Formal speci�cation of beliefs in multi-agentsystemsMassimo Benerecetti1 , Alessandro Cimatti2, Enrico Giunchiglia1Fausto Giunchiglia2, Luciano Sera�ni21 DIST - University of Genoa,Viale Causa 13, 16145 Genova, Italy2 IRST - Istituto Trentino di Cultura,38050 Povo, Trento, Italyfbene,cimatti,fausto,serafinig@irst.itc.itenrico@frege.mrg.dist.unige.itAbstract. The formalization of agents attitudes, and belief in particu-lar, has been investigated in the past by the authors of this paper, alongtwo di�erent but related streams. Giunchiglia and Giunchiglia investigatethe properties of contexts for the formal speci�cation of agents mutualbeliefs, combining extensional speci�cation with (�nite) presentation bymeans of contexts. Cimatti and Sera�ni address the representational andimplementational implications of the use of contexts for representing pro-positional attitudes by tackling a paradigmatic case study. The goal ofthis paper is to show how these two streams are actually complementary,i.e. how the methodology proposed in the former can be successfully ap-plied to formally specify the case study discussed in the latter. In order toachieve this goal, the formal framework is extended to take into accountsome relevant aspects of the case study, the speci�cation of which is thenworked out in detail.1 IntroductionMuch of the work on Multi Agent (MA) systems is concentrated on the formaliza-tion of agents communications and attitudes (see e.g. [3, 20, 14]). In [8, 9, 7, 1, 2],the authors of this paper have proposed theories for the formalization of agentsattitudes, focusing on beliefs in particular. All these works share the intuition thatlarge parts of reasoning happen locally to agents, with few interactions amongthem, and use contexts (as formalized in [5, 8], but see also [16]) as the formaltool to capture locality of reasoning.The approaches followed by the authors can be thought as following two dif-ferent but related streams. Giunchiglia and Giunchiglia [7] study the propertiesof contexts for the formal speci�cation of agents mutual beliefs. The approachfollowed has been to give an extensional speci�cation of agents' beliefs �rst. Ex-tensional speci�cations are useful because of their intuitiveness and independencefrom implementation. Then, it is shown how the extensional speci�cation can be�nitely presented with contexts. As it will be clear later, the main advantage ofa context-based approach is that it allows for modular intensional speci�cations

and implementations. On the other hand, Cimatti and Sera�ni [1, 2] focus on rep-resentational and implementational aspects. By discussing a paradigmatic casestudy, they show how belief and other attitudes (e.g. common belief, ignorance)can be represented in a context based framework, and emphasize how contextsallow for elaboration tolerant speci�cations.The goal of this paper is to show how these two streams are actually comple-mentary, i.e. how the MA scenarios and the properties discussed in the latter canbe formally speci�ed along the lines of the former approach. The formal frame-work proposed in [7] is extended by generalizing the formal notion of \reasoner"with that of \conditional reasoner". The motivation is that reasoners only allowfor the formalization of agents' actual beliefs: what is the case that agents believe.In many cases, including the speci�c MA scenarios speci�ed in this paper, it isalso necessary to formalize agents' \conditional beliefs", i.e. what agents wouldbelieve under certain conditions. Furthermore, two MA scenarios are formallyspeci�ed in detail in the extended speci�cation framework according to the twosteps devised in [7]. We provide �rst an extensional speci�cation of each agent inthe scenario, and then we provide a �nite presentation for the extensional speci�c-ation. The basic idea is to present the reasoning capabilities of each reasoner witha context, formally de�ned as an axiomatic formal system. The desired corres-pondence between di�erent reasoners is enforced with bridge rules, i.e. inferencerules with premises and conclusions belonging to di�erent contexts. Contextsand bridge rules are the components of Multi Context Systems (MC Systems).This paper is structured as follows. In Section 2, we present the theoreticalbasis of the context-based speci�cation formalism. In Section 3 we present ourcase study, consisting of some paradigmatic MA scenarios to be formally spe-ci�ed. In Section 4 we work out a detailed multi context speci�cation of the MAscenarios outlined in Section 3. In Section 6 we draw some conclusions and futuredevelopments.2 Formal Speci�cation of Multi-Agent SystemsIn this section, we �rst de�ne (hierarchical) belief systems as a formal frameworkfor the extensional speci�cation of agents' beliefs. Then, we de�ne (hierarchical)MR systems as the class of MC systems �nitely presenting (hierarchical) beliefsystems.2.1 Extensional Speci�cationsIntuitively, by reasoner we mean anything which is capable of having beliefs, i.e.facts that it is willing to accept as true. Some examples of reasoners are agents(as discussed in, e.g., [12]), views about agents (as discussed in, e.g., [10]), andframes of mind (as discussed in, e.g., [4]). Each reasoner is provided with:� reasoning capabilities, i.e. the ability to derive consequences from a set ofhypothesis, and 2

