Formal specication of beliefs in multi-agent. systems. Viale Causa 13, Genova, Italy Povo, Trento, Italy

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1 Formal specication of beliefs in multi-agent systems Massimo Benerecetti 1, Alessandro Cimatti 2, Enrico Giunchiglia 1 Fausto Giunchiglia 2, Luciano Serani 2 1 DIST - University of Genoa, Viale Causa 13, Genova, Italy 2 IRST - Istituto Trentino di Cultura, Povo, Trento, Italy fbene,cimatti,fausto,serafinig@irst.itc.it enrico@frege.mrg.dist.unige.it Abstract. The formalization of agents attitudes, and belief in particular, has been investigated in the past by the authors of this paper, along two dierent but related streams. Giunchiglia and Giunchiglia investigate the properties of contexts for the formal specication of agents mutual beliefs, combining extensional specication with (nite) presentation by means of contexts. Cimatti and Serani address the representational and implementational implications of the use of contexts for representing propositional attitudes by tackling a paradigmatic case study. The goal of this paper is to show how these two streams are actually complementary, i.e. how the methodology proposed in the former can be successfully applied to formally specify the case study discussed in the latter. In order to achieve this goal, the formal framework is extended to take into account some relevant aspects of the case study, the specication of which is then worked out in detail. 1 Introduction Much of the work on Multi Agent (MA) systems is concentrated on the formalization 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 agents attitudes, focusing on beliefs in particular. All these works share the intuition that large parts of reasoning happen locally to agents, with few interactions among them, and use contexts (as formalized in [5, 8], but see also [16]) as the formal tool to capture locality of reasoning. The approaches followed by the authors can be thought as following two different but related streams. Giunchiglia and Giunchiglia [7] study the properties of contexts for the formal specication of agents mutual beliefs. The approach followed has been to give an extensional specication of agents' beliefs rst. Extensional specications are useful because of their intuitiveness and independence from implementation. Then, it is shown how the extensional specication can be nitely presented with contexts. As it will be clear later, the main advantage of a context-based approach is that it allows for modular intensional specications

2 and implementations. On the other hand, Cimatti and Serani [1, 2] focus on representational and implementational aspects. By discussing a paradigmatic case study, they show how belief and other attitudes (e.g. common belief, ignorance) can be represented in a context based framework, and emphasize how contexts allow for elaboration tolerant specications. The goal of this paper is to show how these two streams are actually complementary, i.e. how the MA scenarios and the properties discussed in the latter can be formally specied along the lines of the former approach. The formal framework proposed in [7] is extended by generalizing the formal notion of \reasoner" with that of \conditional reasoner". The motivation is that reasoners only allow for the formalization of agents' actual beliefs: what is the case that agents believe. In many cases, including the specic MA scenarios specied in this paper, it is also necessary to formalize agents' \conditional beliefs", i.e. what agents would believe under certain conditions. Furthermore, two MA scenarios are formally specied in detail in the extended specication framework according to the two steps devised in [7]. We provide rst an extensional specication of each agent in the scenario, and then we provide a nite presentation for the extensional specication. The basic idea is to present the reasoning capabilities of each reasoner with a context, formally dened as an axiomatic formal system. The desired correspondence between dierent reasoners is enforced with bridge rules, i.e. inference rules with premises and conclusions belonging to dierent contexts. Contexts and bridge rules are the components of Multi Context Systems (MC Systems). This paper is structured as follows. In Section 2, we present the theoretical basis of the context-based specication formalism. In Section 3 we present our case study, consisting of some paradigmatic MA scenarios to be formally specied. In Section 4 we work out a detailed multi context specication of the MA scenarios outlined in Section 3. In Section 6 we draw some conclusions and future developments. 2 Formal Specication of Multi-Agent Systems In this section, we rst dene (hierarchical) belief systems as a formal framework for the extensional specication of agents' beliefs. Then, we dene (hierarchical) MR systems as the class of MC systems nitely presenting (hierarchical) belief systems. 2.1 Extensional Specications Intuitively, 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]), and frames 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 of hypothesis, and 2

