Thesis (Ph.D.) - Indiana University, Computer Sciences, 2009Constraints on different manifestations of data are a central concept in numerous areas of computer science. Examples include mathematical logic, database systems (functional and multivalued dependencies), data mining (association rules), and reasoning under uncertainty (conditional independence statements). One is often interested in a process that derives all or most of the constraints that are entailed by a set of known ones, without the expense and error-proneness of repeatedly analyzing the data. This is what is generally known as the implication problem for data constraints. We present a theoretical framework for disjunctive data constraints and the associated implication problems based on the observation that many instances can be reduced to an implication problem for additive constraints on specific classes of real-valued functions. Furthermore, we provide inference systems and testable properties of classes of real-valued functions which imply the soundness and completeness of these systems. We also derive properties of classes of functions that imply the non-existence of finite, complete axiomatizations. The theoretical framework is applied to derive novel results in the areas of uncertain reasoning and graphical models.