� interaction capabilities, i.e. the ability of deriving consequences from what theother reasoners believe.Formally, a conditional reasoner R as a pair hL;`i, where the language L isa set of sentences, and `� (2L � L). (In the following, for sake of simplicity,we drop the word \conditional" whenever possible. Notationally, Ri stands forthe pair hLi; ii; Ti stands for the set of theorems of the reasoner Ri, i.e. theset of the formulas A such that ; i A.) The interaction capabilities that we areinterested in capturing are those that allow one reasoner R0 to have beliefs aboutanother reasoner R1. R0's beliefs about R1 are expressed via belief sentences, i.e.expressions of the formB(\A"). The two reasoners R0, R1 and the belief predicateB used by R0 to express its own beliefs about R1, characterize a belief system.De�nition1. A belief system is a pair of reasoners hR0; R1iB, where(i ) the parameter B is a unary predicate symbol called the belief predicate,(ii ) R0 is the observer, and(iii ) R1 is the observed reasoner of hR0; R1iB.Intuitively, in a belief system hR0; R1iB, R0 observes R1 and expresses itsbeliefs about R1 using the belief predicate B. In other words, the intended inter-pretation of the expression B(\A") in L0 is that R0 believes that R1 believes A.Reasoners and observers can be categorized depending on their reasoningand interaction capabilities, respectively. We say that reasoners and observersare ideal if they satisfy certain closure properties in the sense made precise bythe following de�nition.De�nition2. Let R0; R1 be two reasoners. We say that:Ri (i = 0; 1) is an Ideal Reasoner if(i ) Li is closed under the formation rules for propositional languages, and(ii ) i is closed under tautological consequence.R0 is an Ideal Observer (of R1) if(i ) L1 = fA j B(\A") 2 L0g, and(ii ) � 1 A =) B(\�") 0 B(\A"), where B(\�") is an abbreviation forthe set fB(\�") j � 2 �g.This de�nition captures a form of ideality with respect to (belief) deriva-tion (i.e. conditional belief) in the observed reasoner. In other words, R0 is anideal observer if it can derive at least all the (conditional) beliefs about those(conditional) beliefs that the observed reasoner can derive.The problem is that ideal reasoners and observers are often inadequate forrepresenting the capabilities of real agents (e.g. humans and programs). Giventhem enough knowledge and resources (e.g. space, time) humans and programstend to converge to an ideal behavior but this may never be the case. Thissituation is modeled by introducing the notion of real reasoner and real observer.When talking of a real reasoner or a real observer we mean that such a reasoneror observer derives too little or too much (i.e. it is incomplete or incorrect) with3

respect to a reasoner or observer taken as reference. Thus, di�erently from whatis the case for ideality, reality is a relative notion which states the absence ofcertain properties with respect to a speci�c reference. The intuition is that,for instance, a reasoner RE is correct with respect to another reasoner RI ifthe inferences of RE are contained in the inferences of RI, i.e. if E� I . If theinferences of RE are strictly contained in the inferences of RI (i.e. E� I) thenRE is incomplete. We say that a reasoner is real, to mean that it is incompleteor incorrect. Real reasoners and/or real observers are the components of a realbelief system.So far we have concentrated on the basic con�guration of one reasoner havingbeliefs about another reasoner. However, we may have more complex con�g-urations with multiple reasoners organized in multiple belief systems, possiblysharing one or both reasoners, or their belief predicate. For example, we mayhave an \isolated" reasoner (a reasoner which is neither being observed by an-other reasoner nor an observer of other reasoners) as well as a reasoner observinganother reasoner which, in turn, is (possibly) observing other reasoners and soon. As a limit case we may also have a reasoner observing itself. It is possible torepresent the structure of a complex belief system as a direct graph, whose nodesrepresent reasoners and whose edges represent the observing relations betweenreasoners. Each edge is labeled with the unary predicate used by the observerto express its own beliefs about the observed reasoner. Thus, for example, thebelief system hR0; R1iB corresponds to Figure 1. For the goals of this paper weconsider con�gurations of reasoners having a tree structure like that of Figure 2.� ��R0?B� ��R1 � ��R�� ��R1� ��R11 � ��R12 ...������= B1 � B2 ...� ��R2 � ��Rn� ��Rn1 � ��Rn2 ...������= B1 � B2������= B1 � B2 ZZZZZZ~BnFig. 1. A belief system. Fig. 2. A hierarchical belief system.Such con�gurations of reasoners are described by hierarchical belief systems.Formally, let I be a set of indexes (each corresponding to a reasoner) such that,given a set I0 = fi1; : : : ; ing, I is a pre�x closed set of string over I03. I containsalso the empty string �. A hierarchical belief system is a pair hfRigi2I ; Bi, wherefRigi2I is a family of reasoners and B is the set of pair h�; �ji such that �j 2 I.We call R� the root reasoner. If h�; �ji is an element of the binary relation B,3 That is, if �i 2 I for any i 2 I0, then � 2 I.4

