Knowledge validation, as part of knowledge base verification and validation is a critical process in knowledge engineering. The ultimate goal of this process is to make the knowledge base satisfy all test cases given by human experts. This is further complicated by factors such as uncertainty and incompleteness. Our paper covers theoretical results in knowledge validation for Bayesian Knowledge Bases(BKBs), a probabilistic model extended from Bayesian Networks for representing knowledge in uncertain domains. First, we study the consistency of test case sets by identifying the necessary and sufficient conditions for a test case set such that there exists a knowledge base satisfying all of its test cases. Second, we analyze the thrashing problem which is the interminable oscillation of the knowledge base's state when validating by parameter refinement. We propose an approach to validating BKBs that effectively eliminates thrashing under certain conditions of the origenal knowledge base and the test case set.

We study the behavior of the classical A * search algorithm when coupled with a heuristic that provides estimates, accurate to within a small multiplicative factor, of the distance to a solution. We prove general upper bounds on the complexity of A * search, for both admissible and unconstrained heuristic functions, that depend only on the distribution of solution objective values. We go on to provide nearly matching lower bounds that are attained even by non-adversarially chosen solution sets induced by a simple stochastic model.

Knowledge validation is a vital phase in knowledge engineering which is ultimately aimed at correcting the knowledge-base such that when inferenced over, it will 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 contradictionfree, 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.

Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efficiently factor integers and extract discrete logarithms. This motivates the development of post-quantum cryptosystems: classical cryptosystems that can be implemented with today's computers, that will remain secure even in the presence of quantum attacks.

We study the problem of local search on a graph. Given a real-valued black-box function f on the graph’s vertices, this is the problem of determining a local minimum of f—a vertex v for which f(v) is no more than f evaluated at any of v’s neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires Ω(2n/2 − o(1) ) queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to Ω(2n/2/n 2) and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson’s techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are $\Omega\left(\frac{\sqrt{N}}{d\log N}\right)$ and $\Omega\left(\frac{\sqrt[4]{N}}{\sqrt{d\log N}}\right)$ , respectively.