L. Catach. Normal multimodal logics. In T. M. Smith and G. R. Mitchell, editor, Proceedings of the 7th National Conference on Artificial Intelligence (AAAI'88), pages 491--495, St. Paul, MN, August 1988. Morgan Kaufmann.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of which has its own knowledge, reasoning style as well as its way of interacting with other agents. This kind of applications is of great interest to the Artificial Intelligence community, which is currently deeply investigating the use of modal logics and, in particular, multimodal logics [12] in agent modelling. Besides their adequacy in supporting reasoning in a multi agent environment and to represent knowledge, beliefs, dynamic changes, actions, and time [34, 24, 50] the interest on the class of multimodal logics is due to their ability to express complex modalities, obtained by ....

....knowledge, beliefs, dynamic changes, actions, and time [34, 24, 50] the interest on the class of multimodal logics is due to their ability to express complex modalities, obtained by composing modal operators even of different types. This is also one of the main features of multimodal logis, see [12]. Let s now get into some more details, in order to explain how this paper places with respect to the work that has already been done on multimodal logics. One of the central points to be faced when one wants to define a body of (multi)modal knowledge is to define the meaning of the modal ....

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L. Catach. Normal Multimodal Logics. In Proc. of the 7th National Conference on Artificial Intelligence, AAAI '88, volume 2, pages 491-- 495. Morgan Kaufmann, 1988.

....relational variables using disjunction and composition. Let # be a set modal formulae in the language of multi modal K (m) and let K (m) # be the extension of K (m) closed under the formulae in #. For example, the axiom schema listed in Figure 4 determine classes of logics considered in Catach [9] and Baldoni [5] Theorem 6.8 Let # be any finite set of instances of formulae in Figure 4, and let # be the set of associated first order properties as specified in Figure 4. Then: For any modal formula #, # is satisfiable in K (m) # i# # #(#) is first order satisfiable. Proof. By noting ....

....the scan algorithm [19] one can prove the properties associated with the modal formulae in Figure 4 are in fact their correspondence properties. Thus, the theorem follows from the well known Sahlqvist Theorem [68] # This theorem is also an easy consequence of a more general theorem by Catach [9]. Theorem 6.9 Let # be any finite set of instances of formulae in Figure 4, with the restriction that in each case # is a relational formula built from relational variables and disjunction only, while # and # denote either a relational variable or a relational formula built from relational ....

L. Catach. Normal multimodal logics. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI'88), pages 491--495. AAAI Press/MIT Press, 1988.

.... of modalities obtained from composition and union (i.e. # 2 A = # 1 # 2 A and # 1 # 2 A = # 1 A # 2 A) Such axioms characterize models satisfying a, b, c, d incestuality, i.e. #(a) 1 #(b) #(c) #(d) 1 , where # is map from string of modalities to accessibility relation, see [6]. In more complex cases we may define an unification schema matching the appropriate a, b, c, d incestuality condition, e.g. the first label ends with a string of world symbols corresponding to a, b while the second corresponds to c, d; then we can apply (fibre) more unifications inside the ....

Laurent Catach. Normal multimodal logics. In Proc. Nat. Conf. on AI (AAAI'88), 491--495, 1988.

....offer preservation of some important logical properties of its elements. In this study we investigate one such method, viz. fibring [5] and use it to reconstruct the logical account of BDI in terms of dovetailing (a special case of fibring) together with the multi modal semantics of Catach [2]. In doing so we identify a set of interaction axioms for BDI, based on the incestual schema G , which covers many of the existing BDI axioms and also make possible the generation of a large class of new ones. Further we identify conditions under which completeness transfers from the component ....

....we need is a set of interaction axioms that can generate a range of multi modal systems for which there is a general characterization theorem so that we could avoid the need for showing it each time a new system is considered. To this end we adopt the class of interaction axioms G of Catach [2] that can account for a range of multi modal systems. 3 Fibring of Modal Logics In this section we present a general semantic methodology, called fibring, for combining modal (BDI) logics and a variant of it called dovetailing. Two theorems stating relationships between dovetailing and BDI ....

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L. Catach. Normal multimodal logics. In In Proc. National Conference on AI(AAAI-88), pages 491--495, 1988.