then R� observes R�j and expresses its beliefs about R�j using the Bj predicate.It is easy to check that a belief system hR0; R1iB corresponds |modulo renamingof indices| to the hierarchical belief system hfR�; R1g; fh�; 1igiB1 .2.2 Finite Presentation of Speci�cationsPresenting a belief system requires representing the reasoning and interactioncapabilities of each reasoner in the belief system. About the reasoning capabil-ity, they can be implemented by providing reasoners with a set of facts whichconstitute their basic knowledge, and with some inference engine for deriving con-sequences. For each reasoner Ri, we thus introduce a corresponding context Ci,de�ned as an axiomatic formal system, i.e. a triple hLi; i;�ii, where Li is thelanguage, i is the set of axioms and �i is the set of inference rules of Ci. (Nota-tionally, in the following, a context Ci is implicitly de�ned as hLi; i;�ii.) Thisallows for a modular speci�cation of the reasoning capabilities of each reasoner.About the interaction capabilities, they can be implemented by providingreasoners with bridge rules, i.e. inference rules with premises and conclusion indi�erent contexts. For instance, the bridge ruleC1 :A1C2 :A2states the derivability of the formula A2 in context C2 (written C2 :A2) from thederivability of A1 in context C1 (C1 :A1). This allows for a exible speci�cationof reasoners' interactions.Contexts and bridge rules are the components of Multi Context systems (MCsystems), de�ned as pairs hfamily-of-contexts, set-of-bridge-rulesi. Derivabilityin a MC system MS, in symbols MS , is de�ned in [8]; roughly speaking, itis a generalization of Prawitz' notion of deduction inside a Natural DeductionSystem [18].The interactions between the observer R0 and the observed reasoner R1 thatwe are interested in capturing are those which enable R0 to contain the derivationB(\�") 0 B(\A") when R1 contains the derivation � 1 A. Hence, the particularclass of MC systems presenting belief systems is characterized as follows:De�nition3. An MC-system hfC0; C1g; BRi is an MR system if BR isC1 :AC0 :B(\A") R1up C0 :B(\A")C1 :A R1dnand the restrictions include:R1up: C1 :A does not depend on any assumption in C1.The notion of presentation of a belief system via an MR system is de�ned byimposing the following relation between the derivability relations of the reasonersand that of an MR system: 5

De�nition4. Let MS = hfC0; C1g; BRi be an MR system. MS presents thebelief system hR0; R1iB if i= f(�;A) j fCi :B j B 2 �g MS Ci :Ag (i = 0; 1).The closure conditions for ideality can be captured by posing appropriaterestrictions on MR systems. Let us consider the following de�nition:De�nition5. An MR system hfC0; C1g; BRi is an MBK system if the followingconditions are satis�ed:(i ) L0 and L1 contain a given set P of propositional letters, the symbol forfalsity ?, and are closed under implication4;(ii ) �0 (�1) is the set of Classical Natural Deduction Rules;(iii ) L1 = fA j B(\A") 2 L0g;(iv ) BR is C1 :AC0 :B(\A") R1up C0 :B(\A")C1 :A R1dnand the only restriction is: R1up: C1 :A does not depend on any assumption in C1.The rule R1up is called re ection up, while R1dn is called re ection down. Therestrictions on R1up is such that a formula can be \re ected up" only if it doesnot depend on any assumptions in C1.We can now state the following theorem:Theorem6. Let hR0; R1iB be the belief system presented by an MBK system.Then(i ) R0 and R1 are ideal reasoners, and(ii ) R0 is an ideal observer.Proof. The �rst statement easily follows from the de�nition of ideal reasoner(De�nition 2) and the consideration that (i ) and (ii ) of De�nition 5 just allowMBK to present ideal reasoners. The second statement can be proved by showingthat � 1 A =) B(\�") 0 B(\A"), where � 2 2L1 . In the following, given anyset of formulas � 2 Li (i = 0; 1), we use the notation Ci :� as an abbreviationfor fCi : � j � 2 �g. Since, by hypothesis, � 1 A, then C1 : � MBK C1 : A(De�nition 4). So there exists in MBK a deduction of C1 :A depending on C1 :� 0,with � 0 � � and �nite. Let us assume B(\�") in C0 for every � 2 � 0. ApplyingR1dn to the assumptions C0 : B(\�") (for every � 2 � 0), allows us to obtain adeduction ofC1 :� depending on the assumptionsC0 :B(\�") and also a deductionof C1 : A depending on C0 : B(\� 0"). Then, applying R1up to A (which onlydepends on assumptions in C0), we obtain a deduction of C0 :B(\A") dependingon C0 :B(\� 0"). Since B(\� 0") � B(\�"), we have C0 :B(\�") MBK C0 :B(\A"),i.e. (De�nition 4) B(\�") 0 B(\A"). 2MC systems presenting a hierarchical belief system are de�ned generalizingDe�nition 3.4 We also use standard abbreviations from propositional logic, such as :A for A � ?,A _B for :A � B, A ^B for :(:A_ :B), > for ?� ?.6

