randomly or depend on the obtained error from the NFIS or RST (regular neuron growth) 
Step (2-2): construction of the granules (crisp). 

Step (3): second granulation (fuzzy or rough IGs) 
by NFIS or RST 
Step (3-1): crisp granules as a new data. Step (3-2): selecting the level of granularity; 
(Error level, number of rules, strength threshold...) 
Step (3-3): checking the suitability. (Closeopen iteration: referring to the real data and reinspect closed world) 
Step (3-4): construction of fuzzy/rough granules. 

Step (4): extraction of knowledge rules Selection of initial crisp granules can be supposed as "Close World Assumption (CWA)" .But in many applications, the assumption of complete information is not feasible, and only cannot be used. In such cases, an "Open World Assumption (OWA)', where information not known by an agent is assumed to be unknown, is often accepted [12]. 

Balancing assumption is satisfied by the close-open iterations: this process is a guideline to balancing of crisp and sub fuzzy/rough granules by some random/regular selection of initial granules or other optimal structures and increment of supporting rules 
(fuzzy partitions or increasing of lower /upper approximations ), gradually. 

The overall schematic of Self Organizing NeuroFuzzy Inference System -Random and Regular neuron growth-: SONFIS-R, SONFIS-AR; has been shown in figure5. 

In first regular granulation, we use a linear relation is given by: 

$$N_{{}_{t+1}}=\alpha N_{{}_{t}}+\Delta_{{}_{t}};\Delta_{{}_{t}}=\beta E_{{}_{t}}+\gamma\tag{5}$$  Where $N_{{}_{t}}=n_{1}\times n_{2};\left|n_{1}-n_{2}\right|=Min.$ is number of neurons in SOM: $E$ is the obtained error (measured
neurons in SOM; Et is the obtained error (measured error) from second granulation on the test data and coefficients must be determined, depend on the used data set. Obviously, one can employ like manipulation in the rule (second granulation) generation part, i.e., number of rules. 

Determination of granulation level is controlled with three main parameters: range of neuron growth, number of rules and error level. The main benefit of this algorithm is to looking for best structure and rules for two known intelligent system, while in independent situations each of them has some appropriate problems such: finding of spurious patterns for the large data sets, extratime training of NFIS or SOM. 

![0_image_0.png](0_image_0.png)

In second algorithm, apart from employing hard computing methods (hard granules), RST instead of NFIS has been proposed (figure 6). Applying of SOM as a preprocessing step and discretization tool is second process. Categorization of attributes (inputs/outputs) is transferring of the attribute space to the symbolic appropriate attributes. In fact for continuous valued attributes, the feature space needs to be discretized for defining indiscernibilty relations and equivalence classes. We discretize each feature in to some levels by SOM, for example 
"low, medium, and high" for attribute "a". 

Finer discretization may lead to better accuracy to recognizing of test data but imposes the higher cost of a computational load. However, to look for best scaled condition and decision attributes, we have developed other SORST system, upon the adaptive scaling of attributes (SORST-AS), gradually [13]. 

Because of the generated rules by a rough set are coarse and therefore need to be fine-tuned, here, we have used the preprocessing step on data set to crisp granulation by SOM (close world assumption). In fact, with referring to the instinct of the human, we understand that human being want to states the events in the best simple words, sentences, rules, functions and so forth. Undoubtedly, such granules while satisfies the mentioned axiom that describe the distinguished initial structure(s) of events or immature data sets. Second SOM, as well as close world assumption, gets such dominant structures on the real data. In other word, condensation of real world and concentration on this space is associated 