3 interaction capabilities, i.e. the ability of deriving consequences from what the other reasoners believe. Formally, a conditional reasoner R as a pair hl; `i, where the language L is a set of sentences, and ` (2 L L). (In the following, for sake of simplicity, we drop the word \conditional" whenever possible. Notationally, R i stands for the pair hl i ; `ii; T i stands for the set of theorems of the reasoner R i, i.e. the set of the formulas A such that ; `i A.) The interaction capabilities that we are interested in capturing are those that allow one reasoner R 0 to have beliefs about another reasoner R 1. R 0 's beliefs about R 1 are expressed via belief sentences, i.e. expressions of the form B(\A"). The two reasoners R 0, R 1 and the belief predicate B used by R 0 to express its own beliefs about R 1, characterize a belief system. Denition1. A belief system is a pair of reasoners hr 0 ; R 1 i B, where (i ) the parameter B is a unary predicate symbol called the belief predicate, (ii ) R 0 is the observer, and (iii ) R 1 is the observed reasoner of hr 0 ; R 1 i B. Intuitively, in a belief system hr 0 ; R 1 i B, R 0 observes R 1 and expresses its beliefs about R 1 using the belief predicate B. In other words, the intended interpretation of the expression B(\A") in L 0 is that R 0 believes that R 1 believes A. Reasoners and observers can be categorized depending on their reasoning and interaction capabilities, respectively. We say that reasoners and observers are ideal if they satisfy certain closure properties in the sense made precise by the following denition. Denition2. Let R 0 ; R 1 be two reasoners. We say that: R i (i = 0; 1) is an Ideal Reasoner if (i ) L i is closed under the formation rules for propositional languages, and (ii ) `i is closed under tautological consequence. R 0 is an Ideal Observer (of R 1 ) if (i ) L 1 = fa j B(\A") 2 L 0 g, and (ii )? `1 A =) B(\? ") `0 B(\A"), where B(\? ") is an abbreviation for the set fb(\") j 2? g. This denition captures a form of ideality with respect to (belief) derivation (i.e. conditional belief) in the observed reasoner. In other words, R 0 is an ideal 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 for representing the capabilities of real agents (e.g. humans and programs). Given them enough knowledge and resources (e.g. space, time) humans and programs tend to converge to an ideal behavior but this may never be the case. This situation 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 reasoner or observer derives too little or too much (i.e. it is incomplete or incorrect) with 3

4 respect to a reasoner or observer taken as reference. Thus, dierently from what is the case for ideality, reality is a relative notion which states the absence of certain properties with respect to a specic reference. The intuition is that, for instance, a reasoner R E is correct with respect to another reasoner R I if the inferences of R E are contained in the inferences of R I, i.e. if `E`I. If the inferences of R E are strictly contained in the inferences of R I (i.e. `E`I) then R E is incomplete. We say that a reasoner is real, to mean that it is incomplete or incorrect. Real reasoners and/or real observers are the components of a real belief system. So far we have concentrated on the basic conguration of one reasoner having beliefs about another reasoner. However, we may have more complex congurations with multiple reasoners organized in multiple belief systems, possibly sharing one or both reasoners, or their belief predicate. For example, we may have an \isolated" reasoner (a reasoner which is neither being observed by another reasoner nor an observer of other reasoners) as well as a reasoner observing another reasoner which, in turn, is (possibly) observing other reasoners and so on. As a limit case we may also have a reasoner observing itself. It is possible to represent the structure of a complex belief system as a direct graph, whose nodes represent reasoners and whose edges represent the observing relations between reasoners. Each edge is labeled with the unary predicate used by the observer to express its own beliefs about the observed reasoner. Thus, for example, the belief system hr 0 ; R 1 i B corresponds to Figure 1. For the goals of this paper we consider congurations of reasoners having a tree structure like that of Figure 2. R 0 B? R 1 Fig. 1. A belief system. = R 11 = R 1 B2.. B1 R 12. R B2 R.. 2 = R n1 B1 Z ZZ. B n Z ZZ~ R n B2.. B1 Fig. 2. A hierarchical belief system. R n2. Such congurations 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 I 0 = fi 1 ; : : :; i n g, I is a prex closed set of string over I 0 3. I contains also the empty string. A hierarchical belief system is a pair hfr i g i2i; Bi, where fr i g i2i 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

