterms hold. Such possible worlds play thus the same role outcomes play in the semantics given in [3]. In fact, each possible world supports the truth of all tautologies of the first order predicate calculus together with the logical consequences (in the sense of classical logic) of any other set of formulae whose truth it already supports. Moreover, each possible world "behaves well" with respect to definition 2 for instance. for any possible world and for any gtams c1. r and c2. (VRoCnuic c1 r 9) is subsumed by cI.

Mom interestingly. tho reflexivity of the accessibility relation 7 guarantees the link between terminology and assertionality by imposing that, for any pair of predicate symbols p and q. if p subsumes q in world w, thcn the formula (FORALL x (IMPLIES (p x) (q x))) be true in w. In fact, if p subsumes q in w. then the extension of p will be a s u p " of the extension of q in any w' accessible from w through 7 and in puticulu. by reflexivity, in w itself, so forcing the truth in w of the above f m l l l a . That the converse is not true in general is guaranteed by the fact that subsumption is evaluated with reference not only to world w but to all worlds accessible from w h g h 7.

S i i l a r l y to the treatment given in [3]. we see a knowledge base as an incomplete specification of what the world and the conceptual dictionary are. and thus identify it with the set of all (completely specified) worlds which arc consistent with it.

As for the "functional" level, the operations for the interaction with knowledge bases that we saw at the end of Section 2 can be redefiied according to the new semantics, with possible worlds replacing outcomes. M m precisely. it can be shown that our alternative semantics for KRYPTON has no impact on the functional level. To do this, we first need a definition (the * superscript identifies operations defiied according to our semantics and KBs obtained through the application of these opcrations):

DEFINITION 4. Let KB and KB* be two knowledge bases. We say that KB is functionally equivalent to KB iff, for all wffs a and for all gtcrms g,. p.

1) ASK [a, KB] =ASK* [a. KB*] 2) SUBSUMES I&. p, KB] = SUBSUMES* [gl. p. KB*l.

We then have the following:

THEOREM. Let KB and KB* be two knowledge bases identified by the same sequence of applications of NEWKB. TELL, DEFINE and NEWKB'. TELL'. DEPINE*.respactively. Then KB is functionally equivalent to KB*.

The proof is given in an extended version of this paper [18]. We notice that, as the interface level operations are the only way a user CM access a KB, the theorem actually states that the two versions of KRYPTON arc indistinguishable to the user. Moreover. because we may see these opcrations 08 actually defining KRYPTON. functional cquivalence is the notion of equivalence that really matters.

EXTENDING KRYPTON

We have s c m that recasting KRYPTON in terms of possible worlds semantics has the significant advantages of 1) providing (what we believe to be) a more simple and orthodox semantic account, and 2) shedding some light on the nature of the relationship between terminology and assertionality. Nevertheless, the main driving force for this endeavour has bcm the extension of the language to allow the expression of modal operators. in particular those modelling belief about the beliefs of multiple agents'.

This should not be taken as the statement that the original semantic account of KRYPTON given by Brachman and colleagues does not allow such extension; we arc only saying that we view PWS as a more natufal framework to accomplish this.

However. before diving into belief and multiple agents we discurs an extension to the base language of KRYPMN that ~ I O W B the expression of what we call "tcnninological wffr". yielding a truly unified hybrid Ianguagc2. ' This result, beside being appealing in itself, will allow us to treat in a uniform fashion belief in propositions with either a terminological. 01 an assertional import, or both. To obtain this we fint extend the language to allow wffs of the kind (IS g1 g2). where g1 and g2 arc gtcms and IS is an operator which is intended to capture the notion of subsumption between gtcrms. Accordingly. we add to defiition 1 the clause

CM, s> I= (IS g1 gZ, iff for all s' such that 67s'

E d 4 d,(Pz) 2 E & , &I)

where gl and g2 arc gtcnns. Besides, we take (SAME g1 g2) as a metalinguistic abbreviation of (AND (IS gl g2) (IS g2 gl)). Henceforth, we will refer to formulae of typc (IS g1 g2) or ( S A M E g1 g2) as terminologicol wffs.

