BASIC NOTIONS

Input vector (In): each choice or demand, must be expressed as a spatial object into a two-dimensional Euclidean space, even if this expression is approximate and refers to empirical data. Each Input must be considered to have a double identity as an act of transformation and its result.

Fuzzy Group: each choice entering the system is an action that transforms the previous choice towards a new object. The final object contains therefore in some degree the initial expression. The group of all choices that preserve an object unchangeable to a specific degree is the fuzzy group of the transformations of this object. Fuzzy group is a hybrid term that derives form the combination of the Group Theory and the idea of the Fuzzy Sets [32], [45], [52], [56]. We should note that the transformations mentioned above might refer to an initial object or to an object that constitutes the common section of all choices. We can also interpret the definition of the Fuzzy Group in a different way, as a group of transformations that each of them preserves unchangeable an object in a certain degree (in our case this degree is determined by "vigilance").

Containment Degree: the basic notion that outlines the function of the plan is that the whole of the expressed demands is contained to some degree in each individual choice. The Containment Degree designates the degree to which the Fuzzy Group of an object is contained in the given object and is measured in a scale from 0 to 1. Mathematically expressed: 0 =< In /\ W / In \/ W =< 1, where /\ is the min operator of fuzzy logic and \/ is the max. [In /\ W= min (In, W) and In \/ W= max (In, W)].

Learning- Preceding-images: learning refers to the systematic change of weights that link artificial neurons with each other. Weights (W) store the connections among Inputs and Outputs, actions and reactions, organizing the "long-term memory" of the network. The adaptability of the plan is due to appropriate alteration of these weights. The weight vector W is updated according to the equation W' = ß(In /\ W) + (1- ß) W, where ß E [0,1] is the learning rate parameter. In our study W is an n x m matrix where each element (ni, mj) is a number between 0 and 1.
Weights are considered to be the preceding-images of the plan as they construct an internal representation of real space and form the initial conditions to activate the system. Each preceding-image is constructed through a continuous procedure of redefinition regarding an initial choice (architectural object). This procedure designates a question regarding the degree in which the initial object preserves a stable identity or it gains a complex quality that can be interpreted in various ways. So, this procedure questions the degree of self-similarity that can be obtained [34].

Vigilance (p): there is a measurement that designates the degree in which a preceding-image can be considered to remain unchangeable after the introduction of a new Input. If the Input is a Fuzzy Subset of W and so meets the criterion of p (In /\ W / In >= p, p E [0,1]), then it is part of the Fuzzy Group of W and resonance occurs. Vigilance therefore is a precondition for resonance and indicates the degree in which the preceding-images are flexible to changes.

Output vector (Out): to each Input corresponds an Output (n x m) where each of its elements is marked by a number between 0 and 1 that represents the containment degree of the whole of the preceding-images, to the local choice. So, we get a three-dimensional object where the two dimensions represent space and the third describes the containment degree.

For the interpretation of the results we should take into account some general remarks.
- When the containment degree reaches near 1, then the Input has been confirmed several times by different preceding-images and so it has a multiple identity. On the contrary, when the containment degree reaches near 0 then the Input has not been influenced by the preceding-images and so it keeps its initial character. In the first case we get a fuzzy proposal that is however well adapted to the whole structure of choices, while in the second case we get a clear proposal that is not tuned to the demands expressed through the preceding-images.
- Input and output should be interpreted in comparison. Input is set as a binary vector (0 or 1): if 1 corresponds to an entity "A", then 0 corresponds to its complement "not-A". Output though is an analog result and therefore tendency towards 1 may mark the confirmation of either "A" or "not-A". So what gets the support of the preceding-images can be either the one or the other.

Defuzzication operated by the second part of the Fuzzy Adaptive Intervention Plan is in fact the procedure that interprets the containment degree in terms of architectural qualities. It shapes rules regarding the actual position and nature of emerging forms through the exploitation of knowledge coming from multiple users. (click to see picture)

In general, the function of the program realized can be described as follows. Architectural choices are designed on a two-dimensional field of n x m pixels that each represents a number from 0 to 1 (0 =< x =< 1). Initially, some pixels are activated and some are not. Knowledge of the system is stored into nods created on the second layer. Each node contains a two-dimensional field of pixels activated in different degrees and receives the sum of common parts among Input and the preceding-images stored into the rest nods. Each nod therefore receives both the Input and the whole of images produced inside the system. If criterion p is met then weights of each nod are altered, if differently the weights are sustained. The result is the cumulative image of the changes. The Output is fed back into the system and is weighed again with the initial Input. The procedure continues repetitively as many times as we define.

For more details you may go to the page: The software step by step