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Symbolic analysis is of paramount importance for parallelizingcompilers and performance estimators to examine symbolic expressions withprogram unknowns such as machine and problem sizes and to solve queriesbased on systems of constraints (equalities and inequalities). This paperdescribes novel techniques for counting the number of solutions to a systemof constraints, simplifying systems of constraints, computing lower andupper bounds of symbolic expressions, and determining the relationshipbetween symbolic expressions. All techniques target wide classes of linearand non-linearsymbolic expressions and systems of constraints. Ourtechniques have been implemented and are used as part of a parallelizingcompiler and a performance estimator to support analysis and optimization ofparallel programs. Various examples and experiments demonstrate theeffectiveness of our symbolic analysis techniques.