A Theory of Interactions: Unifying Qualitative and Quantitative Algebraic Reasoning

Abstract The apparently weak properties of a purely qualitative algebra have led some to conclude that researchers 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, along with symbolic techniques for manipulating this algebra. We have constructed a hybrid algebra, called SR1 which allows intermediate abstractions to be selected between traditional qualitative and quantitative algebras. SR1 and the symbolic algebra system Minima demonstrate substantial progress towards a theory of continuous interactions between quantities—one that allows just the interesting features of interactions to be represented, and that captures skills for composing and comparing interactions. This theory is sufficiently expressive to determine the behaviors that a variety of fluid regulation devices will achieve—not just what is impossible. It embodies in a simple manner many existing algebraic formalisms for describing and individually manipulating interactions, including confluences, inequalities, and monotonicity operators, as well as many of the individual inferences of qualitative arithmetic, composition of monotonicity, inequality algebra, transition analysis, qualitative resolution and traditional algebra.