A Knowledge-Level Account of Abduction

In this paper, we consider a new definition of abduction that makes it depend on an under­ lying formal model of belief. In particular, dif­ ferent models of belief will give rise to differ­ ent forms of abductive reasoning. Based on this definition, we then prove three main the­ orems: first, that when belief is closed under logical implication, the corresponding form of abduction is precisely what is performed by the ATMS as characterized by Reiter and de Kleer; second, that with the more limited "ex­ plicit" belief defined by Levesque, the required abduction is computationally tractable in cer­ tain cases where the ATMS is not; and finally, that something is believed in the implicit sense iff repeatedly applying a limited abduction op­ erator eventually yields something that is be­ lieved in the explicit sense. This last result re­ lates deduction and abduction as well as limited and unlimited reasoning all within the context of a logic of belief. 1 Introduction Using the terminology of C. S. Peirce, given sentences α, β, and , there are three operations one can consider: from α and one might deduce β; from α and β, one might induce 1 and from β and , one might abduce α Of course, characterizing precisely what should be deduced, induced, or abduced in various circumstances is quite another matter, and the last of these is the subject of this paper. Abduction can be thought of as a form of hypothet­ ical reasoning. To ask what can be abduced from β is to ask for an α which, in conjunction with background knowledge, 2 is sufficient to account for β. When α and β are about the physical world, this normally involves 1 More likely, one would want to induce from instances of α and β. 2 The distinction between knowledge and belief is not im­ portant here, and we will use the terms interchangeably. finding a cause α for an observed effect β. For instance, β might say that a symptom of some sort is observed and α might say that a disease is present. We often say in this case, that α explains β. But not all abduction is concerned with cause and effect. If we happen to known that Marc is 3 or 4 years old, the fact that he is not yet 4 does not explain his being 3, although it does imply …