Logic Of Common Knowledge Research Articles

Common Knowledge of Abstract Groups

Epistemic logics typically talk about knowledge of individual agents or groups of explicitly listed agents. Often, however, one wishes to express knowledge of groups of agents specified by a given property, as in ‘it is common knowledge among economists’. We introduce such a logic of common knowledge, which we term abstract-group epistemic logic (AGEL). That is, AGEL features a common knowledge operator for groups of agents given by concepts in a separate agent logic that we keep generic, with one possible agent logic being ALC. We show that AGEL is EXPTIME-complete, with the lower bound established by reduction from standard group epistemic logic, and the upper bound by a satisfiability-preserving embedding into the full µ-calculus. Further main results include a finite model property (not enjoyed by the full µ-calculus) and a complete axiomatization.

Temporal logic of common knowledge and its resolution-based proof method

Temporal logic of common knowledge and its resolution-based proof method

Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The μ-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability and in fact often live within the single variable fragment of the μ-calculus. The family of such flat fixpoint logics includes, e.g., Linear Temporal Logic (LTL), Computation Tree Logic (CTL), and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard μ-calculus including, e.g., flat fragments of the graded μ-calculus and the alternating-time μ-calculus (such as alternating-time temporal logic), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.

Common Knowledge in Well-structured Perfect Recall Systems

We investigate a model checking problem for the logic of common knowledge and fixpoints µPLC n in well-structured multiagent systems with perfect recall. In this paper we show that a perfect recall synchronous environment over a well-structured environment forms a well-structured environment provided with a special PRS-order. This implies that the model checking problem for the disjunctive fragment of µPLC n is decidable.

Common knowledge in well-structured perfect recall systems

In this paper, a model checking problem for the μPLCn logic of common knowledge and fixpoints in well-structured multiagent systems with perfect recall is investigated. It is shown that a synchronous perfect recall environment generated by a well-structured environment and provided with a special PRS order forms a well-structured environment. This implies that the model checking problem for the disjunctive fragment of μPLCn is decidable.

Two complete finitary sequent calculi for reflexive common knowledge

This paper discusses the use of complete sequent calculi for reflexive common knowledge logic. Description of language and complete infinitary calculus for RCL is presented. Then finitary calculi RCLI and RCLL are introduced and completeness of finitary calculi RCLI and RCLL is proven.

Saturation method for reflexive common knowledge logic

This paper discusses the use of saturation procedure in order to check looping sequents in reflexive common knowledge logic. Traditional approach states that common knowledge operator is defined by some induction-like axiom and requires the use of some looping sequents. The loopcheck-free saturation-like procedure lets us obtain special loopfree sequents.

Syntactic cut-elimination for a fragment of the modal mu-calculus

For some modal fixed point logics, there are deductive systems that enjoy syntactic cut-elimination. An early example is the system in Pliuskevicius (1991) [15] for LTL. More recent examples are the systems by the authors of this paper for the logic of common knowledge (Brünnler and Studer, 2009) [5] and by Hill and Poggiolesi (2010) for PDL[8], which are based on a form of deep inference. These logics can be seen as fragments of the modal mu-calculus. Here we are interested in how far this approach can be pushed in general. To this end, we introduce a nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form. This restricted language includes the logic of common knowledge and PDL. There is also a traditional sequent system for the modal mu-calculus by Jäger et al. (2008) [9], without a syntactic cut-elimination procedure. We embed that system into ours and vice versa, thus establishing cut-elimination also for the shallow system, when only the restricted language is considered.

A specialization of definitions in common knowledge logic

It is known that one of main aims of specializations of derivations in nonclassical logics is the various tools which allow us to simplify the searching of termination of derivations. The traditional techniques used to ensure termination of derivations in various non-classical logics are based on loop-checking. In this paper the reflexive common knowledge logic based on modal logic K2 is considered. For considered logic a sequent calculus with specialized non-logical (loop-type) axioms is presented.

