Local Knowledge Assertions in a Changing World

When the state of the world changes due to an action performed by an agent in a multiagent system, the views of other agents, and hence their knowledge, remain unaffected. We describe such situations using a simple modal logic. Traditionally, modal logics of knowledge are interpreted over global states of the multi-agent system. When actions are incorporated into such logics, it leads to high undecidability, whereas, if we see the assertions as made by the agents in the system at their local states, we get a decidable logic, for which we also provide a complete axiomatization. Interestingly, when we consider a corresponding local temporal logic of knowledge and (linear) time, the knowledge modality represents a 'necessarily now' modality of present tense. 1 I n t r o d u c t i o n In reasoning about knowledge in multi-agent systems, whether it is in the context of artificial intelligence, or distributed computing, or economic theory, it is usual to consider dynamic state spaces alongwith the information partition of agents. Typically, dynamic state spaces contain states indexed by time instants. In the theory of distributed systems, one considers the (infinite) runs of a distributed system. Knowledge formulas are interpreted on these runs, or on time-indexed states. Such logics and models have been studied extensively ([HF 89], [HM 90]). There seems to have been less work on capturing changes in state due to actions of agents in the system. While such state changes are usually present in the frames studied, the logical machinery rarely studies how agents' knowledge changes due to actions and conversely how actions cause changes in knowledge states. However, we feel that this is important , because, when the system changes state due to an action by a group of agents in the system, the agents in that group typically know the effect of the action, the knowledge of the rest remains unaffected, and the acting agents know this too. Two of the TARK '94 papers study such systems : the environments of [Me 94] and the knowledge transition systems of [KR 94]. However, the logic studied in the former does not include the effect of actions, and the latter concentrates on the temporal structures defined by these systems. Of particular interest is the case when the set of states is .finite. Then we can consider the ( important) questions of efficiency: how can we compute the knowledge of agents, and how can an agent maintain and update its knowledge ? Once we define an appropriate logic, we can study these questions in terms of the model checking problem for that logic ([HV 91]). Briefly, knowledge transition systems (KTSs) are state transition systems enriched with n equivalence relations (where n is the number of agents in the system). These equivalence relations, as usual, define which states an agent can distinguish, and interact with transitions in the following manner : when an agent i makes a transition s ---+ s', for all other agents j , s and s' are j-equivalent. Further, both the enabling of an action as well as the resulting state are determined only by the views of agents participating in that action. This ensures that when an i-action as well as a j -act ion are both enabled at a state, they can be "commuted" and the resulting state is the same. (See the next section for more detail.) [R94] studied a logic of knowledge interpreted over KTSs. Since propositional dynamic logics (PDL) are suited to reason about state changes, Ki operators were added to a PDLlike logic. Thus the logic had propositional variables, boolean connectives, and modalities for next-state, eventuality and knowledge. Unfortunately, the logic turned out to be highly undecidable, indeed not recursively axiomatizable. This was due to the existence of "commuting" diagrams forced by the frames. (In itself this is no surprise; see [LPRT95] for a variety of negative results of this kind. However, the interesting point is that a seemingly weak next-state modality unindexed by agents or actions can force undecidability.) Can we find a decidable logic of knowledge to reason about KTSs ? It turns out that the solution is to interpret formulas at local states of agents rather than on global states of the system, and build up global formulas from local ones. This requires knowledge assertions at local states. This is a departure from the s tandard fashion, where knowledge is ascribed to agents based on global observations. However, we must hasten to point out that we will continue to use implicit knowledge of agents r a the r than locally computed explicit knowledge, say, as studied by [HMV 94]. Incorporating their notion of algorithmic knowledge into this framework seems to be an interesting question for future study. Thus, in this paper we will be studying a propositional modal logic of knowledge interpreted on local states of KTSs. However, we will confine our at tention to a subclass of KTSs, those in which all communication is only by synchronous handshaking. This is for several reasons : the problem is already quite non-trivial; the model is closely related to concurrent au tomata ([Zie 87]) studied in the literature; the logic is closely related to the partial order based extensions of temporal logics proposed recently ([T94]). 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