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

constraints may be associated with such a behavior, e.g.: T2 during T1 T3 during T1 T3 meets T5, start(T4 ) n after start(T2 ) start(T5 ) m after start(T2 ) specifying that there is a delay of n units of time between the input and output, that the change of state starts m units of time after the input and that the interval for the new state meets the interval for the old state (i.e., that the component must always be in some state). Temporal integrity constraints can be used to impose that d cannot be in two di erent states at the same time. These types of models have been used, e.g., in (Hamscher& Davis 1984 Hamscher 1991). Let us consider now the case of time-varying behavior. The behavior of a device (component) is timevarying if the device (component) can assume di erent modes of behavior across time (Friedrich& Lackinger 1991). This type of information can be expressed in our language using temporal integrity constraints. A powerful formalism used in time-varying systems for expressing how modes can evolve across time is that of Mode Transition Graphs (Console et al. 1994 Nejdl& Gamper 1994), to impose that a set of mutually exclusive behavioral modes could or must follow each other only in a constrained way: e.g. m1 must be directly followed by m2 or m3, but not by other modes. Also these types of constraints can be expressed as temporal integrity constraints between the atoms denoting modes of behavior. The example above can be expressed by the following formula: m1 (T ) ! 9T (T meets T^ (m2 (T ) _ m3 (T ))) (in addition to the constraints that specify that the modes are mutually exclusive). Thus both temporal (non-dynamic and dynamic) and time-varying behavior can be expressed in our formalism with appropriate modeling. In order to provide a formal characterization of temporal diagnosis, a model expressed in terms of a set TBM of temporal behavior formulae and a set TC of temporal integrity constraints can be transformed into a set of logical formulae. These logical formulae thus provide a semantics for our modeling formalism. The transformation involves some technicalities (in particular for expressing in logical terms the meaning of the temporal constraints associated with temporal behavior formulae) and is presented elsewhere (Brusoni et al. 1996). The output of the transformation is formed by the following sets:0 0 0 0

modei (d T1 ) inp(d X T2) s(d Y T3 ) explains out(d f (X Y ) T4 ) s(d g(X Y ) T5 ) fC (T1::: T5 )g which says that if d is in mode modei and in state Y and it receives the input X, it produces the output f (X Y ) and its state changes to g(X Y ). Temporal

TBML: the logical correspondent of the temporal behavioral

model TBM and of the set TC of tem-

poral integrity constraints Abd: the set of abducible symbols, i.e. symbols that can be part of the solution to a diagnostic problem (see next section). These correspond to the assumptions that can be made during the diagnostic process and include at least the symbols denoting modes of behavior of the device (or of the components) being modeled and to be diagnosed. In this section we show how the spectrum of logical definitions of diagnosis introduced in (Console& Torasso 1991b) for the atemporal case can be extended to the temporal case, starting from the modeling formalism introduced in the previous section. A diagnostic problem is characterized by a set of observations to be explained. In particular, diagnosis is relevant when there is a discrepancy between the expected behavior of a system and the observed behavior, so that observations correspond to symptoms gathered during the abnormal behavior of the system. The goal of diagnostic problem solving is to determine which faults of the system (or, in particular, of one or more of its components) can explain the abnormal behavior. From a general point of view, observations can be characterized by four sets: CXT= fa1 a2::: am g: a set of contextual data OBSpos= fo1 o2::: on g: a set of positive atoms corresponding to data that have been observed OBSneg= f:b1:b2::::bk g: a set of negative observations, corresponding to data that are known to be absent (false), at least in some period of time (as speci ed by the item below). TOBSTC (ta1::: tam to1::: ton tnot b1::: tnot ok ) a set of temporal constraints on contextual and observed data. In particular, the set TOBSTC can be partitioned into two subsets: CCXT containing constraints that involve contextual data only and COBS containing all the other constraints. The distinction between contextual data and observations is the same one introduced in (Console& Torasso 1991b): contextual data correspond to data that have not to be accounted for by a diagnosis, unlike observations that have to be accounted for. Typical examples of contextual data are the inputs to the device to be analysed while the outputs from the device are examples of observations to be explained. TOBSTC is a set of temporal constraints (expressed in some language for temporal constraints) and provides temporal information on contextual data and observations. It may contain information on the absolute temporal location (with respect to a reference time point) or duration of observed events, which may be precise, as in

An extended spectrum for diagnosis