This paper investigates the achievable rate and ergodic capacity of an orthogonal amplify-and-forward (OAF) half-duplex multiple-relay network with direct link where multiple relays use channel state information (CSI) to cooperate with the source and destination. The relays are subject to two types of power constraint: the total average power constraint (TAPC) and the individual average power constraint (IAPC). In the first step, by assuming a fixed input covariance matrix at the source, we derive an optimal power allocation (OPA) scheme among the relays via optimal instantaneous power amplification coefficients to maximize the achievable rate. The closed-form optimal solutions are obtained for the considered system under either the TAPC or both the TAPC and IAPC. Next, we derive the ergodic capacity by jointly optimizing the input covariance matrix and the power allocation at the relays. We show that this is a bi-level non-convex problem and solve this using Tammer decomposition method. This approach allows us to convert the original optimization problem to a master problem and a set of sub-problems that have closed-form solutions as obtained in the first step. The ergodic capacity is then obtained using an iterative water-filling-based algorithm.

We study the behavior of the A* search algorithm when coupled with a heuristic h satisfying (1-epsilon1)h*

Knowledge validation is a vital phase in knowledge engineering which is ultimately aimed at correcting the knowledge-base such that when inferenced over, it satisfy all test cases specified by the expert users. Naturally, such a goal is unachievable if there is some contradiction in the given set of test cases. In this work, we analyze this property of test case sets in knowledge validation for knowledge bases that are modelled in terms of Bayesian knowledge bases (BKBs) in order to determine the necessary and sufficient condition for a test case set to be contradiction-free, i.e., "Does there exist a knowledge base satisfying all test cases in the set?" We show that the complexity of deciding if a test case set meets that condition is NP-compete. As such, we also present some special cases in which it is tractable to make this determination.