....combine the axiomatics of L 1 and L 2 . Clearly, the most favourable case is when L 1 L 2 is a conservative extension of both L 1 and L 2 . This is the case e.g. if we combine modal logics of time and knowledge, and more generally when we combine several modal logics into a multimodal logic ((Catach, 1988), Kracht, 1993) A necessary and sufficient condition for L 1 L 2 to be a conservative extension of L i (i = 1; 2) is that every axiom schema of L i is valid in L 1 L 2 , and that every rule schema of L i preserves validity in L 1 L 2 . 1 If we tested validity of instances (and not of ....

L. Catach (1988), Normal multimodal logics. Proc. Nat. Conf. on AI (AAAI-88), pp 491-495.

....can prove now theorems of these monotonic modal logics via our translation into particular multi modal logics. Note that the translations (D) T ) 4) of the standard modal axioms D;T , and 4 become multi modal axioms in the style of Sahlqvist (Sahlqvist 1975) for which completeness results (Catach 1988, Kracht 1993) and automated deduction methods (Ohlbach 1991, 1993, Gasquet 1993, 1995, Nonnengart 1993, Fari nas and Herzig 1993) are known. Note also that the translations of the modal axioms 5 and B would become axioms which have not been studied yet in the literature. Examples Example 1. The ....

....by some axiomatization of classical logic, necessitation rules for [1] and [2] plus the axioms [1 [ 2]F F and : 1 [ 2]F [1 [ 2] 1 [ 2]F . It is well known that KT 5[1[2] is characterized by the class of Kripke frames (W;R1 ; R2 ) where R1 [ R2 is an equivalence relation over W (see e.g. (Catach 1988)) 11 Example 5. The formula (2p 2q) p q) is translated into ( 1]p [2] p [1]q [2] q) p q) In KT 5[1 [ 2] the antecedent is equivalent to ( 1] p q) 2] p q) and now ( 1] p q) 2] p q) p q) is an instance of the T [1 [ 2] axiom. Proofs The proof has been first ....

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L. Catach, "Normal multimodal logics". Proc. Nat. Conf. on AI (AAAI 88), pp. 491495, 1988.

....the first order case by introducing the usual rules for quantifiers. Moreover, it can be extended to deal with a wider class of logics. In particular, in [2] a tableau calculus is developed for the class of multimodal logics characterized by a; b; c; d incestuality axioms (defined by Catach in [7]) and, then, as a special case, also for the multimodal logics characterized by serial, symmetric, and Euclidean accessibility relations. Acknowledgments. The authors would like to thank the referees for the precious advice. ....

L. Catach. Normal Multimodal Logics. In Proc. of the AAAI '88, pages 491--495. Morgan Kaufmann, 1988.

....S5 in which all agents have the same knowledge. This can be modelled by taking an extension of S5n in which the axiom S i;j holds for all i; j 2 A, making all the modalities collapse onto each other. This is a very strong constraint. At the other end of the spectrum is simply S5n . Among others, Catach [Catach 1988] has studied a limited class of such interactions between knowledge of the agents. In this paper, we work with the notion of interpreted system of Halpern and Knowledge in Multi Agent Systems: Initial Con gurations and Broadcast 3 Moses [Halpern and Moses 1990; Fagin et al. 1995] We introduce ....

....expansion of WD. For a discussion of a number of alternative axioms that can be shown to be equivalent to WD, we refer the reader to the thesis of Lomuscio [Lomuscio 1999] One such alternative, for the case n = 2, has appeared in the literature before, as the axiom 3 1 2 2 p ) 2 2 3 1 p due to Catach [Catach 1988], also discussed in [Popkorn 1994] Theorem 4.2. WD in the case n = 2 is S5 2 equivalent to Catach s axiom. Knowledge in Multi Agent Systems: Initial Con gurations and Broadcast 19 Proof. If n = 2 then WD is (3 1 1 3 2 1 ) 3 1 2 3 2 2 ) i;j2f1;2g 3 i 3 j ( 1 2 ) ....

Catach, L. 1988. Normal multimodal logics. In T. M. Smith and G. R. Mitchell Ed., Proceedings of the 7th National Conference on Articial Intelligence (AAAI'88) (St. Paul, MN, Aug. 1988), pp. 491-495. Morgan Kaufmann.

....are purely axiomatic in nature no architectural, correspondence is established between axioms and models that they correspond to. Finally, it is worth noting that there is now a growing body of work addressing the abstract logical properties of multi modal logics, of which VSK is an example [3]. Lomuscio and Ryan, for example, investigates axiomatizations of multi agent epistemic logic (epistemic logics with multiple K operators) 7] The work in this paper can clearly benefit from such work. 8 Conclusions In this paper, we have presented a formalism that allows us to represent ....

