On the projection problem in active knowledge bases with incomplete information

Abstract

The problem of projection has been identified as a fundamental reasoning concern in dynamical domains, where we are to determine whether or not some conditions will hold after a sequence of actions has been performed starting in some initial state. Solving the problem requires, at the very least, effectively reasoning about how actions transform the world, and inferring the logical consequences of the initial knowledge base (KB). For various reasons, tractability one of them, applications often make the closed-world assumption, thereby limiting the scope of these systems for the real world. In this thesis, using the language of the situation calculus, we investigate the computational properties of a number of unsolved reasoning tasks in the context of projection with incomplete information. We first look at inherently incomplete KBs, where the information provided to the agent may not determine every fact about the world. Projection, then, may involve reasoning about what is believed and also, about what is not believed. We then look at physical agents with unreliable hardware, as a result of which actions lead to certain kinds of incomplete knowledge. Intuitively, beliefs should be (periodically) synchronized with this noise. Finally, we consider the presence of other agents in the environment, whose beliefs may differ arbitrarily, and the formalism should incorporate what others sense and learn during actions. To enable a precise mathematical treatment of incomplete KBs, we appeal to a seminal proposal by Levesque, called only knowing. Building on existing work, we investigate projection wrt extensions to the situation calculus for only knowing, noisy hardware and multiple agents. Our central contribution will be to show that, in spite of the additional expressivity, reasoning about knowledge and action reduces to nonepistemic non-dynamic reasoning about the initial KB. More precisely, we show that when the initial KB is an arbitrary first-order theory, we are able to identify conditions under which projection can be solved by progressing the KB to a sentence reflecting the changes due to actions that have already occurred. Moreover, when effectors are unreliable, we allow the system to maintain probabilistic beliefs and then show how projection can be addressed by means of updating these beliefs. Finally, when there are many agents in the picture, we show that queries about the future can be resolved by regressing the query backwards to a formula about the initial KB. Only knowing comes with a significant result that allows us to reduce queries about knowledge to first-order theorem-proving tasks, which is then made use of when solving projection.