We notice that while K R m O N actually consists of two neatly separated languages (the language of assertional formulae and the language of gterms). the availability of terminological wffs renders M-KRYPTON a unified language, where assertional and terminological wffs have the same status. AJ a wnscquence. we are able to express complex formulae. such as

where connectives of the assertional language apply to terminological wffs. or

where both terminological and assertional wffs arc involved. In addition, having a unified language means that it will be sufficient for the interaction language to comprise the NEWKB. ASK and TELL operators only. Notice that ASK and TELL apply now to terminological and assertional wffs alike: thus the effect of the KRYPTON formula DEFINE [t, g. KB] can be achicvcd in M-KRYPMN

becomes ASK [(Is 81 6. KBI.

(OR (IS 8' PZ, (Is 8'

@@UES (N0T-n x (AND @x)(NOT(qx))))) (IS P s))

by TELL [(SAME t g). KB]; U u l o g o ~ l y . SUBSUMES [g2. 81. KB]

Having discussed our modifications to the base language. we may now introduce belief operators and multiple believers. We are interested in the general case where there an n agents (which we call 1, .... n). each believing in a set of propositions either of an assertional or of a terminological nature. Formally. we introduce into the M-KRYPTON language sentence-forming operators B,. .... Bn: if a is a wff then Bla is also a wff, to be read "agent i believes that a". Hence, we may express "nested" beliefs, such as in BiBJa (when i=j we speak of "introspective belier). In other words, an M- KRYPTON knowledge base may contain formulae representing the system's beliefs both about the domain of discourse and about the (possibly nested) beliefs of agents 1. ... , n.

As already anticipated in section 3. a significant body of work on formal treatments of belief has been carried out in the framework of possible worlds semantics: after the seminal works of Hintikka [8. 91. Kripke structures have been widely used as a formal model for a PWS-based analysis of the epistemic notions of knowledge and belief (sec [7] for a review). The intuitive idea is to associate to each agent i an accessibility relation bi, that can be thought as saying which worlds agent i considers as "epistemic alternatives" to a given one; that is. each of these worlds might be, for all agent i believes. the actual world.

To characterize our notion of belief in a formal way. we define a hybrid Kripke structure as a tuple M=<S. D, V. 7 , B 1. .... B,>. where S. D, V and 7 are as defined in section 3 and 8 1 . .... b. are binary relations over S that are serial, transitive and

2 In an extended version of this paper [18] we also deal with the introduction into the language of "hybrid atoms" of the form (g X 1 ... x,). with g a generic gterm.

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Euclidean3. The result of imposing these constraints on the Bi's is a notion of belief according to which the following formulae are valid:

Al. (LMPLIES (Bi a) (NOT CBI (NOT a)))) A2. (IMPLIES (Bi a) (Bi (B1 a))) A3. (IMPLIES (NOT (BI a)) (Bi (NOT (Bi a))))

where a is any M-KRYPTON wff. Intuitively. these formulae mean that agents cannot hold inconsistent beliefs (Al). and that they are conscious of what they believe (A2) and do not believe (A3). This is a fairly common characterization of belief, which corresponds to the modal logic weak S5.

Our definition of the support relation I = is then extended by means of the clause

<M. s> I = (Bi a) iff for all s' such that sBis'. <M, s'> I= a

As we had anticipated, the tight integration between the terminological and the assertional languages that results from the introduction of the IS operator allows us to have a unique operator for expressing an agent's beliefs both on what the world and on what the conceptual vocabulary are like.

For better convenience, we show the definition of I = in its final form: a, s> I = (g x1 ... XJ

(M. s> I= (NOT a) (M, s> I= (OR a p) <M. s> I = (EXISTS x a) <M, s> I= (IS gl g2)

<M, s> I = (Bi a)

iff <V(xl), ..., V(X,,)> E E<M, &) iff it is not the case that <M, s> I = a iff (M. s> I= a or G I , s> I = p iff - 3 4 ~ ~ , ~ l . s> I= a for some dc D iff for all s' such that s'Js'

E<M, a*>(gz) 2 s&l) for all s' such that sBis', <M. s'> I = a

As a consequence of these extensions, there are a number of interesting properties that hold of M-KRYPTON (besides those that already hold of KRYPTON); hereafter, we comment on the more relevant ones. First of all, there is an enriched variety of formulae that are valid according to our semantics (i.e. those formulae a such that ASK [a. NEWKB [I] = yes, or, equivalently, ASK [a, KB] = yes for any KB): in particular, we now have valid formulae with a terminological component such as

(IS (ConGeneric p s) p) or

and valid formulae involving the belief operators such as A l . A2 and A3 above. As for the behaviour of the belief operators, we notice that

