Synthesis of Complex Objects: Rough Mereological Approach

We present a brief account of rough mereology-an approach to problems of approximate reasoning about complex objects and we outline possible applications of the presented formalism. 1 Approximate reasoning based on rough mereology We would like to give here a brief account of the rough mere-ological approach to approximate reasoning. We propose this formalization as a tool for solving multi-agent or distributed applications related to approximate reasoning. Rough mereology ooers the general formalism for the treatment of partial containment. Rough mereology can be regarded as a far-reaching generalization of mereology of Le s-niewski 12]: it does replace the relation of being a (proper) part with a hierarchy of relations of being a part in a degree. The basic notion is the notion of a rough inclusion. A real function (X; Y) on a universe of objects U with values in the interval 0; 1] is called a rough inclusion when it satisses the following conditions: (A) (x; x) = 1 for any x (meaning normalization); (B) (x; y) = 1 implies that (z; y) (z; x) for any triple x; y; z (meaning monotonicity); (C) (x; y) = 1 and (y; x) = 1 imply (x; z) (y; z) for any triple x; y; z (meaning monotonicity); (D) there is n such that (n; x) = 1 for any x. An object n satisfying (D) is a-object. We let x = y ii (x; y) = 1 = (y; x) and x 6 = y ii non(x = y). We introduce other conditions for rough inclusion: (E) if objects x; y have the property : if z 6 = n and (z; x) = 1 then there is t 6 = n with (t; z) = 1 = (t; y) then it follows that: (x; y) = 1. (E) is an inference rule: it is applied to infer the relation of being a part from the relation of being a subpart. (F) For any collection ? of objects there is an object x with the properties: (i) if z 6 = n and (z; x) = 1 then there are t 6 = n; w 2 ? such that (t; z) = (t; w) = (w; x) = 1; (ii) if w 2 ? then (w; x) = 1; (iii) if y satisses the above two conditions then (x; y) = 1. Any x satisfying F(i) is called a set …