L. Catach. Normal multimodal logics. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI-88), pages 491--495, St. Paul, MN, 1988.

....which all agents have the same knowledge. This can be modelled by taking an extension of S5n in which the axiom S i;j holds for all i; j 2 A, making all the modalities collapse onto each other. This is a very strong constraint. At the other end of the spectrum is simply S5n . Among others, Catach [Cat88] has studied a limited class of such interactions between knowledge of the agents. In this paper, we work with the notion of interpreted system of Halpern and Moses [HM90, FHMV95] We introduce and study two special cases of the interpreted systems model, which we call full systems and ....

....of WD. For a discussion of a number of alternative axioms that can be shown to be equivalent to WD, we refer the reader 15 to the thesis of Lomuscio [Lom99] One such alternative, for the case n = 2, has appeared in the literature before, as the axiom 3 1 2 2 p ) 2 2 3 1 p due to Catach [Cat88], also discussed in [Pop94] Theorem 4.2 WD in the case n = 2 is S5 2 equivalent to Catach s axiom. Proof If n = 2 then WD is (3 1 OE 1 3 2 OE 1 ) 3 1 OE 2 3 2 OE 2 ) i;j2f1;2g 3 i 3 j (OE 1 OE 2 ) with OE i i local) WD to Catach: Put OE 1 = 2 1 :p and OE 2 = 2 2 p (note that ....

L. Catach. Normal multimodal logics. In T. M. Smith and G. R. Mitchell, editor, Proceedings of the 7th National Conference on Artificial Intelligence (AAAI'88), pages 491--495, St. Paul, MN, August 1988. Morgan Kaufmann.

....the logic S5 n by: i p ) j p; i j where expresses the order in the computational power at disposal of the agents. In these two cases, some information is being shared among the agents of the group. A third example of sharing in the literature is the axiom i j p ) j i p; i 6= j [2] which says that: if agent i considers possible that agent j knows p then agent j must know that agent i considers possible that p is the case. It is easy to imagine other meaningful axioms that express interactions between the agents in the system; clearly there is a spectrum of possible degrees ....

....with i p , j p; for all i; j 2 A; saying that the agents have precisely the same knowledge (total sharing) The three examples mentioned above exist somewhere in the (partially ordered) spectrum between these two extremes. Some instances of such systems have already been identified in [2, 1, 15] and in other papers. Our aim in this paper is to explore the spectrum systematically. We restrict our attention to the case of two agents (i.e. to extensions of S5 2 ) and explore axiom schemas of the forms p ) p p ) p p ) p p ) p where each occurrence of is in the set f 1 ; 1 ; 2 ; ....

[Article contains additional citation context not shown here]

L. Catach. Normal multimodal logics. In T. M. Smith and G. R. Mitchell, editor, Proceedings of the 7th National Conference on Artificial Intelligence (AAAI'88), pages 491--495, St. Paul, MN, August 1988. Morgan Kaufmann.

....functions we propose that make new modalities. The functions we found, with another second order function, recursion, are safe for bisimulation in this sense. These constructs were originally developed in the framework of Dynamic Logic, Pratt 1990) The functions we consider are also suggested by (Catach 1988) as ways of making new modalities. Bisimulation, and its relation to nonwellfounded set theory are explored in detail in (Barwise and Moss 1996) Non wellfounded set theory has also been considered as a natural model of situation theory (Barwise and Perry 1983) which bears much similarity with ....

Catach, L. 1988. Normal Multimodal Logics. In Proc. National Conference on Artificial Intelligence (AAAI '88), 491--495, Saint Paul, MI. AAAI.

....established. Furthermore, they define a conditional formula f g to denote Pi(f ) Pi(g) and show that f g can be transalted into a wff of PL P . However, the given translation is not truth preserving. That is, it is possibile that f g is false but the resultant wff is true. Third, Catach[4] considers some axiomatic schemata for general multimodal logics. Though his results can be applied to different intensional logics, such as epistemic, doxastic, temporal, dynamic logics and their combination, however, it seems that he only allows a finite set of atomic modal operators and then ....

Catach, L., Normal multimodal logics, Proc. of the 7th AAAI, Morgan Kaufmann Publishers, 491-495, 1988.

