Aggregate information,common knowledge and agreeing not to bet

This note considers gambles that take place even if only some—but not all—individuals agree to participate. I show that the bet cannot take place if it is commonly known how many individuals are willing to participate.     

Barwise's information frames and modal logics

 The paper studies Barwise's information frames and answers the John Barwise question: to find axiomatizations for the modal logics generated by information frames. We find axiomatic systems for (i) the modal logic of all complete information frames, (ii) the logic of all sound and complete information frames, (iii) the logic of all hereditary and complete information frames, (iv) the logic of all complete, sound and hereditary information frames, and (v) the logic of all consistent and complete information frames. The notion of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4 respectively. To develop general theory, we prove that (i) any Kripke complete modal logic is the modal logic of a certain class of information frames and that (ii) the modal logic generated by any given class of complete, rarefied and fully classified information frames is Kripke complete. This paper is dedicated to the memory of talented mathematician John Barwise. Received: 7 May 2000 Published online: 10 October 2002 Key words or phrases: Knowledge presentation – Information – Information flow – Information frames – Modal logic-Kripke model     

Approximate common knowledge revisited

Suppose we replace “knowledge” by “belief with probability p” in standard definitions of common knowledge. Very different notions arise depending on the exact definition of common knowledge used in the substitution. This paper demonstrates those differences and identifies which notion is relevant in each of three contexts: equilibrium analysis in incomplete information games, best response dynamics in incomplete information games, and agreeing to disagree/no trade results.     

Fuzzy events and fuzzy logics in classical information systems

Classical information systems are introduced in the framework of measure and integration theory. The measurable characteristic functions are identified with the exact events while the fuzzy events are the real measurable functions whose range is contained in the unit interval. Two orthogonality relations are introduced on fuzzy events, the first linked to the fuzzy logic and the second to the fuzzy structure of partial a Baer¹-ring. The fuzzy logic is then compared with the “empirical” fuzzy logic induced by the classical information system. In this context, quantum logics could be considered as those empirical fuzzy logics in which it is not possible to have preparation procedures which provide physical systems whose “microstate” is always exactly defined.     

The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting”.     

The generation of formulas held in common knowledge

Common knowledge of a finite set of formulas implies a special relationship between syntactic and semantic common knowledge. If S, a set of formulas held in common knowledge, is implied by the common knowledge of some finite subset of S, and A is a non-redundant semantic model where exactly S is held in common knowledge, then the following are equivalent: (a) S is maximal among the sets of formulas that can be held in common knowledge, (b) A is finite, and (c) the set S determines A uniquely; otherwise there are uncountably many such A. Even if the knowledge of the agents are defined by their knowledge of formulas, 1) there is a continuum of distinct semantic models where only the tautologies are held in common knowledge and, 2) not assuming that S is finitely generated (a) does not imply (c), (c) does not imply (a), and (a) and (c) together do not imply (b). Received November 1999/Revised version January 2000     

Sequent calculi for branching time temporal logics of knowledge and beliefwith awareness: Completeness and decidability

In this paper, we consider branching time temporal logic CT L with epistemic modalities for knowledge (belief) and with awareness operators. These logics involve the discrete-time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. In addition, the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”). In a language of these logics, there are awareness operators. For these logics, we present sequent calculi with a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for these calculi are proved. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 328–340, July–September, 2007.     

Combining dependent evidential bodies that share common knowledge

We establish a formula for combining dependent evidential bodies that are conditionally independent given their shared knowledge. Markov examples are provided to illustrate various aspects of our combination formula, including its practicality. We also show that the Dempster–Shafer formula and the conjunctive rule of the Transferable Belief Model can be recovered as special cases of our combination formula.     

The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.     

Inclusion and exclusion dependencies in team semantics — On some logics of imperfect information

We introduce some new logics of imperfect information by adding atomic formulas corresponding to inclusion and exclusion dependencies to the language of first order logic. The properties of these logics and their relationships with other logics of imperfect information are then studied. As a corollary of these results, we characterize the expressive power of independence logic, thus answering an open problem posed in Grädel and Väänänen, 2010 [9].     

Characterizing solution concepts in terms of common knowledge of rationality

Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known. Analogous characterizations of sequential equilibrium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using earlier results of Halpern characterizing these solution concepts using non-Archimedean fields.     

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(～) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.     

Algebraization of logics defined by literal‐paraconsistent or literal‐paracomplete matrices

We study the algebraizability of the logics constructed using literal‐paraconsistent and literal‐paracomplete matrices described by Lewin and Mikenberg in [11], proving that they are all algebraizable in the sense of Blok and Pigozzi in [3] but not finitely algebraizable. A characterization of the finitely algebraizable logics defined by LPP‐matrices is given. We also make an algebraic study of the equivalent algebraic semantics of the logics associated to the matrices ??³_2,2, ??³_2,1, ??³_1,1, ??³_1,3, and ??⁴ appearing in [11] proving that they are not varieties and finding the free algebra over one generator. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-valued reductions of infinite-valued logics

In this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to ?ukasiewicz, Gödel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP.     

Conuclear images of substructural logics

Our work proposes to study the conuclear image of a substructural logic and in particular to investigate the relationship between a substructural logic and its conuclear image. We analyze some axioms familiar to substructural logics and we check if they: (a) are preserved under conuclear images, (b) never hold in a conuclear image, or (c) are compatible with conuclear images but are not necessarily preserved under conuclear images. Moreover, we prove that the conuclear image of any substructural logic has the disjunction property. We finally give a sufficient condition in order that an inequality is preserved under conuclear images and observe that if we slightly relax this condition, we meet counterexamples of inequalities that are not preserved under conuclear images.     

Methods for knowledge reduction in inconsistent ordered information systems

Knowledge reduction is one of the most important problems in the study of rough set theory. However, in real-world, most of information systems are based on dominance relations in stead of the classical equivalence relation because of various factors. The ordering of properties of attributes plays a crucial role in those systems. To acquire brief decision rules from the systems, knowledge reductions are needed. The main objective of this paper is to deal with this problem. The distribution reduction and maximum distribution reduction are proposed in inconsistent ordered information systems. Moreover, properties and relationship between them are discussed. Furthermore, judgment theorem and discernibility matrix are obtained, from which an approach to knowledge reductions can be provided in inconsistent ordered information systems.