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Related Work

Other researchers have proposed extensions to FMVE as a decision method for dependence analysis [IJT91][WT92][MHL91a]. Lam et al. [MHL91a] extend FMVE to integers by computing a sample solution, using branch and bound techniques if needed. Michael Wolfe and Chau-Wen Tseng [WT92] discuss how to recognize when FMVE may produce a conservative result, but do not give a method to verify the existence of integer solutions. These methods are decision tests and cannot return symbolic answers.

Corinne Ancourt and François Irigoin [AI91] describe the use of FMVE for quantified variable elimination. They use this to generate loop bounds that scan convex polyhedra. They extend FMVE to integers by introducing floor and ceiling operators. Although makes their elimination exact, it may not be possible to eliminate additional variables from a set of constraints involving floor and ceiling operators. This limits their ability to check for the existence of integer solutions and remove redundant constraints.

Cooper [Coo72] describes a complete algorithm for verifying and/or simplifying Presburger formulas. His method for quantified variable elimination always introduces disjunctions, even if the result is convex. We have not yet performed a head-to-head comparison of the Omega test with Cooper's algorithm. However, I believe that the Omega test will prove better for quantified variable elimination when the result is convex and better for verification of a formula already in disjunctive normal form. Cooper's algorithm does not require formulas to be transformed into disjunctive normal form and may be better for formulas that would be expensive to put into disjunctive normal form (although our methods for handling negation address this as well).

The SUP-INF method [Sho77][Ble75] is a semi-decision procedure. It sometimes detects solutions when only real solutions exist and it cannot be used for symbolic quantified variable elimination.

Jean-Louis Lassez [HLL92][LL92][LHM89] gives an alternative to FMVE for elimination of existentially quantified variables. However, his methods work over real variables, are optimized for dense constraints (constraints with few zero coefficients) and are inefficient when the final problem contains more than a few variables since they build a convex hull in the space of variables remaining after all quantified variables have been eliminated.



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