A Logic for Reasoning about Agents with Finite Explicit Knowledge

It is widely accepted that there is a need for theories describing what agents explicitly know and can act upon, as opposed to what they know only implicitly. Most approaches to modeling explicit knowledge describe rules governing closure conditions on explicit knowledge. Our approach is different; we model static knowledge at a point in time and assume that knowledge is represented syntactically rather than as propositions. Another concept which has been suggested is called only knowing, and describes all that an agent knows. We argue that a proper logic for reasoning about knowledge must combine this concept with the concept of explicit knowledge, and present such a logic. The logic is weakly complete with respect to a simple and natural semantics; we present a characterization of sets of formulae, called finitary theories, for which we also have completeness when used as premises and thus can be used to extend the logic with e.g. epistemic properties. Incorporating axioms which describe purely epistemic properties corresponds to simply removing states from the set of legal epistemic states for each agent. An interesting property of our semantic model, if we require that knowledge must be true, is an incompleteness of knowledge: it is impossible to know “all I know”. 1 Finite Explicit Knowledge It is widely accepted that there is a need for theories describing what agents actually or explicitly know and can act upon, as opposed to what they implicitly know, i.e. what follows logically from their explicit knowledge. This distinction is closely related to the logical omniscience problem [9]; in classical modal epistemic logics agents know all the logical consequences of their knowledge – a description of implicit knowledge. Levesque has proposed to formalize the rules governing explicit knowledge [13], and his approach has been extended by or inspired others [10, 4]. Our model of explicit knowledge in this paper takes a different approach. First, instead of describing closure conditions on explicit knowledge, we assume that explicit knowledge has been obtained, and we construct a logic for reasoning about static explicit knowledge in a group of agents. A framework for reasoning about static knowledge is useful for analyzing the knowledge in a group of agents at an instant in time (or a time span when no epistemic changes are made), for example between computing deductions. Second, we consider ascribing knowledge of propositions to agents in multiagent systems to be unrealistic. Instead, we assume that the agents posses and process syntactical objects. (Of course, neither of these ideas are new; we briefly discuss related work in the last section). The need for reasoning about explicit knowledge is illustrated by the following example. Consider an agent sending an encrypted version of a secret message to agent through a public channel, that it is possible to decipher the message using two (large) prime numbers and , and that the product is publicly known. Particularly, and are known by agent . If we use the “implicit” knowledge concept, the sentence “agent knows ” (1) could be derived from the sentences “agent knows ” (2) “agent knows ” (3) assuming agent knows the rules of arithmetic, since the values of and follows logically from the value of . However, even if we use the “explicit” knowledge concept, the sentence “agent does not know ” (4) does not follow logically from sentences (2) and (3). Information about what the agent explicitly knows, does not make us able to deduce what he (explicitly) does not know. For example, agent could have gotten to know even before the message was sent. But if we add the sentence “sentences (2) and (3) describe all that is known by ” (5) then sentence 4 follows from 2, 3 and 5. The concept of only knowing has been suggested to capture “all an agent knows”, but most approaches are in the context of implicit knowledge. As illustrated by our example, using the “implicit” knowledge concept does not allow us to deduce sentence 4 from 2, 3 and 5, because is implicitly included in what is “only known” since it follows logically from other known facts. Thus, a proper logic for reasoning about knowledge combines reasoning about explicit knowledge with the concept of only knowing. We construct a logic for explicit knowledge. We are not concerned with how the agents obtain their explicit knowledge, nor in the relation of this knowledge to reality (an agent can e.g. know false facts or contradictions). Particularly, we do not in general assume anything about the completeness or consistency of the deliberation mechanism, i.e. we do not assume any closure condition of explicit knowledge. We want to capture the concept “all an agent explicitly knows”. Everything an agent implicitly knows may be possible to describe by one single formula ; the agent implicitly knows everything that is logically entailed by the formula. If the operator denotes the concept of implicit knowledge, we can then express the agent’s knowledge by . Everything an agent explicitly knows, however, cannot be described by a single formula (if it knows more than one formula), because there is no closure condition for explicit knowledge. If the operator denotes explicit knowledge, the formulae and say that the agent knows and , but does not say anything about e.g. whether the agent knows the formula . We therefore need to express knowledge about sets of formulae. We describe the fact that the formulae are all that is explicitly known by agent ! by the formula