5 then R observes R j and expresses its beliefs about R j using the B j predicate. It is easy to check that a belief system hr 0 ; R 1 i B corresponds modulo renaming of indices to the hierarchical belief system hfr ; R 1 g; fh; 1igi B Finite Presentation of Specications Presenting a belief system requires representing the reasoning and interaction capabilities of each reasoner in the belief system. About the reasoning capability, they can be implemented by providing reasoners with a set of facts which constitute their basic knowledge, and with some inference engine for deriving consequences. For each reasoner R i, we thus introduce a corresponding context C i, dened as an axiomatic formal system, i.e. a triple hl i ; i ; i i, where L i is the language, i is the set of axioms and i is the set of inference rules of C i. (Notationally, in the following, a context C i is implicitly dened as hl i ; i ; i i.) This allows for a modular specication of the reasoning capabilities of each reasoner. About the interaction capabilities, they can be implemented by providing reasoners with bridge rules, i.e. inference rules with premises and conclusion in dierent contexts. For instance, the bridge rule C 1 : A 1 C 2 : A 2 states the derivability of the formula A 2 in context C 2 (written C 2 : A 2 ) from the derivability of A 1 in context C 1 (C 1 : A 1 ). This allows for a exible specication of reasoners' interactions. Contexts and bridge rules are the components of Multi Context systems (MC systems), dened as pairs hfamily-of-contexts, set-of-bridge-rulesi. Derivability in a MC system MS, in symbols `MS, is dened in [8]; roughly speaking, it is a generalization of Prawitz' notion of deduction inside a Natural Deduction System [18]. The interactions between the observer R 0 and the observed reasoner R 1 that we are interested in capturing are those which enable R 0 to contain the derivation B(\? ") `0 B(\A") when R 1 contains the derivation? `1 A. Hence, the particular class of MC systems presenting belief systems is characterized as follows: Denition3. An MC-system hfc 0 ; C 1 g; BRi is an MR system if BR is C 1 : A C 0 :B(\A") R1 up C 0 :B(\A") C 1 : A R 1 dn and the restrictions include: R 1 up: C 1 : A does not depend on any assumption in C 1. The notion of presentation of a belief system via an MR system is dened by imposing the following relation between the derivability relations of the reasoners and that of an MR system: 5

6 Denition4. Let MS = hfc 0 ; C 1 g; BRi be an MR system. MS presents the belief system hr 0 ; R 1 i B if `i= f(?; A) j fc i : B j B 2? g `MS C i : Ag (i = 0; 1). The closure conditions for ideality can be captured by posing appropriate restrictions on MR systems. Let us consider the following denition: Denition5. An MR system hfc 0 ; C 1 g; BRi is an MBK system if the following conditions are satised: (i ) L 0 and L 1 contain a given set P of propositional letters, the symbol for falsity?, and are closed under implication 4 ; (ii ) 0 ( 1 ) is the set of Classical Natural Deduction Rules; (iii ) L 1 = fa j B(\A") 2 L 0 g; (iv ) BR is C 1 : A C 0 :B(\A") R1 up C 0 :B(\A") C 1 : A and the only restriction is: R 1 up : C 1 : A does not depend on any assumption in C 1. The rule R 1 up is called reection up, while R1 dn is called reection down. The restrictions on R 1 up is such that a formula can be \reected up" only if it does not depend on any assumptions in C 1. We can now state the following theorem: Theorem 6. Let hr 0 ; R 1 i B be the belief system presented by an MBK system. Then (i ) R 0 and R 1 are ideal reasoners, and (ii ) R 0 is an ideal observer. Proof. The rst statement easily follows from the denition of ideal reasoner (Denition 2) and the consideration that (i ) and (ii ) of Denition 5 just allow MBK to present ideal reasoners. The second statement can be proved by showing that? `1 A =) B(\? ") `0 B(\A"), where? 2 2 L1. In the following, given any set of formulas? 2 L i (i = 0; 1), we use the notation C i :? as an abbreviation for fc i : j 2? g. Since, by hypothesis,? `1 A, then C 1 :? `MBK C 1 : A (Denition 4). So there exists in MBK a deduction of C 1 : A depending on C 1 :? 0, with? 0? and nite. Let us assume B(\") in C 0 for every 2? 0. Applying R 1 dn to the assumptions C 0 : B(\") (for every 2? 0 ), allows us to obtain a deduction of C 1 : depending on the assumptions C 0 :B(\") and also a deduction of C 1 : A depending on C 0 : B(\? 0 "). Then, applying R 1 up to A (which only depends on assumptions in C 0 ), we obtain a deduction of C 0 :B(\A") depending on C 0 :B(\? 0 "). Since B(\? 0 ") B(\? "), we have C 0 :B(\? ") `MBK C 0 :B(\A"), i.e. (Denition 4) B(\? ") `0 B(\A"). 2 MC systems presenting a hierarchical belief system are dened generalizing Denition 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??. R 1 dn 6

