Symbolic Reasoning with Relative Orders of Magnitude

Abstract

Reasoning with relative orders of magnitude is particularly appropriate when it is necessary to capture commonsense reasoning and tackle engineering tasks involving complex physical systems. The ftrst systems to be proposed, FOG [Raiman, 1986] and O(M) [Mavrovouniotis and Stephanopoulos, 1987], are however limited and thus not widely used. In this ftrst paper, only symbolic reasoning is considered, and the limitation due to the impossibility of expressing a gradual change in order of magmtude is tackled. A new relation "to be distant from" is introduced, followed by the 15 axioms of the ROM(K) system and 45 or so properties which have a clear qualitative meaning. A simple algebraic formalization of this perfectly symmetric framework with two degrees of freedom and a set structure overlap is then given, its consistency being proved by building a family of models in the fteld of real numbers of non standard analysis. The result is 15 primitive relations which, thanks to their overlapping, allow gradual transitions in order of magnitude to be expressed smoothly. The conclusion leads into the second paper [Dague, 1993] which concerns the sound transposition of the rules of O(M) in ~ for connection with the numeric world and control of the inferences.