Calculi of Granules for Adaptive Distributed Synthesis over Schemes of Intelligent Agents Founded on Rough Mereology

A notion of a synthesis granule is proposed based on rough mereological theory of multi-agent systems developed by the authors. The metafor "granule of knowledge" has appeared recently 6], 7] in the research literature devoted to the problems of approximate reasoning and soft computing and the importance of granulation of knowledge for these problems has been stressed therein. The meaning of the phrase "knowledge granulation" is related to the phenomenon that reasoning under incomplete or uncertain knowledge is carried out by means of knowledge about objects decomposed into indivisible units i.e. "granules". The division of knowledge into granules is by no means of static character, to the contrary, the changing character of our knowledge due to innux of a new information causes currently existing granules to divide or merge into new granules. Dynamic processes of granule of knowledge formation are very important for problems related to synthesis of approximate solutions to speciications formulated in an uncertain or incomplete way e.g. in the problems of synthesis of adaptive decision algorithms 3], 5]. 2. BASIC GRANULE STRUCTURES We refer to rough set literature (cf. 1,3,4]) for the basic notions. For an information system (U; A) and a set B A; the indiscernibility relation IND B (u; w) induces the B-algebra CG(B) of pre-granules as the boolean algebra generated over the set of atoms U=IND B by means of set-theoretic operations of union, intersection and complement. For a set of attributes B A, the logic L B is deened: elementary formulae of L B are of the form (a = v) where a 2 B and v 2 V a : Formulae of L B are built from elementary formulae by means of logical connectives _; ^: The meaning ] is the set of objects that satisfy : Clearly, given a pre-granule G 2 CG(B); there exists a formula (unique in DNF) G of L B such that G ] = G. In the same way as above we can treat the decision attribute d as well as the whole set A 0 = Afdg of attributes which would lead to algebras CG(A 0); CG(d); CG(B) where B A 0 and logics, respectively, For given non-empty sets B; C A; a pair (G B ; G C) where G B 2 CG(B) and G C 2 CG(C) is called a (B; C)? granule of knowledge. There exist formulae GB 2 L B ; …