7 Denition7. Let I be a prex closed set of indices. An MC system hfc i g i2i; BRi is a hierarchical MR I system if BR consists of C j : A C :B j (\A") Rj up C :B j (\A") C j : A R j dn (j 2 I), and the restrictions include: R j up : C j : A does not depend on any assumption in C j. Denition8. Let MS = hfc i g i2i; BRi be a hierarchical MR I system. MS presents the hierarchical belief system hfr i g i2i; Bi, if for all i 2 I `i= f(?; A) j fc i : B j B 2? g `MS C i : Ag. 3 Description of the Scenarios In this section we present the case study of this paper. We start from the \Three Wise Men Puzzle" as informally described in [15]: A certain King wishes to test his three wise men. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their forehead but that at least one spot will be white. In fact all three spots are white. He then repeatedly asks 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 thorough analysis can be found in [1]). Here we provide an informal denition of dierent MA scenarios, which will be formally specied using the theoretical notions presented in the previous section. Scenario 1 (the original TWM): as it can be intuitively understood from the statement of the puzzle. Scenario 2 (the \not so wise" man): as Scenario 1, but where the second wise man is \not so wise", in that he is not able to reason about other agents' beliefs. There are two dierent 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 specication of these scenarios involve the representation of a number of issues which are relevant in the context of MA systems. Indeed, there are dierent agents, each with his own beliefs, which may or may not be consistent with the beliefs of the other agents, and his own reasoning capabilities. The agents have to be able to reason about other agents by ascribing beliefs and inferential abilities to them. These forms of ascription may or may not 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 dierent levels. For instance, it is possible that the third wise man can reason about the beliefs ascribed by the second wise to the rst. 7

8 4 Specication of the Scenarios In this section we provide a formal specication of the sample scenarios informally presented in the previous section. Each of the scenarios are extensionally characterized by means of a hierarchical belief systems, and nitely presented by a hierarchical MR system. This approach is based on the intuition that each reasoner in the tree-structure represents a certain point of view on the scenario. For instance, the reasoner R represents the (point of) view of an external reasoner looking at the scenario, R i formalizes the view of (wise man) i, and R ij formalizes the view ascribed by i to j. 4.1 Specication of Scenario 1 Starting from the informal description of Scenario 1, the class of hierarchical belief systems which extensionally specify it, is characterized as follows: { as there are three agents, the set I 0 is f1; 2:3g (in the following, i; j; k 2 I 0 ); { being wise, the agents are able to iterate the process of belief ascription to arbitrary depth, and therefore I = I0 (in the following, 2 I); { under every view it is possible to express statements about the the color of the spots and the agents' beliefs. Without loss of generality, we impose that L = L TWM for every. L TWM is the smallest language containing the propositions W i, B i (\A") for all A 2 L TWM, and closed under application of propositional connectives. Intuitively, W i express that i has a white spot, while B i (\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 is both an ideal reasoner and an ideal observer of R i ; { as the king utterance is considered true under each point of view, T contains W 1 _ W 2 _ W 3 King-Utterance { as each wise can see his colleagues, for any i 6= j T contains and, if does not contain i (W i B j (\W i ")) ^ (:W i B j (\:W i ")) i-sees-j W i can-see-i { the wise men speak in numerical order 5, and they are aware of this fact. Therefore any point of view about the second situation contains the statement of ignorance of wise man 1, and any point of view about the third situation contains the statement of the ignorance of both wise men 1 and 2. This is formalized by the following constraint: :B i (\W i ") 2 T, if 2 fj 2 I 0 : j > ig 5 The answers being \I don't know" for 1 and 2, and \My spot is white" for 3 (see [1]). 8

