Logic Of Belief Research Articles

Research on Concurrent Dynamic Logic of Knowledge, Belief and Certainty for Multi-Agent Systems

This paper extends the logic of Knowledge,Belief and Certainty(KBC) for one agent to that for multi-agent systems.It presents a Concurrent Dynamic logic of Knowledge,Belief and Certainty for MAS,which is called CDKBC logic,with a good combination between logic of KBC and action modalities that have concurrent and dynamic properties in multi-agent systems.Furthermore,a CDKBC model is given for interpreting this logic.The authors construct a CDKBC proof system for the logic and show the proof system is sound and complete,where the validity problem for the system will be proved to be EXPTIME-complete and give an application of the CDKBC logic.

SIGACT news logic column 21

In this issue, Kramer, Gore, and Okamoto investigate how to use a logic of belief and knowledge to define the concepts of weak and strong trust relations as well as trust domains, and apply these definitions to security applications, such as trusted-third parties, the Web of Trust, public-key infrastructures, and identity-based cryptography. I am always looking for contributions. If you have any suggestion concerning the content of the Logic Column, or if you would like to contribute by writing a column yourself, feel free to get in touch with me.

Behind the Scenes of K&CK

I was honored to receive the 2009 Dijkstra Prize, joint with Joe Halpern. The circumstances under which the prize was awarded did not allow for more than a brief response. I have since wondered what I might have said in a longer address. I concluded that, rather than discuss the future of the field, I would try to tell the story behind the award-winning paper. This note is a draft of this story, from my very biased and subjective point of view. It is presented here in the style of an added clip in a new DVD of an old movie providing footage of some behind the scenes developments, bloopers etc. In contrast to my other papers, the KC it does not set out to improve, extend, or refine pre-existing work. This is why I think that tracing its origins is of interest. The story of how it came to be touches on some of the issues and topics covered in the paper. Moreover, the published versions of the paper do not properly acknowledge some of the people who have influenced our work on the paper. Perhaps this note can compensate. As a PhD student at Stanford, I attended a series of lectures on logics of programs given by Joe Halpern, who had just moved west to IBM Almaden Research Center, following his post-doc at MIT. I was looking for an advisor, and Joe was likewise seeking a first graduate student. We agreed to try working together toward my PhD, and decided to start by considering logics of knowledge and belief. We knew, based on the work of Kripke and Hintikka in the late Fifties and early Sixties, that these logics are technically similar to the Temporal and Dynamic logics that Joe had been involved with. One new element in our approach was going to be the study of logics with many “knowers,” whereas most of the classical treatment of knowledge and belief focused on the epistemics of a single agent. In a logic for many knowers one could make (and make sense of) statements about what one agent, say Alice, knows about the world and about what another agent (Bob) may know, or be ignorant of. Indeed, knowledge could be further nested, and statements could involve any number of agents. We started looking at the literature on modal logics of programs trying to see how they may apply to logics of knowledge. We also considered other logical issues such as characterizing ignorance by way of what it might mean for an agent to “know only” some set of facts and nothing more. I spent the summer of 1983 doing a student internship at IBM Almaden, working with Joe. Danny Dolev was there, and heard that we were working on logics of knowledge. He told us about a talk he had attended by Robert Aumann at the Hebrew University, who described the “cheating wives” puzzle, and explained that Common Knowledge played an important role. (Roughly at the same time, Joe’s colleague at IBM Shel

A logic of trust and reputation

The aim of this paper is to present a logical framework in which the concepts of trust and reputation can be formally characterized and their properties studied. We start from the definition of trust proposed by Castelfranchi & Falcone (C&F). We formalize this definition in a logic of time, action, beliefs and choices. Then, we provide a refinement of C&F’s definition by distinguishing two general types of trust: occurrent trust and dispositional trust. In the second part of the paper we present a definition of reputation that is structurally similar to the definition of trust but moves the basic concept of belief to a collective dimension of group belief.

First-order belief and paraconsistency

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language L QB by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since φ ∧ ¬φ does not “explode” with arbitrary syntactic consequences.

