*Brian C. Williams*

The apparently weak properties of a qualitative algebra have lead some to conclude that we must turn instead to extra-mathematical properties of physical systems. We propose instead that a more powerful qualitative algebra is needed, one that merges the algebras on signs and reals. We have invented a hybrid algebra, called Ql, allows us to select abstractions intermediate between traditional qualitative and quantitative algebras. The power of our algebra is demonstrated in three ways: First, analysis of Ql shows that the algebra is robust, sharing many properties of reals, but including several that are unique. Second, these properties enable symbolic manipulation techniques for canonicalization and factorization distinct from those applied to the reals. Finally, these manipulation techniques hold much promise for tasks like design and verification, as suggested by a simple design example.

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