Reference: Nayak, P. P. Order of Magnitude Reasoning Using Logarithms. 1992.
Abstract: Converting complex equations into simpler, more tractable equations usually involves approximation. Approximation is usually done by identifying and removing insignificant terms, while retaining significant ones. The significance of a term can be determined by order of magnitude reasoning. In this paper we describe NAPIER, an implemented order of magnitude reasoning system. NAPIER defines the order of magnitude of a quantity on a logarithmic scale, and uses a set of rules to propagate orders of magnitudes through equations. A novel feature of NAPIER is the way it handles non-linear simultaneous equations, using linear programming in conjunction with backtracking. We show that order of magnitude reasoning in NAPIER is, in general, intractable and then discuss an approximate reasoning technique that allow it to run fast in practice. Some of NAPIER's inference rules are heuristic, and we estimate the error introduced by their use.
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