De�nition7. Let I be a pre�x closed set of indices. AnMC system hfCigi2I ; BRiis a hierarchical MRI system if BR consists ofC�j :AC� :Bj(\A") R�jup C� :Bj(\A")C�j :A R�jdn(�j 2 I), and the restrictions include:R�jup: C�j :A does not depend on any assumption in C�j.De�nition8. Let MS = hfCigi2I ; BRi be a hierarchical MRI system. MSpresents the hierarchical belief system hfRigi2I ; Bi, if for all i 2 I i= f(�;A) jfCi :B j B 2 �g MS Ci :Ag.3 Description of the ScenariosIn this section we present the case study of this paper. We start from the \ThreeWise Men Puzzle" as informally described in [15]:A certain King wishes to test his three wise men. He arranges them in acircle so that they can see and hear each other and tells them that he willput a white or black spot on each of their forehead but that at least onespot will be white. In fact all three spots are white. He then repeatedlyasks them \Do you know the color of your spot?". What do they answer?Discussing the solution of the puzzle is out of the scope of this paper (a thoroughanalysis can be found in [1]). Here we provide an informal de�nition of di�er-ent MA scenarios, which will be formally speci�ed using the theoretical notionspresented in the previous section.Scenario 1 (the original TWM): as it can be intuitively understood fromthe statement of the puzzle.Scenario 2 (the \not so wise" man): as Scenario 1, but where the secondwise man is \not so wise", in that he is not able to reason about otheragents' beliefs. There are two di�erent sub-cases:2A. the other agents are aware that the second wise is not so wise.2B. the other agents are not aware that the second wise is not so wise.Despite their simplicity, the speci�cation of these scenarios involve the repres-entation of a number of issues which are relevant in the context of MA systems.Indeed, there are di�erent agents, each with his own beliefs, which may or maynot be consistent with the beliefs of the other agents, and his own reasoning cap-abilities. The agents have to be able to reason about other agents by ascribingbeliefs and inferential abilities to them. These forms of ascription may or maynot be faithful to the actual agent and can be subordinated to certain hypotheses.Finally, the process of reasoning about other agents can be nested at di�erentlevels. For instance, it is possible that the third wise man can reason about thebeliefs ascribed by the second wise to the �rst.7

4 Speci�cation of the ScenariosIn this section we provide a formal speci�cation of the sample scenarios inform-ally presented in the previous section. Each of the scenarios are extensionallycharacterized by means of a hierarchical belief systems, and �nitely presentedby a hierarchical MR system. This approach is based on the intuition that eachreasoner in the tree-structure represents a certain point of view on the scen-ario. For instance, the reasoner R� represents the (point of) view of an externalreasoner looking at the scenario, Ri formalizes the view of (wise man) i, and Rijformalizes the view ascribed by i to j.4.1 Speci�cation of Scenario 1Starting from the informal description of Scenario 1, the class of hierarchicalbelief systems which extensionally specify it, is characterized as follows:{ as there are three agents, the set I0 is f1; 2:3g (in the following, i; j; k 2 I0);{ being wise, the agents are able to iterate the process of belief ascription toarbitrary depth, and therefore I = I�0 (in the following, � 2 I);{ under every view it is possible to express statements about the the colorof the spots and the agents' beliefs. Without loss of generality, we imposethat L� = LTWM for every �. LTWM is the smallest language containing thepropositions Wi, Bi(\A") for all A 2 LTWM, and closed under applicationof propositional connectives. Intuitively,Wi express that i has a white spot,while Bi(\A") expresses that i believes (the proposition expressed by) A;{ the view of the external observer is consistent, i.e. ? 62 T�;{ as each man is considered wise from any point of view, each reasoner R� isboth an ideal reasoner and an ideal observer of R�i;{ as the king utterance is considered true under each point of view, T� containsW1 _W2 _W3 King-Utterance{ as each wise can see his colleagues, for any i 6= j T� contains(Wi � Bj(\Wi")) ^ (:Wi � Bj(\:Wi")) i-sees-jand, if � does not contain i Wi can-see-i{ the wise men speak in numerical order5, and they are aware of this fact.Therefore any point of view about the second situation contains the statementof ignorance of wise man 1, and any point of view about the third situationcontains the statement of the ignorance of both wise men 1 and 2. This isformalized by the following constraint::Bi(\Wi") 2 T�, if � 2 fj 2 I0 : j > ig�5 The answers being \I don't know" for 1 and 2, and \My spot is white" for 3 (see [1]).8