1 ) all agents believe in all valid propositions: that is. for each i and for each KB. ASK [a, NEWKB [I] = yes implies that ASK [(Bi a). KB] = yes; 2)however, agents do not necessarily believe in all true propositions: that is. ASK [a, KB] = yes does not imply in general that ASK [(B, a), KB] = yes; 3) agents' beliefs are closed with respect to implication: that is, for each KB. ASK [(B1 a), KB] = yes and ASK [(Bi (IMPLIES a B)), KB] = yes implies that ASK [(Bi p), KB] = yes; 4 ) agents may believe in false propositions: that is. ASK [(Bi a), KB] = yes does not imply in general that ASK [a, KB] = yes.

Properties 2) and 4) tell us that the belief set of an agent is

3 We recall that a binary relation B is serial if. for all SE S , there is some s' such that sBis'; is transitive if, for all s, s'. S"E S, if sBis' and s'Bis", then sBis"; is Euclidean if, for all s. s'. S"E S . if &is'

and sBis", then s'Bis".

iff

(LMPLIB (Is P e) " O L L x w m s (P x) (q x))))

independent from what is true in the real world (recall footnote 1 ) . However, properties 1) and 3) tell us that our agents are perfect reasoners on their belief set: that is, they believe in all logical consequences of their beliefs according to the hybrid logic.

CONCLUSION

In this paper we have described the M-KRYPTON knowledge representation system, according to which the belief set of an agent may comprise:

1 ) beliefs about the nature of the entities of the domain of discourse and of the conceptual relationships between them; 2) beliefs about facts about these entities that are believed true in the current state of affairs; 3) beliefs about the belief sets of other agents.

This has been achieved by combining insights from the disciplines of knowledge representation and epistemic logic. In particular, we have recast KRYPTON in terms of possible worlds semantics, obtaining a semantic account that, besides being "functionally" equivalent to the original one, has allowed an easy extension to the representation of beliefs about the beliefs of multiple agents and of beliefs in propositions of a terminological nature.

Moreover, other concepts that have traditionally been the subject of modal and related logics, such as other "propositional attitude" notions (e.g. "wanting that". "intending to", etc.) could be embedded in this framework. We are investigating this last possibility in view of extending M-KRYPTON to deal with the representation of plans. This would rest on what Pollack [16] calls "the mental phenomenon view of plans", where having a plan is analyzed as having a particular configuration of beliefs and intentions. This extension would find an immediate application to the dialogue system hinted at in the introduction, along the line of those (see for instance 14. 151) who have shown that dialogue participation can best be seen in terms of plan formation and recognition.

M-KRYPTON is a knowledge representation language that, though absolutely general. is particularly geared to applications where multiple interacting agents are to be modelled. In particular, the whole endeavour of user modelling may be seen as a special case of this problem, user and system being the interacting agents.

It may be interesting to note, however, that if we subscribe to this view of user modelling, we are adopting a perspective which is. from a theoretical viewpoint, slightly different from the one which is customary in artificial intelligence. That is, it seems implicit in most research in AI that a user model is to be embedded in a system that is to act as an "artificial partner in the interaction". In the view presented here, instead. we are interested in a model of the interaction between a "SYSTEM" and a "USER": the computer system which will implement this model can be thought as an "external observer". and no link is supposed to exist, in principle. between the observing system and the observed SYSTEM. In practice, the user interacts with a physical computing system (the observer) that simulates an agent of our logic (the observed SYSTEM). The payoff of adopting this view is that non-trivial models of the interaction (e.g. the ones cited in this section), with agents possibly holding complex types of belief, can be. dealt with.

Unfortunately, when human believers are involved, the notion of belief we have adopted suffers from the so-called problem of logical omniscience[lO]. This means that our agents are modelled as believing every logical consequence of what F y believe; this is a gross inadequacy (somehow intrinsic to the notion of possible world) when we deal with real, resource-bounded agents (such as human users of the system). and has been the object of many studies [5, 141. In another paper [19] we have shown how the ideas underlying M-KRYPTON may also be applied to provide some new insights into this problem.

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A c k n o w l e d g e m e n t s This research has bcen carried out in the frame of ESPRIT project P527 "Communication Failure in Dialogue: Techniques for Detection and Repair". We are grateful to Francesco Orilia for many enlightening discussions. Giacomo Ferrari. Irina Rodanof and Ronan Reilly have been constant sources of encouragement and productive criticism. Finally. we thank agents 1, ..., n for believing all kinds of wondrous things.

