Sophisticated software systems are inherently complex. Understanding, debugging and maintaining such systems requires inferring high-level characteristics of the system's behavior from a myriad of low-level details. For large systems, this quickly becomes an extremely difficult task. MrSpidey is a static debugger that augments the programmers ability to deal with such complex systems. It statically analyzes the program and uses the results of the analysis to identify and highlight any program operation may cause a run-time fault. The programmer can then investigate each potential fault site and, using the graphical explanation facilities of MrSpidey, determine if the fault will really happen or whether the corresponding correctness proof is beyond the analysis's capabilities. In practice, MrSpidey has proven to be an effective tool for debugging program under development and understanding existing programs. The key technology underlying MrSpidey is componential set-based analysis. This is a constraint-based, whole-program analysis for object-oriented and functional programs. The analysis first processes each program component (eg. module or package) independently, generating and simplifying a constraint system describing the data flow behavior of that component. The analysis then combines and solves these simplified constraint systems to yield invariants characterizing the run-time behavior of the entire program. This component-wise approach yields an analysis that handles significantly larger programs than previous analyses of comparable accuracy. The simplification of constraint systems raises a number of questions. In particular, we need to ensure that simplification preserves the observable behavior, or solution space, of a constraint system. This dissertation provides a complete proof-theoretic and algorithmic characterization of the observable behavior of constraint systems, and establishes a close connection between the observable equivalence of constraint systems and the equivalence of regular tree grammars. We exploit this connection to develop a complete algorithm for deciding the observable equivalence of constraint systems, and to adapt a variety of algorithms for simplifying regular tree grammars to the problem of simplifying constraint systems. The resulting constraint simplification algorithms yield an order of magnitude reduction in the size of constraint systems for typical program expressions.