Systems are subject to faults in their components, affecting their overall behaviour. In a 'black-box' system, such faults only become apparent in the output when appropriate inputs are given, which poses a number of satisfaction and optimisation problems regarding both testing and diagnosing. In this dissertation we address such problems, developing models with multi-valued logics that we formalise and generalise to multiple faults. Such logics extend Boolean logic by encoding dependencies on faults, thus allowing the modelling of an arbitrary number of diagnostic theories. The effectiveness of constraint solving over finite domains and sets is shown on several of the above problems.