Information sharing among ideal agents

Multi-agent systems operating in complex domains crucially require agents to interact with each other. An important result of this interaction is that some of the private knowledge of the agents is being shared in the group of agents. This thesis investigates the theme of knowledge sharing from a theoretical point of view by means of the formal tools provided by modal logic. More specifically this thesis addresses the following three points. First, the case of hypercube systems, a special class of interpreted systems as defined by Halpern and colleagues, is analysed in full detail. It is here proven that the logic S5WDn constitutes a sound and complete axiomatisation for hypercube systems. This logic, an extension of the modal system S5n commonly used to represent knowledge of a multi-agent system, regulates how knowledge is being shared among agents modelled by hypercube systems. The logic S5WDn is proven to be decidable. Hypercube systems are proven to be synchronous agents with perfect recall that communicate only by broadcasting, in separate work jointly with Ron van der Meyden not fully reported in this thesis. Second, it is argued that a full spectrum of degrees of knowledge sharing can be present in any multi-agent system, with no sharing and full sharing at the extremes. This theme is investigated axiomatically and a range of logics representing a particular class of knowledge sharing between two agents is presented. All the logics but two in this spectrum are proven complete by standard canonicity proofs. We conjecture that these two remaining logics are not canonical and it is an open problem whether or not they are complete. Third, following a influential position paper by Halpern and Moses, the idea of refining and checking of knowledge structures in multi-agent systems is investigated. It is shown that, Kripke models, the standard semantic tools for this analysis are not adequate and an alternative notion, Kripke trees, is put forward. An algorithm for refining and checking Kripke trees is presented and its major properties investigated. The algorithm succeeds in solving the famous muddy-children puzzle, in which agents communicate and reason about each other's knowledge. The thesis concludes by discussing the extent to which combining logics, a promising new area in pure logic, can provide a significant boost in research for epistemic and other theories for multi-agent systems

Failure of interpolation in combined modal logics

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalisation of Humberstone's inaccessibility logic. Viewing first order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. It's use is exemplified in a wide range of cases

Failure of Interpolation in Combined Modal Logics

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone’s inaccessibility logic. Viewing first order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases. In 1957, W. Craig proved the interpolation theorem for first order logic [Cra57]. Comer [Com69] showed that the property fails for all finite variable fragments except the onevariable fragment. The n-variable fragment of first order logic –for short Ln – contains all first order formulas using just n variables and containing only predicate symbols of arity not higher that n (we assume the language has only variables as terms.) Here we will show that the axiom which makes the quantifiers commute can be seen as the reason for this failure. Since Craig’s paper, interpolation has become one of the standard properties that one investigates when designing a logic, though it hasn’t received the status of a completeness or a decidability theorem. One of the main reasons why a logic should have interpolation is because of “modular theory building”. As we will see below interpolation in modal logic is equivalent to the following property (which is the semantical version of Robinson’s consistency lemma.