Epistemic logic and logical omniscience: A survey

This survey brings together a collection of epistemic logics and discusses their approaches in alleviating the logical omniscience problem. Of particular note is the logic of implicit and explicit belief. Explicit belief refers to information actively held by an agent, while implicit belief refers to the logical consequence of explicit belief. Ramifications of Levesque's logic include nonstandard epistemic logic and the logics of awareness and local reasoning. Models of nonstandard epistemic logic are defined with respect to nonstandard proportional logic to weaken its semantics. In the logic of awareness, an agent can only believe a concept that it is aware of. Closely related to awareness are S-1 and S-3 epistemic operators which can be used to model skeptical and credulous agents. The logic of local reasoning provides a semantics for representing the fact that agents can have different clusters of beliefs which may contradict each other. Other variations include epistemic structures which are generalizations of the logic of local reasoning and fusion epistemic models which provide an account that agents can combine information conjunctively or disjunctively. Another closely related approach is the logic of explicit propostions which captures the insight that agents can hold beliefs independently without putting them together. © 1997 John Wiley & Sons, Inc.     

Reasoning about knowledge,belief and certainty in hierarchical multi-agent systems

Multi-agent systems (MAS) have received extensive studies in the last decade. However, little attention is paid to investigation on reasoning about logics in MAS with hierarchical structures. This paper proposes a complete quantified temporal KBC (knowledge, belief and certainty) logic and corresponding reasoning in hierarchical multi-agent systems (HMAS). The key point is that internal beliefs and certainty, and external belief and certainty are considered in our logic. The internal beliefs and certainty show every agent is autonomous, while the external belief and certainty indicate the mutual influence of mental attitudes between two different agents on different layers in HMAS. To interpret this logic, we propose four classes of corresponding quantified interpreted systems, and define first-order KBC axiomatisations over HMAS, which are sound and complete with respect to the corresponding semantical classes. Finally, we give a case study to show the advantages in terms of expressiveness of our logic.     

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.      

Multiagent reasoning with probability,time, and beliefs

Any agent interacting with the real world must be able to reason about uncertainty in the world, about the actions that may occur in the world (either due to the agent or those initiated by other agents), about the (probabilistic) beliefs of other agents, and how these (probabilistic) beliefs are changing over time. In this article, we develop a family of logics that a reasoning agent may use to perform successively more sophisticated types of reasoning in such environments. We also characterize different types of agents. Furthermore, we provide a logic that enables a systems designer (who may have populated an environment with a collection of such autonomous agents) to reason about the system of agents as a whole. © 1995 John Wiley & Sons, Inc.     

Requirements for belief models in cooperative dialogue

Models of rationality typically rely on underlying logics that allow simulated agents to entertain beliefs about one another to any depth of nesting. Such models seem to be overly complex when used for belief modelling in environments in which cooperation between agents can be assumed, i.e., most HCI contexts. We examine some existing dialogue systems and find that deeply-nested beliefs are seldom supported, and that where present they appear to be unnecessary except in some situations involving deception.Use of nested beliefs is associated with nested reasoning (i.e., reasoning about other agents' reasoning). We argue that for cooperative dialogues, representations of individual nested beliefs of the third level (i.e., what A thinks B thinks A thinks B thinks) and beyond are in principle unnecessary unless directly available from the environment, because the corresponding nested reasoning is redundant.Since cooperation sometimes requires that agents reason about what is mutually believed, we propose a representation in which the second and all subsequent nesting levels are merged into a single category. In situations affording individual deeply-nested beliefs, such a representation restricts agents to human-like referring and repair strategies, where an unrestricted agent might make an unrealistic and perplexing utterance.     

Multi-Agent Dynamic Logics with Informational Test

This paper investigates a family of logics for reasoning about the dynamic activities and informational attitudes of agents, namely the agents' beliefs and knowledge. The logics are based on a new formalisation and semantics of the test operator of propositional dynamic logic and a representation of actions which distinguishes abstract actions from concrete actions. The new test operator, called informational test, can be used to formalise the beliefs and knowledge of particular agents as dynamic modalities. This approach is consistent with the formalisation of the agents' beliefs and knowledge as K(D)45 and S5 modalities. Properties concerning informativeness, truthfulness and preservation of beliefs are proved for a derivative of the informational test operator. It is shown that common belief and common knowledge can be expressed in the considered logics. This means, the logics are more expressive than propositional dynamic logic with an extra modality for belief or knowledge. The logics remain decidable and belong to 2EXPTIME. Versions of the considered logics express natural additional properties of beliefs or knowledge and interaction of beliefs or knowledge with actions. It is shown that a simulation of PDL can be constructed in one of these extensions.     

Active logic semantics for a single agent in a static world

For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general (and of beliefs about the current time in particular), very tight controls on what can be derived from direct contradictions ($P\mathrm{\&}\neg P$), and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Furthermore, inspired by the notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons with it as if it were consistent), we introduce an “apperception function” that represents an agent's limited awareness of its own beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules remain sound in our new sense. However, not everything that is classically sound remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active logic not everything follows from a contradiction.     