R e f e r e n c e s

[11 BRACHMAN. Ronald. FIKES. Richard and LEVESQUE, Hector (1983a). Krypton: integrating terminology and assertion. In Proceedings of AAAI-83. Washington DC, pp. 31-35.

[?I BRACHMAN. Ronald, FIKES. Richard and LEVESQUE. Hector (1983b). Krypton: a fnnstiond approach to knowledge representation. IEEE Computer 16. pp. 67-73.

[3] BRACHMAN, Ronald. PIGMAN GILBERT, Victoria and LEVESQUE, Hector (1985). An essential hybrid reasoning system: knowledge and symbol level accounts of KRYPTON. In Proceedlngs of IJCAI-85, Los Angclw CA, pp. 532-539.

[4] COHEN. Philip md LEVESQUE. Hector (1985). Spctch Acts and Rationality. In Proceedings of the u r d Anuurl Meetlng of the ACL, Chicago IL. pp. 49-60.

[SI FAGIN, Ronald and HALPERN. Joseph (1986). Belief, a w m e s s and limited rcaronhg. Research Report 4657 (49665). IBM Research Laboratory, San Jose CA (also forthcoming in Artiflclal Intelligence).

[a] HALPERN. J O S H and MOSES, Yoram (1984). Knowledge and common knowledge in a distributed environment. In Proceedings of the 3rd ACM Conference ou Prlnclples of Dlstrlbuted Computing, pp. 50-61.

[71 HALPERN. Joseph and MOSES, Yoram (1985). A guide to the modal logics of knowledge and beliek preliminary draft. In Proceedlngs of UCAI-85. Los Angeles CA, pp. 480-490.

[81 HINTIKKA. Jaakko (1962). Knowledge and bellef. An lutroductlon to the lo& of the two notlons. Ithaca N Y Come11 University Press.

[91 HINTIKKA, Jaakko (1969). Semantics for propositional attituder. In LINSKY. Leonard (ea.). Reference a n d modality. London GB: Oxford University Press. 1971, pp. 145- 167.

[lo] HINTIKKA, Jaakko (1975). Impossible possible worlds vindicated. Journal of Phllosophiul Loglc 4. pp. 475-484.

1111 HUGHES, G.E. and CRESSWELL, Maxwell (1968). A U lntroductlon to modal logic. London GB: Methuen.

[12] KRIPKE. Saul (1963). Semantical analysis of modal logic. Zeltschrlft flir Mathematlsche Loglk und Grundlagen der Mathematlk 9, pp. 67-96.

(131 LEVESQUE. Hector (1984.). Foundations of a functional approach to knowledge representation. A r t l f l c l r l Intelllgence 23, pp. 155-212.

[14] LEVESQUE. Hector (1984b). A logic of implicit and explicit belief. Fairchild Technical Rcport n. 653, Palo Alto CA.

[15] PERRAULT. Raymond and ALLEN, James (1980). A plan based analysis of indirect speech acts. American Journa l of Computatlonrl Lingulstlcs 6. pp. 167-182.

[I61 POLLACK, Martha (1986). A model of plan inference that distinguishes betwcm the beliefs of actors and obsmcrs. In Proceedings of the 24th Meetlng of the ACL, New York

[17] REILLY. Ronan (ed.) (1987). Communlcatlon failure In dialogue. Amstcrdun N L North Holland.

[18] SAFFIOTTI. Alessmdro and SEBASTIANI. Fabrizio (1988.). M- KRYPTON extending KRYPTON with hybrid formulae and multiple believers. Forthcoming technical report. Department

NY. pp. 207-214.

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of Linguistics, University of Pis.. [19] SAFFIOTII. Alessandro and SEBASTIANI. Fabrizio (1988b).

Steps towards a hybrid logic of awareness. Submittcd to the 3rd Annual Symposium on Logic in Computer Science, Edinburgh OB. 1988.

The restricted language architecture of a hybrid representation system. In Proceedings of IJCAI- 85, Los Angeles CA, pp. 547-551.

Dialogue- based user models. Proceedlugs of the IEEE74. pp. 948- 960.

[20] VILAKN. M m (1985).

[21] WAHLSTER. Wolfgang and KOBSA. Alfred (1986).