Any hierarchical system satisfying the above requirements is said to be a BTWMsystem.Some remarks. The extensional speci�cation of the scenario (i.e. the hierarch-ical belief system) is simply obtained by composing di�erent constraints derivingfrom the rationalization of the scenario. This process has the consequence thatmany underlying assumptions are forced to be made explicit, making thereforethe features of the speci�ed system more precise from the beginning of the designphase. Notice also that the speci�cation given above is a partial speci�cation ofthe scenario. The above constraints, indeed, give some (not all) conditions for ahierarchical belief systems to formalize Scenario 1. There are, indeed, facts whichare true in the informal scenario, but which are not forced to hold by the par-tial speci�cation given above. For instance, the speci�cation does not imply that:B1(\:W1") 2 T�, which corresponds to the fact that in the view of the externalobserver wise man 1 does not believe that his spot is black. However partial, thisspeci�cation is strong enough to enforce the properties we are interested in, i.e.wise 1 and wise 2 do not believe that their spot is white, while 3 does believethat his spot is white. Furthermore, there is at least a hierarchical belief systemswhich satis�es the speci�cation above. The following theorem substantiates theabove claims.Theorem9. In any BTWM system(i ) W1 62 T1,(ii ) W2 62 T2,(iii ) W3 2 T3.The class of BTWM systems is not empty.Proof. Item (i) is provable by contradiction. Suppose that W1 2 T1, this im-plies by ideality conditions that B1(\W1") 2 T�. Since :B1(\W1") is also inT� and R� is an ideal reasoner, then T� = LTWM. This contradicts the consist-ency condition of R�. Item (ii) can be proved in the same way. Item (iii ): theproof follows from the ideal properties of each reasoner. Indeed, we have that:W2;:W3 321 W1 (King-Utterance). The ideal properties of 32 (w.r.t. 321) allowto obtain B1(\:W2");B1(\:W3") 32 B1(\W1"). Moreover from i-sees-j, we alsohave both :W2 32 B1(\:W2") and :W3 32 B1(\:W3"). This allows to obtain:W2;:W3 32 B1(\W1"). Since also 32 :B1(\W1") holds (Utterance-1), we canconclude that :W3 32 W2. By ideality of 3 (w.r.t. 32), B2(\:W3") 3 B2(\W2").By i-sees-j, :W3 3 B2(\:W3") and then :W3 3 B2(\W2"). By Utterance-2,3 :B2(\W2") and, �nally, 3 W3.Now let us prove the second statement of the above theorem: i.e. that thereexists at least a BTWM system. Consider a BTWM system where for each reasonerR� = hLTWM; �i, � is de�ned as follows (� 2 I):� � C () � [ fCg � LTWM and � 2 Inf�; 1; 2; 21g� 21 C () �;W3;Bj(\LTWM") j= A, for j 2 I0� 1 C () �;W2;W3;Bj(\LTWM") j= A, for j 2 I0� 2 C () �;:B1(\W1");W1;W3;Bj(\�") � Bj(\C"); 1-sees-2 j= A;9

