We investigate efficient algorithms for learning the structure of a Markov network from data using the independence-based approach. Such algorithms conduct a series of conditional independence tests on data, successively restricting the set of possible structures until there is only a single structure consistent with the outcomes of the conditional independence tests executed (if possible). As Pearl has shown, the instances of the conditional independence relation in any domain are theoretically interdependent, made explicit in his well-known conditional independence axioms. The first couple of algorithms we discuss, GSMN and GSIMN, exploit Pearl's independence axioms to reduce the number of tests required to learn a Markov network. This is useful in domains where independence tests are expensive, such as cases of very large data sets or distributed data. Subsequently, we explore how these axioms can be exploited to "correct" the outcome of unreliable statistical independence tests, such as in applications where little data is available. We show how the problem of incorrect tests can be mapped to inference in inconsistent knowledge bases, a problem studied extensively in the field of non-monotonic logic. We present an algorithm for inferring independence values based on a sub-class of non-monotonic logics: the argumentation framework. Our results show the advantage of using our approach in the learning of structures, with improvements in the accuracy of learned networks of up to 20%. As an alternative to logic-based interdependence among independence tests, we also explore probabilistic interdependence. Our algorithm, called PFMN, takes a Bayesian particle filtering approach, using a population of Markov network structures to maintain the posterior probability distribution over them given the outcomes of the tests performed. The result is an approximate algorithm (due to the use of particle filtering) that is useful in domains where independence tests are expensive.