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1.2. FUNDAMENTAL TOOLS: VERBAL AND FORMAL MATHEMATICAL METHODS

What are the basic characteristics of our mental representations? The brain collects and stores relevant ensembles of phenomena. These have different names in various disciplines; here, the name pattern will be used, with a reference to the analogy of visual patterns. The patterns represent objects (any kind, living, not living), situations (some coincidences of objects), actions of these objects, ourselves included. The actions create new objects, situations, i.e., patterns are remembered as some sequences like consecutive frames of a movie. The brain patterns will be discussed later in relation to the pattern representation in computers.

The consecution, sequence of identical patterns, happen with some frequency. What identical more precisely means, comes later. This frequency is remembered, too. Certain consecution of certain patterns happens more frequently, sometimes nearly always. This particular pattern dynamics itself emerges as a pattern. These are the patterns of pattern relations, which we call metapatterns. The linguistic representation of this metapattern pattern relations developed to be the logic. Logical sentences are represented by verbal, formulary, and graphical schemes, i.e., patterns, the variables of the scheme are the original patterns.

The frequency issue is much more dubious, just because of the fact that identity and the nature of similarity, their relation to frequency occurrence of these identical or similar patterns is always vague. Probability theory and statistics are the major representations of these pattern relations, joined with several other uncertainty representation philosophies and methods, but all related to the basic impression of frequency of patterns. Representations denying the direct statistical frequency origin return really to the frequency-related memories of pattern impressions. These impressions are weighted by the relevance of the experience; this relevance is context dependent and will be a critical estimation issue in knowledge representation for expert systems.

Identity, or more permissively similarity, is the most open problem for representations. This is the ever open problem of knowledge, on the one hand, consolidated and canonized by rules, laws, standards, accepted textbooks, professional procedures, on the other hand, ever revised by new patterns, new knowledge. Identity and similarity are based on frequencies of identical or similar pattern consecutions, originally and still now based on relevant effects on changes related to the events and necessities of our life. The verbal representation of these identities and similarities are concepts. Conceptual thinking developed different levels, as concepts themselves behaved somehow in a similar way. A multilevel hierarchy of concepts was built where sometimes the higher levels forget completely the original factual patterns but the line can be traced back in most cases. This tracing back on the conceptual level ladder is a very important necessity for expert system representations; they provide the control of logical reasoning and frequency hypotheses of metapattern usage. The antinomies of similarities, differences, play a similar role: they provide the bifurcations, other branching, backtracks of the reasoning control.

Now, we are ready with the enumeration of the basic mental and linguistic-philosophical tools of representation. This reference to the origins was important from the practical point of view; the comments on the bottom-up-top-down procedures of conceptual thinking were visible demonstrations of the necessity for progressing in and returning to this theoretical-pragmatic ladder. Unfortunately, these metapatterns were all that was available for representation, a limited scope of representational tools for the unlimited variance of the real world. It was collected in millennia of human development, and only refinements are added, though very delicate ones. This means that representation is always a limited picture, no method can offer the "final" truth, and all the results of representations should be taken care of with continuous critics and checks if they are to work in any new situation.

These tools -- logic, frequency relations of uncertainty, conceptual generalizations leading primarily to abstract algebra, and set theories -- are the frames of mathematical tools for representation, and these mathematical tools are the direct bridges from verbal to computer representation because of their unambiguously defined character. Mathematical formulae are the representations that can be interpreted by a machine in a straightforward way.


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