A concurrent dynamic logic of knowledge, belief and certainty for multi-agent systems

This paper extends the logic of knowledge, belief and certainty from one agent to multi-agent systems, and gives a good combination between logic of knowledge, belief, certainty in multi-agent systems and actions that have concurrent and dynamic properties. Based on it, we present a concurrent dynamic logic of knowledge, belief and certainty for MAS, which is called CDKBC logic. Furthermore, a CDKBC model is given for interpreting this logic. We construct a CDKBC proof system for the logic and show that the proof system is sound and complete, and prove that the validity problem for the system is EXPTIME-complete.

Probability Logic of Finitely Additive Beliefs

Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage's postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ+ that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ+ iff it is satisfied in a finitely additive type space. Although we can characterize Σ+-theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics.

Multiagent systems: algorithmic, game-theoretic, and logical foundations

This exciting and pioneering new overview of multiagent systems, which are online systems composed of multiple interacting intelligent agents, i.e., online trading, offers a newly seen computer science perspective on multiagent systems, while integrating ideas from operations research, game theory, economics, logic, and even philosophy and linguistics. The authors emphasize foundations to create a broad and rigorous treatment of their subject, with thorough presentations of distributed problem solving, game theory, multiagent communication and learning, social choice, mechanism design, auctions, cooperative game theory, and modal logics of knowledge and belief. For each topic, basic concepts are introduced, examples are given, proofs of key results are offered, and algorithmic considerations are examined. An appendix covers background material in probability theory, classical logic, Markov decision processes and mathematical programming. Written by two of the leading researchers of this engaging field, this book will surely serve as THE reference for researchers in the fastest-growing area of computer science, and be used as a text for advanced undergraduate or graduate courses.

Keep ‘hoping’ for rationality: a solution to the backward induction paradox

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.

Analysis of Authentication Protocols in Agent-Based Systems Using Labeled Tableaux

The study of multiagent systems (MASs) focuses on systems in which many intelligent agents interact with each other using communication protocols. For example, an authentication protocol is used to verify and authorize agents acting on behalf of users to protect restricted data and information. After authentication, two agents should be entitled to believe that they are communicating with each other and not with intruders. For specifying and reasoning about the security properties of authentication protocols, many researchers have proposed the use of belief logics. Since authentication protocols are designed to operate in dynamic environments, it is important to model the evolution of authentication systems through time in a systematic way. We advocate the systematic combinations of logics of beliefs and time for modeling and reasoning about evolving agent beliefs in MASs. In particular, we use a temporal belief logic called TML (+) for establishing trust theories for authentication systems and also propose a labeled tableau system for this logic. To illustrate the capabilities of TML (+), we present trust theories for several well-known authentication protocols, namely, the Lowe modified wide-mouthed frog protocol, the amended Needham-Schroeder symmetric key protocol, and Kerberos. We also show how to verify certain security properties of those protocols. With the logic TML (+) and its associated modal tableaux, we are able to reason about and verify authentication systems operating in dynamic environments.

Agent Communication for Dynamic Belief Update

Thus far, various formalizations of rational / logical agent model have been proposed. In this paper, we include the notion of communication channel and belief modality into update logic, and introduce Belief Update Logic (BUL). First, we discuss that how we can reformalize the inform action of FIPA-ACL into communication channel, which represents a connection between agents. Thus, our agents can send a message only when they believe, and also there actually is, a channel between him / her and a receiver. Then, we present a static belief logic (BL) and show its soundness and completeness. Next, we develop the logic to BUL, which can update Kripke model by the inform action; in which we show that in the updated model the belief operator also satisfies K45. Thereafter, we show that every sentence in BUL can be translated into BL; thus, we can contend that BUL is also sound and complete. Furthermore, we discuss the features of CUL, including the case of inconsistent information, as well as channel transmission. Finally, we summarize our contribution and discuss some future issues.

Probabilistic dynamic belief revision

We investigate the discrete (finite) case of the Popper-Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of "knowledge". We develop a probabilistic version of this concept ("degree of safety") and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic "action models" and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws.

Justification logics, logics of knowledge, and conservativity

Several justification logics have been created, starting with the logic LP, (Artemov, Bull Symbolic Logic 7(1):1---36, 2001). These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.