for j 2 I0 and � 2j C� � C () �;:Bi(\W1");:B2(\W2");Wi;Bj(\�") � Bj(\C"); i-sees-j j= A;for i; j 2 I0 and � j C:(j= is the classical logical consequence relation). For all � 2 I and j 2 I0, R� isboth an ideal reasoner and an ideal observer of R�j. Furthermore it can be easilyveri�ed that, for each � 2 I, T� contains all the formulas given in the extensionalspeci�cation of BTWM.It remains to prove the consistency of R� (i.e. ? 62 T�). We prove it byproviding a model for T�. Let us consider the interpretationM of LTWM in whichIM (Wi) = True, and IM (Bi) = fA j ; 1 Ag, for i = 1; 2; 3. It can be proved thatM satis�es all formulas in T�, and in particular :B1(\W1") and :B2(\W2"). 2We perform now the second step of the speci�cation process, i.e. we �nitelypresent (extensional) BTWM systems with hierarchical MRI systems. A hierarch-ical MRI system hfCigi2I ; BRi is an MBKTWM system if{ I0 = f1; 2; 3g and I = I�0 ;{ the language L� of every context C� is LTWM;{ the set of inference rules �� of every context C� is the set of Classical NaturalDeduction Rules;{ the set of axioms � of each context C� contains� King-Utterance and j-sees-i, for i; j = 1; 2; 3 such that i 6= j; and� W1 ^W2 ^W3, if � = �; and� :Bi(\Wi"), if � 2 fj 2 I0 : j > ig�.The following theorem states that the hierarchical belief system presented byan MBKTWM system is a BTWM system.Theorem10. An MBKTWM system presents a BTWM system.Corollary 11 (of Theorems 9, 10). For any MBKTWM system MS(i ) 6 MS C1 :W1,(ii ) 6 MS C2 :W2,(iii ) MS C3 :W3.Figure 3 reports the corresponding MC proof ofC3 :W3 in a MBKTWM system.4.2 Speci�cation of Scenario 2We extensionally specify Scenario 2 starting from Scenario 1. The �rst di�erenceto be accounted for is the limited inferential capability of wise 2. In particular, 2 isunable to reason under the other agents point of view. Therefore, 2 is representedas a reasoner unable to observe other reasoners.For the two Scenarios 2A and 2B have to consider a hierarchical belief systemhfRigi2I ; Bi, where I is the set of indices in f1; 2; 3g� such that10

1 C3 :W3 (1) By assumption2 :W3 � B2(\:W3") From 2-sees-33 B2(\:W3") (1) From 1 and 2 by �E4 C32 :W3 (1) From 3 by R32dn5 :W3 � B1(\:W3") From 1-sees-36 B1(\:W3") (1) From 4 and 5 by �E7 :W2 (7) By assumption8 :W2 � B1(\:W2") From 1-sees-29 B1(\:W2") (7) From 7 and 8 by �E10 C321 :W3 (1) From 6 by R321dn11 :W2 (7) From 9 by R321dn12 W1 _W2 _W3 From King-Utterance13 W1 (1; 7) From 10, 11 and 1214 C32 B1(\W1") (1; 7) From 13 by R321up15 :B1(\W1") From Utterance-116 ? (1; 7) From 14 and 15 by �E17 W2 (1) From 16 by ?c18 C3 B2(\W2") (1) From 17 by R32up19 :B2(\W2") From Utterance-220 ? (1) From 18 and 19 by �E21 W3 From 20 by ?cFig. 3. In context C3 can be proved W3.2A. (1 and 3 are aware of the lack of wisdom of man 2.) I does not contain labelsof the form �2� with � 6= � (i.e. I does not contain strings in which 2 occursbefore some other character). This has the e�ect that under no point of viewa reasoner representing wise 2 is able to observe any other reasoner6 .2B. (1 and 3 are not aware of the lack of wisdom of 2.) I does not contain labelsof the form 2� with � 6= �. In this case 1 and 3 are modeled exactly as inScenario 1.A hierarchical belief system is a BNWM system [BNANWM system], if it satis�esthe constraints given for BTWM systems except that I is de�ned according to item2A [2B].From an extensional point of view, the reasoner R2 generated by an BNWM andBNANWM system is (in general) correct but incomplete with respect to the samereasoner as generated by a BTWM system. Indeed, for instance, in a BNWM system2 does not necessarily contain all the derivations which would follow if R2 were6 To be very precise, this restriction formalizes the limit case where 1 and 3 are awarethat 2 is not so wise, that each of them is aware that the other is aware of this fact,and so on at arbitrary depth. Intermediate cases can be easily captured by imposingthe suitable constraints on I. 11

