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Modal logics and agents in Software Get barcode 3 of 9 in Software Modal logics and agents

5 Modal logics and agents generate, create barcode 3 of 9 none on software projects datamatrix Scenario 2. The Code 3 of 9 for None father rst announces that at least one of them is muddy which is something they know already; and then, as before, he repeatedly asks them Does any of you know whether you have mud on your own forehead The rst time they all answer no. Indeed, they go on answering no to the rst k 1 repetitions of that same question; but at the kth those with muddy foreheads are able to answer yes.

At rst sight, it seems rather puzzling that the two scenarios are di erent, given that the only di erence in the events leading up to them is that in the second one the father announces something that they already know. It would be wrong, however, to conclude that the children learn nothing from this announcement. Although everyone knows the content of the announcement, the father s saying it makes it common knowledge among them, so now they all know that everyone else knows it, etc.

This is the crucial di erence between the two scenarios. To understand scenario 2, consider a few cases of k. k = 1, i.

e. just one child has mud. That child is immediately able to answer yes, since she has heard the father and doesn t see any other child with mud.

k = 2, say only the children Ramon and Candy have mud. Everyone answers no the rst time. Now Ramon thinks: since Candy answered no the rst time, she must see someone with mud.

Well, the only person I can see with mud is Candy, so if she can see someone else it must be me. So Ramon answers yes the second time. Candy reasons symmetrically about Ramon and also answers yes the second time round.

k = 3, say only the children Alice, Bob, and Charlie have mud. Everyone answers no the rst two times. But now Alice thinks: if it was just Bob and Charlie with mud, they would have answered yes the second time; making the argument for k = 2 above.

So there must be a third person with mud; since I can see only Bob and Charlie having mud, the third person must be me. So Alice answers yes the third time. For symmetrical reasons, so do Bob and Charlie.

And similarly for other cases of k. To see that it was not common knowledge before the father s announcement that one of the children was muddy, consider again k = 2, with Ramon and Candy. Of course, Ramon and Candy both know someone is muddy they see each other; but, for example, Ramon doesn t know that Candy knows that someone is dirty.

For all Ramon knows, Candy might be the only dirty one and therefore not be able to see a dirty child.. 5.5 Reasoning about knowledge in a multi-agent system 5.5.2 The modal Software 3 of 9 logic KT45n We now generalise the modal logic KT45 given in Section 5.

3.4. Instead of having just one 2, it will have many, one for each agent i from a xed set A = {1, 2, .

. . , n} of agents.

We write those modal connectives as Ki (for each agent i A); the K is to emphasise the application to knowledge. We assume a collection p, q, r, . .

. of atomic formulas. The formula Ki p means that agent i knows p; so, for example, K1 p K1 K2 K1 p means that agent 1 knows p, but knows that agent 2 doesn t know he knows it.

We also have the modal connectives EG , where G is any subset of A. The formula EG p means everyone in the group G knows p. If G = {1, 2, 3, .

. . , n}, then EG p is equivalent to K1 p K2 p Kn p.

We assume similar binding priorities to those put forward on page 307.. Convention 5.22 The binding priorities of KT45n are the ones of basic modal logic, if we think of each modality Ki , EG and CG as being 2. One might think that could not be more widely known than everyone knowing it, but this is not the case.

It could be, for example, that everyone knows , but they might not know that they all know it. If is supposed to be a secret, it might be that you and your friend both know it, but your friend does not know that you know it and you don t know that your friend knows it. Thus, EG EG is a state of knowledge even greater than EG and EG EG EG is greater still.

We say that is common knowledge among G, written CG , if everyone knows and everyone knows that everyone knows it; and everyone knows that; and knows that etc. So we may think of CG as an in nite conjunction E G EG EG EG EG EG . .

. . However, since our logics only have nite conjunctions, we cannot reduce CG to something which is already in the logic.

We have to express the in nite aspect of CG via its semantics and retain it as an additional modal connective. Finally, DG means the knowledge of is distributed among the group G; although no-one in G may know it, they would be able to work it out if they put their heads together and combined the information distributed among them. De nition 5.

23 A formula in the multi-modal logic of KT45n is de ned by the following grammar: ::= . p . ( ) . ( ) . ( ) . ( ) . ( ) . (Ki ) . (EG ) . (CG ) . (DG ).
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