Belief–logic conflict resolution in syllogistic reasoning: Inspection-time evidence for a parallel-process model

An experiment is reported examining dual-process models of belief bias in syllogistic reasoning using a problem complexity manipulation and an inspection-time method to monitor processing latencies for premises and conclusions. Endorsement rates indicated increased belief bias on complex problems, a finding that runs counter to the “belief-first” selective scrutiny model, but which is consistent with other theories, including “reasoning-first” and “parallel-process” models. Inspection-time data revealed a number of effects that, again, arbitrated against the selective scrutiny model. The most striking inspection-time result was an interaction between logic and belief on premise-processing times, whereby belief – logic conflict problems promoted increased latencies relative to non-conflict problems. This finding challenges belief-first and reasoning-first models, but is directly predicted by parallel-process models, which assume that the outputs of simultaneous heuristic and analytic processing streams lead to an awareness of belief – logic conflicts than then require time-consuming resolution.

Modal tableaux for verifying stream authentication protocols

To develop theories to specify and reason about various aspects of multi-agent systems, many researchers have proposed the use of modal logics such as belief logics, logics of knowledge, and logics of norms. As multi-agent systems operate in dynamic environments, there is also a need to model the evolution of multi-agent systems through time. In order to introduce a temporal dimension to a belief logic, we combine it with a linear-time temporal logic using a powerful technique called fibring for combining logics. We describe a labelled modal tableaux system for the resulting fibred belief logic (FL) which can be used to automatically verify correctness of inter-agent stream authentication protocols. With the resulting fibred belief logic and its associated modal tableaux, one is able to build theories of trust for the description of, and reasoning about, multi-agent systems operating in dynamic environments.

Formalising theories of trust for authentication protocols

This paper discusses a formal approach for establishing theories of trust for authentication systems which can be used to reason about how agent beliefs evolve through time. The goal of an authentication system is to verify and authorise users in order to protect restricted data and information, so trust is a critical issue for authentication systems. After authentication, two principals (people, computers, services) should be entitled to believe that they are communicating with each other and not with intruders. So, it is important to express such beliefs precisely and to capture the reasoning that leads to them. In this paper, we focus on analysis of agent beliefs in dynamic environments using a temporalised belief logic, obtained by adding a temporal logic onto a belief logic. Working through a well-known authentication protocol, namely Kerberos, we discuss how to express principal beliefs involved in authentication protocols and the evolution of those beliefs based on a series of observations of agents as a consequence of communication. Our approach could be used for designing, verifying and implementing authentication protocols.

Specifying and reasoning about uncertain agents

Logical formalisation of agent behaviour is desirable, not only in order to provide a clear semantics of agent-based systems, but also to provide the foundation for sophisticated reasoning techniques to be used on, and by, the agents themselves. The possible worlds semantics offered by modal logic has proved to be a successful framework in which to model mental attitudes of agents such as beliefs, desires and intentions. The most popular choices for modeling the informational attitudes involves annotating the agent with an S5-like logic for knowledge, or a KD45-like logic for belief. However, using these logics in their standard form, an agent cannot distinguish situations in which the evidence for a certain fact is ‘equally distributed’ over its alternatives, from situations in which there is only one, almost negligible, counterexample to a ‘fact’. Probabilistic modal logics are a way to address this, but they easily end up being both computationally and conceptually complex, for example often lacking the property of compactness. In this paper, we propose a probabilistic modal logic P F KD45, in which the probabilities of the possible worlds range over a finite domain of values, while still allowing the agent to reason about infinitely many options. In this way, the logic remains compact, implying that the agent still has to consider only finitely many possibilities for probability distributions during a reasoning task. We demonstrate a sound, compact and complete axiomatization for P F KD45 and show that it has several appealing features. Then, we discuss an implemented decision procedure for the logic, and provide a small example.

A Grim Semantics For Logics of Belief

Patrick Grim has presented arguments supporting the intuition that any notion of a totality of truths is incoherent. We suggest a natural semantics for various logics of belief which reflect Grim’s intuition. The semantics is a topological semantics, and we suggest that the \(T_{1} \) condition can be interpreted to reflect Grim’s intuition. Beyond this, we present a natural canonical topological model for K4 and KD4.

WITHDRAWN: Non-monotonic modal logic of belief

WITHDRAWN: Non-monotonic modal logic of belief

Sequent Calculi for Branching Time Temporal Logics of Knowledge and Belief

In this paper we consider branching time temporal logics of knowledge and belief. These logics involve the discrete time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. The latter operator is interpreted with respect to a version of the bundle semantics. In addition the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”) and it contains the modality of common knowledge (belief). 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.