The Atom-Association-Aggregation approach, or AAA approach, or A3 approach, treats the logical mind as a machine operating with indivisible entities called atoms, upon which it does two operations:

  • Associating two atoms by connecting them with a directed link, which is an atom said to be their association link — an operation, which we ascribe to the left hemisphere of human brain, specialized on dealing with time;
  • Aggregating a set of atoms resulting in an atom, said to be their aggregation link — an operation, which we ascribe to the right hemisphere of brain, specialized on dealing with space.

The results of both operations are said to be links — the association link, which has direction and connects two atoms presented to the mind one after another in the order of their presentation, and the aggregation link, which is non-directed and puts together in a whole the atoms simultaneously presented to mind. Coincidentally or not, the word “intellect” comes from Lat. “inter-lego”, which can be translated as “I link (things between themselves)”.

Because there is no strict definition of intelligence or of intelligent agent, we found it appropriate to define intelligence as the ability to process linked data structures, where the links are of two types — association link and aggregation link, and regard an agent as intelligent, if it has this ability and can pass the intelligence test formulated by Turing.

Each of the two operations imposes its type of logic, but in order for an agent to be intelligent, it must necessarily have the stereo logic provided by the concurrent use of the two operations. Despite tremendous anatomic differences, like the number of limbs or senses, animals of all species including people, have two hemispheres of brain. We treat this matter of fact as support by the Nature of our approach.

Sequentiality and simultaneity are two fundamental orders in matter and mind. Sequentiality is the temporal proximity between events. Mind represents the sequentiality by creating an association link, which it can later traverse in the process of reasoning or recollection of events. A set of atoms together with associations between them can be represented as a directed multi-graph. Simultaneity is the spatial proximity between physical bodies — the mind represents it by aggregating their images into a whole. A set of atoms together with aggregation can be represented as a multi-hypergraph.

The condition, that only atoms serve as parameters of operations, is essential for the A3 approach and this condition is the first distinguishing feature from relational model of data of Codd or entity-relationship approach of Chen, which both allow to apply relationships and operations to complex objects of any structure.

Even though the term structure is used in different domains of mathematics, we are aware of no definition of this notion which would be both strict and general enough to apply to any phenomenon called a structure. Therefore, we say a structure to be any object obtained by multiple application of the association and integration operations. We have formalized the notion of structure in terms of set theory. A structure can be represented as a graph which is more complex than a directed multigraph and a multi hypergraph and it has no name in graph theory .

The formalization of mathematics started with naive set theory which, according the A3 approach, is specific to the right brain hemisphere specialized on aggregation. A finite set is defined if all its elements are present before the definition — a finite set is a typical aggregate. We call this logic presentation logic, or informally — visual logic. This type of logic “resides” in patterns, forms and other kinds of objects which are matter of pattern recognition and perception of the whole. Aggregation is possible only when all the entities to be aggregated are present simultaneously. This is one of the reasons for the name “presentation logic”. Another reason is in the opposition between describing something and presenting it. Also, even though the difference is subtle, we make distinction between “representation” as an action and “presentation” as the source or result of this action.

Axiomatic set theory made a “hemisphere shift” by starting dealing with sets via associations relationship called “membership” — a specialty of the left brain hemisphere. Sets became entities linked by directed links of membership relationship, so that you can only discuss about sets, but you have no mechanism for their (re)presentation or such a mechanism is external to the theory. The abstract operator of creating a set, even though used in the practice of an axiomatic set theory, did not enter the signature of the language of any axiomatic theory. We say this to be discourse logic, or informally, and to keep the parallel with the informal name of logics of the other hemisphere — verbal logic.

It is commonly accepted that left hemisphere deals with speech, listening and “logic”, and the right hemisphere deals with intuition, insight, and similar phenomena, which though are not said to be “logic”. In the context of A3 approach, both hemispheres are said to deal with logic, but the logic is regarded of two different types — presentation logic and discourse logic. Presentation logic of the right brain hemisphere operates with objects, and discourse logic of the left brain hemisphere operates with statements, questions and other forms of discourse.

Most of the difficulties of semantic technologies are due exactly to the non-discriminatory treatment of presentation and representation, and between presentation logic and discourse logic. We overcame these difficulties by using the A3 approach which we also say to be the wholebrain approach to the web.