Reasoning about agent deliberation

We present a family of sound and complete logics for reasoning about deliberation strategies for SimpleAPL programs. SimpleAPL is a fragment of the agent programming language 3APL designed for the implementation of cognitive agents with beliefs, goals and plans. The logics are variants of PDL, and allow us to prove safety and liveness properties of SimpleAPL agent programs under different deliberation strategies. We show how to axiomatise different deliberation strategies for SimpleAPL programs, and, for each strategy we prove a correspondence between the operational semantics of SimpleAPL and the models of the corresponding logic. We illustrate the utility of our approach with an example in which we show how to verify correctness properties for a simple agent program under different deliberation strategies.     

Nonmonotonic logic and temporal projection

Nonmonotonic formal systems have been proposed as an extension to classical first-order logic that will capture the process of human “default reasoning” or “plausible inference” through their inference mechanisms, just as modus ponens provides a model for deductive reasoning. But although the technical properties of these logics have been studied in detail and many examples of human default reasoning have been identified, for the most part these logics have not actually been applied to practical problems to see whether they produce the expected results.We provide axioms for a simple problem in temporal reasoning which has long been identified as a case of default reasoning, thus presumably amenable to representation in nonmonotonic logic. Upon examining the resulting nonmonotonic theories, however, we find that the inferences permitted by the logics are not those we had intended when we wrote the axioms, and in fact are much weaker. This problem is shown to be independent of the logic used; nor does it depend on any particular temporal representation. Upon analyzing the failure we find that the nonmonotonic logics we considered are inherently incapable of representing this kind of default reasoning.The first part of the paper is an expanded version of one that appeared in the 1986 AAAI proceedings. The second part reports on several responses to our result that have appeared since the original paper was published.     

The dynamics of default reasoning

In this paper we study default reasoning from a dynamic, agent-oriented, semantics-based point of view. In a formal framework used to specify and to reason about rational agents, we introduce actions that model the (attempted) jumping to conclusions that is a fundamental part of reasoning by default. Application of such an action consists of three parts. First it is checked whether the formula that the agent tries to jump to is a default, thereafter it is checked whether the default formula can consistently be incorporated by the agent, and if this is the case the formula is included in the agent's beliefs. As for all actions in our framework, we define the ability and opportunity of agents to apply these actions, and the states of affairs following application. To formalise formulae being defaults, we introduce the modality of common possibility. This modality is related to, but not reducible to, the notions of common knowledge and ‘everybody knows’-knowledge. To model the qualitative difference that exists between hard, factual knowledge and beliefs derived by default, we employ different modalities to represent these concepts, thus combining knowledge, beliefs, and defaults in one framework. Based on the concepts used to model the default reasoning of agents, we look into the dynamics of the supernormal fragment of default logic. We show in particular that by sequences of jumps to conclusions agents can end up with extensions in the sense of default logic of their belief.     

The relationship between knowledge,belief, and certainty

We consider the relation between knowledge and certainty, where a fact isknown if it is true at all worlds an agent considers possible and iscertain if it holds with probability 1. We identify certainty with belief, interpreted probabilistically. We show that if we assume one fixed probability assignment, then the logic KD45, which has been identified as perhaps the most appropriate for belief, provides a complete axiomatization for reasoning about certainty. Just as an agent may believe a fact although is false, he may be certain that a fact is true although is false. However, it is easy to see that an agent can have such false (probabilistic) beliefs only at a set of worlds of probability 0. If we restrict attention to structures where all worlds have positive probability, then S5 provides a complete axiomatization. If we consider a more general setting, where there might be a different probability assignment at each world, then by placing appropriate conditions on thesupport of the probability function (the set of worlds which have non-zero probability), we can capture many other well-known modal logics, such as T and S4. Finally, we considerMiller's principle, a well-known principle relating higher-order probabilities to lower-order probabilities, and show that in a precise sense KD45 characterizes certainty in those structures satisfying Miller's principle.     

Minimal Belief and Negation as Failure in Multi-Agent Systems

We propose an epistemic, nonmonotonic approach to the formalization of knowledge in a multi-agent setting. From the technical viewpoint, a family of nonmonotonic logics, based on Lifschitz's modal logic of minimal belief and negation as failure, is proposed, which allows for formalizing an agent which is able to reason about both its own knowledge and other agents' knowledge and ignorance. We define a reasoning method for such a logic and characterize the computational complexity of the major reasoning tasks in this formalism. From the practical perspective, we argue that our logical framework is well-suited for representing situations in which an agent cooperates in a team, and each agent is able to communicate his knowledge to other agents in the team. In such a case, in many situations the agent needs nonmonotonic abilities, in order to reason about such a situation based on his own knowledge and the other agents' knowledge and ignorance. Finally, we show the effectiveness of our framework in the robotic soccer application domain.     