Two Ways to Common Knowledge

It is not clear what a system for evidence-based common knowledge should look like if common knowledge is treated as a greatest fixed point. This paper is a preliminary step towards such a system. We argue that the standard induction rule is not well suited to axiomatize evidence-based common knowledge. As an alternative, we study two different deductive systems for the logic of common knowledge. The first system makes use of an induction axiom whereas the second one is based on co-inductive proof theory. We show the soundness and completeness for both systems.

Common knowledge does not have the Beth property

Common knowledge is an essential notion for coordination among agents. We show that the logic of common knowledge does not have the Beth property and thus it also lacks interpolation. The proof we present is a variant of Maksimova's proof that temporal logics with ‘the next’ do not have the Beth property. Our result also provides an explanation why it is so difficult to find ‘nice’ deductive systems for common knowledge.

Mechanizing common knowledge logic using COQ

This paper presents a formalization in Coq of Common Knowledge Logic and checks its adequacy on case studies. Those studies allow exploring experimentally the proof-theoretic side of Common Knowledge Logic. This work is original in that nobody has considered Higher Order Common Knowledge Logic from the point of view of proofs performed on a proof assistant. As a matter of facts, it is experimental by nature as it tries to draw conclusions from experiments.

Sequent Calculi for Temporal Logics of Common Knowledge and Belief

In this paper we consider two logics: temporal logic of common knowledge and temporal logic of common belief. These logics involve the discrete time linear temporal logic operators “next” and “until”. In addition the first logic contains an indexed set of unary modal operators “agent i knows”, the second one contains an indexed set of unary modal operators “agent i believes”. Also the first logic contains the modality of common knowledge and the second one contains the modality of common belief. For these logics we present sequent calculi with an analytic cut rule. The soundness and completeness for these calculi are proved.

Deduction chains for common knowledge

Deduction chains represent a syntactic and in a certain sense constructive method for proving completeness of a formal system. Given a formula ϕ, the deduction chains of ϕ are built up by systematically decomposing ϕ into its subformulae. In the case where ϕ is a valid formula, the decomposition yields a (usually cut-free) proof of ϕ. If ϕ is not valid, the decomposition produces a countermodel for ϕ. In the current paper, we extend this technique to a semiformal system for the Logic of Common Knowledge. The presence of fixed point constructs in this logic leads to potentially infinite-length deduction chains of a non-valid formula, in which case fairness of decomposition requires special attention. An adequate order of decomposition also plays an important role in the reconstruction of the proof of a valid formula from the set of its deduction chains.

Decision procedures for quantified fragments of reflexive common knowledge logic

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A Map of Common Knowledge Logics

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole.

Common knowledge logic and game logic

AbstractWe show the faithful embedding of common knowledge logic CKL into game logic GL, that is, CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pure strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and its cut-elimination. The faithful embedding theorem is proved for the KD4–type propositional CKL and GL, but it holds for some variants of them.

On the logic of common belief and common knowledge

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge — whether individual or common — is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought — onlymonotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the “infinitary” nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they aredecidable in the logician's sense.

Interpreting logics of knowledge in propositional dynamic logic with converse

Abstract : A natural propositional logic of knowledge, common knowledge, and branching time appropriate for reasoning about distributed systems is presented. This logic may be interpreted in Propositional Dynamic Logic with Converse (PDLC) making the relationship between protocol models and general Kripke models precise and showing that PDLC already suffices for a certain amount of reasoning about knowledge in distributed systems. It follows that satisfiability for propositional logic of branching time remains EMPTIME complete with the addition of any combination of knowledge and common knowledge operators. Finally, the validity or satisfiability of a formula involving two or more participants is not affected by restricting attention to protocol involving just those participants explicitly mentioned in the formula. Keywords: Logic of knowledge; Distributed system; Modal logic; Propositional dynamic logic.