observing the reasoners R21; R22; R23 (as it is the case in a BTWM system).Theorem12. In any BNWM [BNANWM] system,(i ) W1 62 T1, [W1 62 T1; ](ii ) W2 62 T2, [W2 62 T2; ](iii ) W3 62 T3. [W3 2 T3:]The class of BNWM [BNANWM] systems is not empty.The proof proceeds in the same way as for the case of BTWM systems.Note that, from the extensional speci�cations, BTWM, BNWM and BNANWM di�eronly for the set I of reasoners. For instance, the reasoners of a BNWM systemsare obtained by deleting the reasoners of a BTWM system with index of the form�2� for some � 6= �. The modularity of our approach allows us to obtain the�nite presentation of a BNWM and BNANWM system, by deleting the correspondingcontexts in the �nite presentation of a BTWM.An MBKNWM [MBKNANWM] system is de�ned as an MBKTWM system withthe exception of the set I of indexes, which is equal to the setfi 2 f1; 2:3g�j i 6= �2�, for any � 6= �g.[fi 2 f1; 2:3g�j i 6= 2�, for any � 6= �g:]Theorem13. An MBKNWM [MBKNANWM] system presents a BNWM [BNANWM]system.Corollary 14 (of Theorems 12, 13). For any MBKNWM [MBKNANWM] systemMS(i ) 6 MS C1 :W1, [6 MS C1 :W1; ](ii ) 6 MS C2 :W2, [6 MS C2 :W2; ](iii ) 6 MS C3 :W3. [ MS C3 :W3:]5 Related workIn the �eld of formal speci�cation of multi agent systems a lot of approaches arebased on modal logics (see for instance [4, 11, 13]). An alternative approach is theBDI architecture and the relative BDI logics proposed by Rao and George� [19].The main di�erence with our approach is that we exploit distinct theories andrelations between them to represent agents' internal state and agents interactionsrespectively. This allows for a modular speci�cation which, as pointed out in [2]and [1], gives advantages in terms of elaboration tolerance and implementation.An application of MC system for the speci�cation of a multi agent scenariocan be found in [17]. In this paper authors provide a formal speci�cation of a setof agents able to perform \cognitive tasks" such as communication and reasoning.In this framework each agent is modelled by a set of contexts, and communicationbetween agents is formalized via bridge rules.In [21] the author provides a formal semantics for multi agent systems spe-ci�ed in concurrent MetateM. Concurrent MetateM allows to extensionally12

specify a set of executable agents. However, it is possible to �nitely present aconcurrent MetateM multi agent system with an MC system. For lack of spacewe omit here the technical description of the construction of the MC system.Comparing the resulting MC system with Wooldridge's initial speci�cation, itseems that our speci�cation is more elaboration tolerant. In our MC system eachbridge rule formalizes a single aspect of a concurrent MetateM multi agent sys-tem. For instance a bridge rule, called sync, formalizes synchronous execution.This implies elaboration tollerance [1]. For instance to formalize system withasynchronous computation, we simply drop bridge rule sync. Furthermore, in aMC system a proposition may have independent meaning in two di�erent con-texts. Since di�erent agents are formalized by di�erent contexts, this allows usto drop the technical assumption made in [21] that agents has disjoint languages.6 Conclusion and Future WorkIn this paper we have proposed a formal framework for the extensional speci�c-ation of MA's conditional beliefs and for their �nite presentation. We have intro-duced the notions of conditional reasoner |extending that of reasoner in [7]|and of observer. We have shown how the MA scenarios introduced in [2] can beformally speci�ed using the above framework.As evidentiated by the case study, the extensional speci�cation allows for adescription independent from implementation; the context-based speci�cation al-lows for modular implementations. Needless to say that the theory proposed isstill limited in many ways and needs extensions. From a theoretical point of view,the characterization of ideal and real reasoners developed in [7] needs to be ex-tended to conditional reasoners. Furthermore, concepts like elaboration tolerance|as informally introduced and discussed in [2]| need to be more precisely ad-dressed and studied. From a representational point of view, many propositionalattitudes other than belief (e.g. common belief, ignorance, desire, intention) needto be incorporated in the framework. Finally, from an implementational point ofview, we have a prototypical implementation of the whole theory, but it de�nitelyneeds enhancements.References1. A. Cimatti and L. Sera�ni. Multi-Agent Reasoning with Belief Contexts: the Ap-proach and a Case Study. In M. Wooldridge and N. R. Jennings, editors, IntelligentAgents: Proceedings of 1994 Workshop on Agent Theories, Architectures, and Lan-guages, number 890 in Lecture Notes in Computer Science, pages 71{85. SpringerVerlag, 1995. Also IRST-Technical Report 9312-01, IRST, Trento, Italy.2. A. Cimatti and L. Sera�ni. Multi-Agent Reasoning with Belief Contexts II: Elab-oration Tolerance. In Proc. 1st Int. Conference on Multi-Agent Systems (ICMAS-95), pages 57{64, 1996. Also IRST-Technical Report 9412-09, IRST, Trento, Italy.Commonsense-96, Third Symposium on Logical Formalizations of CommonsenseReasoning, Stanford University, 1996.13