9 Any hierarchical system satisfying the above requirements is said to be a B TWM system. Some remarks. The extensional specication of the scenario (i.e. the hierarchical belief system) is simply obtained by composing dierent constraints deriving from the rationalization of the scenario. This process has the consequence that many underlying assumptions are forced to be made explicit, making therefore the features of the specied system more precise from the beginning of the design phase. Notice also that the specication given above is a partial specication of the scenario. The above constraints, indeed, give some (not all) conditions for a hierarchical belief systems to formalize Scenario 1. There are, indeed, facts which are true in the informal scenario, but which are not forced to hold by the partial specication given above. For instance, the specication does not imply that :B 1 (\:W 1 ") 2 T, which corresponds to the fact that in the view of the external observer wise man 1 does not believe that his spot is black. However partial, this specication 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 believe that his spot is white. Furthermore, there is at least a hierarchical belief systems which satises the specication above. The following theorem substantiates the above claims. Theorem9. In any B TWM system (i ) W 1 62 T 1, (ii ) W 2 62 T 2, (iii ) W 3 2 T 3. The class of B TWM systems is not empty. Proof. Item (i) is provable by contradiction. Suppose that W 1 2 T 1, this implies by ideality conditions that B 1 (\W 1 ") 2 T. Since :B 1 (\W 1 ") is also in T and R is an ideal reasoner, then T = L TWM. This contradicts the consistency condition of R. Item (ii) can be proved in the same way. Item (iii ): the proof follows from the ideal properties of each reasoner. Indeed, we have that :W 2 ; :W 3 `321 W 1 (King-Utterance). The ideal properties of 32 (w.r.t. 321) allow to obtain B 1 (\:W 2 "); B 1 (\:W 3 ") `32 B 1 (\W 1 "). Moreover from i-sees-j, we also have both :W 2 `32 B 1 (\:W 2 ") and :W 3 `32 B 1 (\:W 3 "). This allows to obtain :W 2 ; :W 3 `32 B 1 (\W 1 "). Since also `32 :B 1 (\W 1 ") holds (Utterance-1), we can conclude that :W 3 `32 W 2. By ideality of 3 (w.r.t. 32), B 2 (\:W 3 ") `3 B 2 (\W 2 "). By i-sees-j, :W 3 `3 B 2 (\:W 3 ") and then :W 3 `3 B 2 (\W 2 "). By Utterance-2, `3 :B 2 (\W 2 ") and, nally, `3 W 3. Now let us prove the second statement of the above theorem: i.e. that there exists at least a B TWM system. Consider a B TWM system where for each reasoner R = hl TWM ; `i, ` is dened as follows ( 2 I):? ` C ()? [ fcg L TWM and 2 Inf; 1; 2; 21g? `21 C ()?; W 3 ; B j (\L TWM ") j= A, for j 2 I 0? `1 C ()?; W 2 ; W 3 ; B j (\L TWM ") j= A, for j 2 I 0? `2 C ()?; :B 1 (\W 1 "); W 1 ; W 3 ; B j (\") B j (\C"); 1-sees-2 j= A; 9