A FRAMEWORK FOR LOGICS OF EXPLICIT BELIEF

The epistemic notions of knowledge and belief have most commonly been modeled by means of possible worlds semantics. In such approaches an agent knows (or believes) all logical consequences of its beliefs. Consequently, several approaches have been proposed to model systems of explicit belief, more suited to modeling finite agents or computers. In this paper a general framework is developed for the specification of logics of explicit belief. A generalization of possible worlds, called situations, is adopted. However the notion of an accessibility relation is not employed; instead a sentence is believed if the explicit proposition expressed by the sentence appears among a set of propositions associated with an agent at a situation. Since explicit propositions may be taken as corresponding to "belief contexts" or "frames of mind," the framework also provides a setting for investigating such approaches to belief. The approach provides a uniform and flexible basis from which various issues of explicit belief may be addressed and from which systems may be contrasted and compared. A family of logics is developed using this framework, which extends previous approaches and addresses issues raised by these earlier approaches. The more interesting of these logics are tractable, in that determining if a belief follows from a set of beliefs, given certain assumptions, can be accomplished in polynomial time.     

Uniform semantic treatment of default and autoepistemic logics

We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise one-to-one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.     

Towards a Model of Learning through Communication

Communication is an interactive, complex, structured process involving agents that are capable of drawing conclusions from the information they have available about some real-life situations. Such situations are generally characterized as being imperfect. In this paper, we aim to address learning from the perspective of the communication between agents. To learn a collection of propositions concerning some situation is to incorporate it within one's knowledge about that situation. That is, the key factor in this activity is for the goal agent, where agents may switch role if appropriate, to integrate the information offered with what it already knows. This may require a process of belief revision, which suggests that the process of incorporation of new information should be modeled nonmonotonically. We shall employ for reasoning a three-valued based nonmonotonic logic that formalizes some aspects of revisable reasoning and it is accessible to implementation. The logic is sound and complete. A theorem-prover of the logic has successfully been implemented. Received 3 August 1999 / Revised 17 April 2000 / Accepted 6 May 2000     

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.

Logics and Multi-agents: towards a new symbolic model of cognition

The last edition of CLIMA, held in 2001 in Paphos (Cyprus) ended with a panel session on the role of Computational Logic (CL) in Multi-Agent Systems (MAS).Two dimensions in MAS development were singled out and discussed: on the one hand reactivity vs. rationality, and on the other hand individuals vs. societies. Most of the points discussed aimed at justifying and motivating the application of CL techniques to MAS development: should be logics used to implement the individuals, or the society, or both? should be logics used to model the reactive part, or the pro-active part, or both? what do we want to achieve in terms of properties, openness to integration, etc.?A most intuitive reply to these questions could be that logic should be used for what logic is good at. For instance, logic programming-based techniques such as abductive and inductive logic programming seem suitable for modelling agent hypothetical reasoning and adaptability. Modal logic operators such as those adopted by a BDI agent model [3] could be a powerful and synthetic way to describe the agent behaviour and to put it into relationship with the other agents in a society. Model checking-based techniques can be applied to the verification of agent systems. A combination of multiple approaches, like modal and temporal logics, or abduction and induction in a logic programming framework, could be the key to achieve a more comprehensive agent and agent system architecture. But in this case, to determine which properties of the chosen combinations hold is not an easy task.At the time of this new edition of CLIMA, while the debate about the role of CL in MAS is still open, from within the CL community we are witnessing a growth of interest for Multi-Agent Systems considered per se as an interesting cognitive model. This is due to many reasons, among which, we would say, the need to put “abstract” reasoning in the context of a “concrete” environment, and to use logic not only to solve problems in a virtual world, but in a real arena. The multi-agent metaphor of intelligent individuals that are situated into dynamic and unpredictable environments and that can interact with each other by updating their beliefs, can be regarded then as the basis for a new symbolic model of cognition.     

Epistemic Logic for Rule-Based Agents

The logical omniscience problem, whereby standard models of epistemic logic treat an agent as believing all consequences of its beliefs and knowing whatever follows from what else it knows, has received plenty of attention in the literature. But many attempted solutions focus on a fairly narrow specification of the problem: avoiding the closure of belief or knowledge, rather than showing how the proposed logic is of philosophical interest or of use in computer science or artificial intelligence. Sentential epistemic logics, as opposed to traditional possible worlds approaches, do not suffer from the problems of logical omniscience but are often thought to lack interesting epistemic properties. In this paper, I focus on the case of rule-based agents, which play a key role in contemporary AI research but have been neglected in the logical literature. I develop a framework for modelling monotonic, nonmonotonic and introspective rule-based reasoners which have limited cognitive resources and prove that the resulting models have a number of interesting properties. An axiomatization of the resulting logic is given, together with completeness, decidability and complexity results.