In this paper we present an extension of the spectrum of logical de nitions of model-based diagnosis introduced in (Console &Torasso 1991b). The extended spectrum considers the case of temporal model-based diagnosis and generalizes the logical characteriza

where TOBSE is the whole set of observations) moreover also all the intermediate cases (corresponding to all the other de nitions of TOBSE ) can be considered. The spectrum is extended in this paper also to the temporal dimension in the sense that di erent de nitions of TOBSTE lead to di erent notions of explaining the temporal constraints on the observations. Also in this case a lattice can be singled out: at one extreme case, where TOBSTE=, it is only required that the temporal constraints on the observations are consistent with those in the part of the model involved in the explanation. At the other extreme, where TOBSTE contains all the constraints on the observations, then all the temporal information on the observations must follow from the explanation. Thus de nition 2 points out that there are two orthogonal dimensions in the characterization of the notion of temporal diagnosis: selecting the notion of\explaining" the observations selecting the notion of\temporally explaining" the observations Actually, the two notions are not completely independent in the sense that the choice of TOBSTE depends in some sense of the choice of TOBSE . In fact, suppose that a 62 TOBSE, i.e., the explanation must only be consistent with the observation a and it is not required that a is explained abductively. In this case it is not reasonable to include tempor

al constraints on a in TOBSTE . In fact, it would be strange to impose that although the explanation must not entail (predict) the presence of a it must entail (predict) the temporal location of a, i.e., that a must be true on some interval T respecting some constraints. Thus in this sense the temporal notion of explanation depends on the logical notion of explanation being adopted. The analysis in (Console& Torasso 1991b) shows that the completeness of the model is the criterion to choose abductive over consistency-based diagnosis. In particular, since abduction can be regarded as deduction on a completed theory (Console, Theseider Dupre,& Torasso 1991), its use is recommended when the model of the system to be diagnosed is\complete" (i.e., when all the causes of the observations are included in the model). In particular, it is reasonable (and recommended) to include an observation a in the set of observations to be explained abductively in case all the causes of a are in the model. In other cases, the requirement that the explanation must entail an observation may be too restrictive and may lead to loosing relevant solutions. Similar considerations apply also for the temporal dimension: requiring that the temporal constraints on the observations are entailed is reasonable only in case the temporal constraints in the model are precise (or, better, at least as precise as those that may occur in the observations) if this is not the case, the requirement may be too restrictive and may lead to loosing relevant

solutions. Let us consider, as an example, the explanatory formula a explains b fta overlaps tb g (1) Suppose that the observation to be explained is: tb Since 10 Until 20 (2) Assuming a with the constraint ta Since 5 Until 15 (3) is intuitively a reasonable explanation, but note that (1) and (3) do not entail (2), since it would not be reasonable to predict (2) given (1) and (3) since the constraints in the model are not very precise (with respect to those in the observations). Speci cally, there is no set of assumptions that entails b( 10 20]), and then there is no abductive explanation for it. In case the temporal constraints in the model are precise enough (that is, they are such that assumptions entail information on observable states that is at least as precise as the constraints that may be available on the observations), it may make sense to require that the constraints on the observations are entailed by those in the explanation. For example, consider the following explanatory formula: a explains b fta contains tb g (4) and the observation tb Lasting AtMost 10 (5) Given (4), the assumption ta Lasting AtMost 10 is the weakest assumption that entails (5) notice that, on the other hand, the assumption ta Lasting 20 is consistent with (5) but does not entail it. Precise observations (such as tb Lasting 10) cannot be explained also in this case. In this section we analyse some of the approaches to temporal model-based diagnosis showing

how they can be cast into the spectrum of de nition introduced in the previous section. The analysis, summarized in gure 1, is not exhaustive and only aims at showing that there are approaches that t into the various alternative possibilities showed by our general characterization. Notice, moreover, that each one of the approaches in the literature deals only with some of the aspects mentioned in the introduction (i.e., temporal, dynamic, time-varying behavior). Most of the temporal extensions of consistencybased atemporal diagnosis make use of a notion of temporal explanation which is also based on consistency. This is the case, for example, of the xde system (Hamscher 1991), which deals (even if using abstract forms of

Towards a classi cation of the approaches in the literature