10 for j 2 I 0 and `2j C? ` C ()?; :B i (\W 1 "); :B 2 (\W 2 "); W i ; B j (\") B j (\C"); i-sees-j j= A; for i; j 2 I 0 and `j C: (j= is the classical logical consequence relation). For all 2 I and j 2 I 0, R is both an ideal reasoner and an ideal observer of R j. Furthermore it can be easily veried that, for each 2 I, T contains all the formulas given in the extensional specication of B TWM. It remains to prove the consistency of R (i.e.? 62 T ). We prove it by providing a model for T. Let us consider the interpretation M of L TWM in which I M (W i ) = True, and I M (B i ) = fa j ; `1 Ag, for i = 1; 2; 3. It can be proved that M satises all formulas in T, and in particular :B 1 (\W 1 ") and :B 2 (\W 2 "). 2 We perform now the second step of the specication process, i.e. we nitely present (extensional) B TWM systems with hierarchical MR I systems. A hierarchical MR I system hfc i g i2i; BRi is an MBK TWM system if { I 0 = f1; 2; 3g and I = I 0 ; { the language L of every context C is L TWM ; { the set of inference rules of every context C is the set of Classical Natural Deduction 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 W 1 ^ W 2 ^ W 3, if = ; and :B i (\W i "), if 2 fj 2 I 0 : j > ig. The following theorem states that the hierarchical belief system presented by an MBK TWM system is a B TWM system. Theorem 10. An MBK TWM system presents a B TWM system. Corollary 11 (of Theorems 9, 10). For any MBK TWM system MS (i ) 6`MS C 1 : W 1, (ii ) 6`MS C 2 : W 2, (iii ) `MS C 3 : W 3. Figure 3 reports the corresponding MC proof of C 3 : W 3 in a MBK TWM system. 4.2 Specication of Scenario 2 We extensionally specify Scenario 2 starting from Scenario 1. The rst dierence to be accounted for is the limited inferential capability of wise 2. In particular, 2 is unable to reason under the other agents point of view. Therefore, 2 is represented as a reasoner unable to observe other reasoners. For the two Scenarios 2A and 2B have to consider a hierarchical belief system hfr i g i2i; Bi, where I is the set of indices in f1; 2; 3g such that 10

11 1 C3 :W3 (1) By assumption 2 :W3 B2(\:W3") From 2-sees-3 3 B2(\:W3") (1) From 1 and 2 by E 4 C32 :W3 (1) From 3 by R 32 dn 5 :W3 B1(\:W3") From 1-sees-3 6 B1(\:W3") (1) From 4 and 5 by E 7 :W2 (7) By assumption 8 :W2 B1(\:W2") From 1-sees-2 9 B1(\:W2") (7) From 7 and 8 by E 10 C321 :W3 (1) From 6 by R 321 dn 11 :W2 (7) From 9 by R 321 dn 12 W1 _ W2 _ W3 From King-Utterance 13 W1 (1; 7) From 10, 11 and C32 B1(\W1") (1; 7) From 13 by R 321 up 15 :B1(\W1") From Utterance-1 16? (1; 7) From 14 and 15 by E 17 W2 (1) From 16 by? c 18 C3 B2(\W2") (1) From 17 by R 32 up 19 :B2(\W2") From Utterance-2 20? (1) From 18 and 19 by E 21 W3 From 20 by? c Fig. 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 labels of the form 2 with 6= (i.e. I does not contain strings in which 2 occurs before some other character). This has the eect that under no point of view a reasoner representing wise 2 is able to observe any other reasoner 6. 2B. (1 and 3 are not aware of the lack of wisdom of 2.) I does not contain labels of the form 2 with 6=. In this case 1 and 3 are modeled exactly as in Scenario 1. A hierarchical belief system is a B NWM system [B NANWM system], if it satises the constraints given for B TWM systems except that I is dened according to item 2A [2B]. From an extensional point of view, the reasoner R 2 generated by an B NWM and B NANWM system is (in general) correct but incomplete with respect to the same reasoner as generated by a B TWM system. Indeed, for instance, in a B NWM system `2 does not necessarily contain all the derivations which would follow if R 2 were 6 To be very precise, this restriction formalizes the limit case where 1 and 3 are aware that 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 imposing the suitable constraints on I. 11

