The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. We show that any quantum algorithm in this model whose total query time is T can be simulated by a quantum algorithm in the discrete-time query model that makes O(T log T / log log T ) ⊂Õ(T ) queries. This is the first such upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of Ω(T log log T / log T ) ⊂Ω(T ) in the continuous-time query model.

Most model-based approaches to fault diagnosis of discrete-event systems (DESs) require a complete and accurate model of the system to be diagnosed. However, the discrete-event model may have arisen from abstraction and simplification of a continuous time system or through model building from input-output data. As such, it may not capture the dynamic behavior of the system completely. In this paper, we address the problem of diagnosing faults, given an incomplete model of the discrete-event system. When the model is incomplete, discrepancies will arise between the actual output and the output predicted by the model. We introduce learning into the diagnoser construction by forming hypotheses that explain these discrepancies. We view the process of generating and evaluating hypotheses about the model of the system as an instance of the set-cover problem, which we formalize using parsimonious covering theory. We describe in detail the construction of the learning diagnoser, which not only performs fault diagnosis but also attempts to learn the missing model information. If the model is complete, the learning diagnoser reduces to the standard state-based diagnoser. Examples are provided to illustrate how learning and diagnosis can be simultaneously achieved through the learning diagnoser.

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We present a bounded-error quantum algorithm for evaluating Min-Max trees with N 1 2 +o(1) queries, where N is the size of the tree and where the allowable queries are comparisons of the form [x j < x k ]. This is close to tight, since there is a known quantum lower bound of Ω(N 1 2 ).

Most model-based approaches to fault diagnosis of discrete-event systems require a complete and accurate model of the system to be diagnosed. However, the discrete-event model may have arisen from abstraction and simplification of a continuous time system, or through model building from input-output data. As such, it may not capture the dynamic behavior of the system completely. In a previous paper, we addressed the problem of diagnosing faults given an incomplete model of the discrete-event system. We presented the learning diagnoser which not only diagnoses faults, but also attempts to learn missing model information through parsimonious hypothesis generation. In this paper, we study the properties of learnability and diagnosability. Learnability deals with the issue of whether the missing model information can be learned, while diagnosability corresponds to the ability to detect and isolate a fault after it has occurred. We provide conditions under which the learning diagnoser can learn missing model information. We define the notions of weak and strong diagnosability and also give conditions under which they hold.