....message passing, showing MU to be even more powerful than the formalism of [MP90] In the conventional OO notation, this clause could be written as: object v is . method p is q; self:r, z:s . 4. 2 Formal In the formal presentation, we adhere to the terminology and denotations from [Che80] and [Cat88]. Proofs and technicalities can be found in [Uus91] To simplify the presentation, we shall consider propositional fragments everywhere. Section 4.3 gives hints for generalizations into the full predicate logic. For any modal operator [a] a will stand for [a] Language For any u U, the ....

....rules and axioms of the classical propositional logic. Modal rules and axioms. For any u, v, z U, we have: N u . A [u]A, K u . u] A B) u]A [u]B) D u . u]A u A, K v u . v]A [u]A if u isa v, 4 u,z . z]A [u] z]A, 5 u,z . z A [u] z A. MU is normal in the sense of [Cat88]. Semantics A complete Kripke semantics of MU is defined by the following restrictions on the accessibility relations: d u . seriality: for any w W there exist w W such that wR u w, k v u . inclusion: if u isa v then for any w,w W if wR u w then wR v w, iv u,z . quasi transitivity: ....

[Article contains additional citation context not shown here]

L. Catach. Normal multimodal logics. In AAAI-88: Proc. 7th Nat'l Conf. on AI, St Paul, Aug 1988, Vol 2, pp 491-5. 1988.

....of a fact known by another agent. This is not surprising once we remember that all the known facts must be true at the real world. Quite surprisingly little work has been done to analyse systematically such interaction schemas the only exceptions to this that we are aware of are [3] and [1] in which a limited class of interactions between the agents is proposed. In this paper we isolate and study a special class of interpreted systems that can be modelled by an extension of S5n that falls into the above described spectrum. The systems we investigate, that we call hypercube systems ....

....(nWD) if 8w; w1 ; wn 2 W , such that wR i w i ; i = 1; n, and 8(x1 ; xn) 2 Pn , 9w such that w i Rx i w, for all i = 1; n. When n is clear from the context we just use we refer to nWD just as WD. The property nWD for n = 2 was discussed in [16] and [1]; nWD is a generalisation of it. It is immediate to note that: Lemma 3.4 If a frame is directed then it is weakly directed. We analyse extensions of S5n with respect to the axiom: x 1 ; x n )2Pn ( Sigma 1 x 1 p1 Delta Delta Delta Sigma n Gamma1 x n Gamma1 pn Gamma1 ) n Sigma ....

L. Catach, `Normal multimodal logics', in Proceedings of the 7th National Conference on Artificial Intelligence, ed., Tom M. Smith, Reid G.; Mitchell, pp. 491--495, St. Paul, MN, (August 1988). Morgan Kaufmann.

....logics. An implementation of these decision procedures, written in Eiffel, is described in [4] There are many ways in which we hope to extend this work in the future: i) undertake a systematic analysis of interaction axioms, which characterise interactions between time and knowledge belief (cf. [2]) and extend the decision procedure to deal with these axioms; ii) use the implementation developed to further investigate the efficiency of the method; iii) extend the decision procedures to the first order case; iv) develop different proof techniques for KLn and BLn (such as translation ....

L. Catach. Normal multimodal logics. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI-88), pages 491--495, St. Paul, MN, 1988.

....of time do not provide sufficient expressive power. For such applications, it is necessary to provide connectives that allow us to represent the properties of different modal dimensions in the same logic. Logics which contain more than one different type of modality are called multi modal logics [3]. In this paper, we consider a multi modal logic which contains connectives for representing both time and belief. The obvious approach to defining the semantics of a temporal belief logic involves adapting possible worlds semantics for belief [11] one might define a world to be a sequence of ....

....in the study of theoretically perfect believers, it is clearly at odds with any reasonable understanding of how belief works in resource bounded reasoners. The second problem is that belief and time would interact in such a way as as to make the development of an automatic proof method awkward [3]. In this paper, we develop a temporal belief logic called L TB , in which the semantics of belief are not based on possible worlds, but on a simple new model of belief which is outlined in 2. The logic L TB is then developed in 3, which also includes a discussion of its applications. Since time ....

L. Catach. Normal multimodal logics. In Proceedings of the National Conference on Artificial Intelligence (AAAI '88), St. Paul, MN, 1988.

No context found.

L. Catach. Normal multimodal logics. In T. M. Smith and G. R. Mitchell, editor, Proceedings of the 7th National Conference on Artificial Intelligence (AAAI'88), pages 491--495, St. Paul, MN, August 1988. Morgan Kaufmann.

No context found.

L. Catach. Normal multimodal logics. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI'88), pages 491--495. AAAI Press/MIT Press, 1988.

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