12 observing the reasoners R 21 ; R 22 ; R 23 (as it is the case in a B TWM system). Theorem 12. In any B NWM [B NANWM ] system, (i ) W 1 62 T 1, [W 1 62 T 1 ; ] (ii ) W 2 62 T 2, [W 2 62 T 2 ; ] (iii ) W 3 62 T 3. [W 3 2 T 3 :] The class of B NWM [B NANWM ] systems is not empty. The proof proceeds in the same way as for the case of B TWM systems. Note that, from the extensional specications, B TWM, B NWM and B NANWM dier only for the set I of reasoners. For instance, the reasoners of a B NWM systems are obtained by deleting the reasoners of a B TWM system with index of the form 2 for some 6=. The modularity of our approach allows us to obtain the nite presentation of a B NWM and B NANWM system, by deleting the corresponding contexts in the nite presentation of a B TWM. An MBK NWM [MBK NANWM ] system is dened as an MBK TWM system with the exception of the set I of indexes, which is equal to the set fi 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:] Theorem 13. An MBK NWM [MBK NANWM ] system presents a B NWM [B NANWM ] system. Corollary 14 (of Theorems 12, 13). For any MBK NWM [MBK NANWM ] system MS (i ) 6`MS C 1 : W 1, [6`MS C 1 : W 1 ; ] (ii ) 6`MS C 2 : W 2, [6`MS C 2 : W 2 ; ] (iii ) 6`MS C 3 : W 3. [`MS C 3 : W 3 :] 5 Related work In the eld of formal specication of multi agent systems a lot of approaches are based on modal logics (see for instance [4, 11, 13]). An alternative approach is the BDI architecture and the relative BDI logics proposed by Rao and George [19]. The main dierence with our approach is that we exploit distinct theories and relations between them to represent agents' internal state and agents interactions respectively. This allows for a modular specication which, as pointed out in [2] and [1], gives advantages in terms of elaboration tolerance and implementation. An application of MC system for the specication of a multi agent scenario can be found in [17]. In this paper authors provide a formal specication of a set of agents able to perform \cognitive tasks" such as communication and reasoning. In this framework each agent is modelled by a set of contexts, and communication between agents is formalized via bridge rules. In [21] the author provides a formal semantics for multi agent systems specied in concurrent MetateM. Concurrent MetateM allows to extensionally 12

13 specify a set of executable agents. However, it is possible to nitely present a concurrent MetateM multi agent system with an MC system. For lack of space we omit here the technical description of the construction of the MC system. Comparing the resulting MC system with Wooldridge's initial specication, it seems that our specication is more elaboration tolerant. In our MC system each bridge rule formalizes a single aspect of a concurrent MetateM multi agent system. For instance a bridge rule, called sync, formalizes synchronous execution. This implies elaboration tollerance [1]. For instance to formalize system with asynchronous computation, we simply drop bridge rule sync. Furthermore, in a MC system a proposition may have independent meaning in two dierent contexts. Since dierent agents are formalized by dierent contexts, this allows us to drop the technical assumption made in [21] that agents has disjoint languages. 6 Conclusion and Future Work In this paper we have proposed a formal framework for the extensional specication of MA's conditional beliefs and for their nite presentation. We have introduced 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 be formally specied using the above framework. As evidentiated by the case study, the extensional specication allows for a description independent from implementation; the context-based specication allows for modular implementations. Needless to say that the theory proposed is still 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 extended to conditional reasoners. Furthermore, concepts like elaboration tolerance as informally introduced and discussed in [2] need to be more precisely addressed and studied. From a representational point of view, many propositional attitudes other than belief (e.g. common belief, ignorance, desire, intention) need to be incorporated in the framework. Finally, from an implementational point of view, we have a prototypical implementation of the whole theory, but it denitely needs enhancements. References 1. A. Cimatti and L. Serani. Multi-Agent Reasoning with Belief Contexts: the Approach and a Case Study. In M. Wooldridge and N. R. Jennings, editors, Intelligent Agents: Proceedings of 1994 Workshop on Agent Theories, Architectures, and Languages, number 890 in Lecture Notes in Computer Science, pages 71{85. Springer Verlag, Also IRST-Technical Report , IRST, Trento, Italy. 2. A. Cimatti and L. Serani. Multi-Agent Reasoning with Belief Contexts II: Elaboration Tolerance. In Proc. 1st Int. Conference on Multi-Agent Systems (ICMAS- 95), pages 57{64, Also IRST-Technical Report , IRST, Trento, Italy. Commonsense-96, Third Symposium on Logical Formalizations of Commonsense Reasoning, Stanford University,

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