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Diversity of Agents
 University of Amsterdam
, 2006
"... Diversity of agents is investigated in the context of standard epistemic logic, dynamic information update, and belief revision. We provide a systematic discussion of different sources of diversities, such as introspection ability, powers of observation, memory capacity, and revision policies. In ea ..."
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Diversity of agents is investigated in the context of standard epistemic logic, dynamic information update, and belief revision. We provide a systematic discussion of different sources of diversities, such as introspection ability, powers of observation, memory capacity, and revision policies. In each case, we show how this diversity can be encoded in a logical system allowing for individual variation among rational agents. We conclude by raising some general issues concerning this view of a logic as a system for encoding a society of diverse agents and their interaction. 1 Diversity Inside Logical Systems Logical systems seem to prescribe one norm for an “idealized agent”. Any discrepancies with actual human behavior are then irrelevant, since the logic is meant to be normative, not descriptive. But logical systems would not be of much appeal if they did not have a plausible link with reality. And this is not just a matter of confronting one ideal norm with one kind of practical behavior. The striking fact is that human and virtual agents are not all the same: actual reasoning takes place in societies of diverse agents. This diversity shows itself particularly clearly in epistemic logic. There have been long debates about the appropriateness of various basic axioms, and they have to do with agents ’ different powers. In particular,
Products Of `transitive' Modal Logics Without The (abstract) Finite Model Property
"... It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstr ..."
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It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstract) finite model property either. These are the first known examples of 2D modal product logics without the finite model property where both components are natural unimodal logics having the finite model property.
Onevariable firstorder linear temporal logics with counting
"... Firstorder temporal logics are notorious for their bad computational behaviour. It is known that even the twovariable monadic fragment is highly undecidable over various timelines. However, following the introduction of the monodic formulas (where temporal operators can be applied only to subformu ..."
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Firstorder temporal logics are notorious for their bad computational behaviour. It is known that even the twovariable monadic fragment is highly undecidable over various timelines. However, following the introduction of the monodic formulas (where temporal operators can be applied only to subformulas with at most one free variable), there has been a renewed interest in understanding extensions of the onevariable fragment and identifying those that are decidable. Here we analyse the onevariable fragment of temporal logic extended with counting (to two), interpreted in models with constant, decreasing, and expanding firstorder domains. We show that over most classes of linear orders these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator ‘eventually’. A more general result says that the bimodal logic of commuting linear and pseudoequivalence relations is undecidable. The proofs are by reductions of various counter machine problems. 1 Introduction and
The Topological Product of S4 and S5
"... The most obvious bimodal logic generated from unimodal logics L1 and L2 is their fusion, L1⊗L2, axiomatized by the theorems of L1 for □1 and of L2 for □2, and by the rules of modus ponens, substitution and necessitation for □1 and for □2. Shehtman introduced the frame product L1×L2, as the logic of ..."
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The most obvious bimodal logic generated from unimodal logics L1 and L2 is their fusion, L1⊗L2, axiomatized by the theorems of L1 for □1 and of L2 for □2, and by the rules of modus ponens, substitution and necessitation for □1 and for □2. Shehtman introduced the frame product L1×L2, as the logic of the products of certain Kripke frames. Typically, L1 ⊗L2 � L1 ×L2, e.g. S4⊗S4 � S4×S4. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product L1 ×t L2, as the logic of the products of certain topological spaces: they showed, in particular, that S4×t S4 = S4⊗S4. In this paper, we axiomatize S4×t S5, which is strictly between S4 ⊗ S5 and S4 × S5. We also apply our techniques to proving a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.