Integrating a modal logic of knowledge into terminological logics

If we want of group of autonomous agents to act and to cooperate in a world, each of them needs knowledge about this world, about the knowledge of other agents, and about his own knowledge. To describe such knowledge we introduce the language ALC_{K} which extends the concept language ALC by a new operator Box i. Thereby, Box i,phi is to be read as "agent i knows varphi". This knowledge operator is interpreted in terms of possible worlds. That means, besides the real world, agents can imagine a number of other worlds to be possible. An agent is then said to know a fact phi if phi is true in all worlds he considers possible. In this paper we use an axiomatization of the knowledge operator which has been proposed by Moore. Thereby, knowledge of agents is interpreted such that (i) agents are able to reason on the basis of their knowledge, (ii) anything that is known by an agent is true, and (iii) if an agent knows something then he knows that he knows it. We will give tableaux-based algorithms for deciding whether a set of ALC_{K} sentences is satisfiable, and whether such a set entails a given ALC_{K} sentence

Representing belief in multi-agent worlds via terminological logics

In multi-agent systems a group of autonomous intelligent systems, called agents, acts and cooperates in a world in order to achieve certain goals. Such systems are in general assumed to have no central control structure and hence each agent can only perform actions that are based on his local knowledge and on his local beliefs. In the literature knowledge of agents is mostly represented under the view that knowledge is true belief. On the other hand, if agents are acting in a (real) world their knowledge often is obtained by perception and communication, and hence typically is not true. Thus, the use of belief - where agents may have false beliefs - seems more appropriate than the use of knowledge in multi-agent systems. Terminological logics provide a well-investigated and decidable fragment of first-order logics that is much more expressive than propositional logic and well suited to describe a world agents are acting in. However, knowledge or belief of agents can only be represented in a very limited way. In this paper we investigate how terminological logics can be extended in such a way that belief of agents can be represented in an adequate manner. We therefore exemplarily extend the concept language mathcal{ALC} by a modal operator square, which is indexed by agents. Thereby, square_{i}varphi represents the fact "agent i believes varphi';. This belief operator will be interpreted in terms of possible worlds using the well-known modal logic KD45. This extended language mathcal{ALC_{B}} provides a uniform formalism to describe both, a world agents are acting in and the beliefs agents have about this world and about their own and other agents' beliefs. Thus, it can be seen as a two-dimensional extension of mathcal{ALC} which allows both, reasoning about objective facts that hold in the world and reasoning on the level of possible worlds. We will give sound and complete algorithms to check consistency of the represented beliefs and to decide whether an mathcal{ALC_{B}}-sentence is logically entailed by the beliefs of agents. Hence, when acting in a world agents can use beliefs which are explicitly represented as well as implicit beliefs that are entailed by their knowledge base