3. P.R. Cohen and H.J. Levesque. Intention Is Choice with Commitment. Arti�cialIntelligence, 42:213{261, 1990.4. R. Fagin and J.Y. Halpern. Belief, awareness, and limited reasoning. Arti�cialIntelligence, 34:39{76, 1988.5. F. Giunchiglia. Multilanguage systems. In Proceedings of AAAI Spring Symposiumon Logical Formalizations of Commonsense Reasoning, 1991. Also IRST-TechnicalReport 9011-17, IRST, Trento, Italy.6. F. Giunchiglia and C. Ghidini. A Local Models Semantics for Modal Logics. Tech-nical Report 9607-13, IRST, Trento, Italy, 1996.7. F. Giunchiglia and E. Giunchiglia. Ideal and Real Belief about Belief: Some in-tuitions. In Agents Breaking Away, Proc. MAAMAW'96, number 1038 in LectureNotes in Arti�cial Intelligence, pages 1{12. Springer Verlag, 1996.8. F. Giunchiglia and L. Sera�ni. Multilanguage hierarchical logics (or: how we can dowithout modal logics). Arti�cial Intelligence, 65:29{70, 1994. Also IRST-TechnicalReport 9110-07, IRST, Trento, Italy.9. F. Giunchiglia, L. Sera�ni, E. Giunchiglia, and M. Frixione. Non-Omniscient Be-lief as Context-Based Reasoning. In Proc. of the 13th International Joint Con-ference on Arti�cial Intelligence, pages 548{554, Chambery, France, 1993. AlsoIRST-Technical Report 9206-03, IRST, Trento, Italy.10. K. Konolige. A deduction model of belief. Pitman, London, 1986.11. K. Konolige. Explanatory Belief Ascription: notes and premature formalization. InProc of the third Conference on Theoretical Aspects of Reasoning about Knowledge,pages 85{96, 1990.12. A. Lansky. Localized Event-Based Reasoning for Multiagent Domains. Computa-tional Intelligence, 4:319{339, 1988.13. A. Lomuscio and M. Colombetti. Qlb: a quanti�ed logic for belief. In J. P. M�uller,M. J. Wooldridge, and N.R. Jennings, editors, Intelligent Agents III Prooceed-ings of the Third International Workshop on Agent, Theories, Architectures, andLanguages (ATAL-96), Lecture Notes of Arti�cial Intelligence, Heidelberg, 1996.Springer-Verlag. In this volume.14. M. Luck and M. d'Inverno. A formal framework for agency and autonomy. In Proc.1st Int. Conference on Multi-Agent Systems (ICMAS-95), pages 254{260, 1996.15. J. McCarthy. Formalization of Two Puzzles Involving Knowledge. In V. Lifschitz,editor, Formalizing Common Sense - Papers by John McCarthy, pages 158{166.Ablex Publishing Corporation, 1990.16. J. McCarthy. Notes on Formalizing Context. Unpublished, 1991.17. P. Noriega and C. Sierra. Towards rayered dialogical agents. In J. P. M�uller, M. J.Wooldridge, and N.R. Jennings, editors, Intelligent Agents III Prooceedings of theThird International Workshop on Agent, Theories, Architectures, and Languages(ATAL-96), Lecture Notes of Arti�cial Intelligence, Heidelberg, 1996. Springer-Verlag. In this volume.18. D. Prawitz. Natural Deduction - A proof theoretical study. Almquist and Wiksell,Stockholm, 1965.19. A. S. Rao and M. George�. BDI-Agents, from Theory to Practice. In Proc. 1stInt. Conference on Multi-Agent Systems (ICMAS-95), pages 312{319, 1996.20. A. S. Rao and M. P. George�. Modeling rational agents within a BDI architec-ture. In J. Allen, R. Fikes, and E. Sandewall, editors, Proceedings of the 2nd In-ternational Conference on Principle of Knowledge Representation and Reasoning.Morgan Kaufmann Publishers, 1991. 14

21. M. J. Wooldridge. A knowledge-theoretic semantics for concurrent metatem. InJ. P. M�uller, M. J. Wooldridge, and N.R. Jennings, editors, Intelligent Agents IIIProoceedings of the Third International Workshop on Agent, Theories, Architec-tures, and Languages (ATAL-96), Lecture Notes of Arti�cial Intelligence, Heidel-berg, 1996. Springer-Verlag. In this volume.

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