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Arkadij Zakrevskij (National Academy of Science, Belarus)

Copyright: © 2013
|Pages: 12

DOI: 10.4018/978-1-4666-1900-5.ch001

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TopOne of the most important problems of artificial intelligence is the problem of pattern recognition (Bongard, 1970; Hunt, 1975). To solve it, various formal methods were applied, usually based on the theory of Boolean functions (Triantaphyllou, 1994; Zakrevskij, 1988). However, they become insufficient when dealing with objects described in terms of multi-valued attributes, so other means should be involved in this case, finite predicates for example (Zakrevskij, 1993).

The finite predicates are two-valued functions, which arguments are variables with restricted number of values. Denote these variables by *x*_{1}, *x*_{2}, …, *x _{n}.* Let them receive values accordingly from finite sets

When solving practical problems related to the usage of finite predicates, it is useful to represent the latter whenever possible in a more compact form. Here it is possible to use experience of the theory of Boolean functions, developed chiefly for the case when the considered functions are represented in the disjunctive normal form (DNF). The most efficient methods of minimization of Boolean functions and solution of logical equations are designed just for that form. It is reasonable to extend these methods onto finite predicates.

According to tradition, let us assume that an *elementary conjunction k* represents the characteristic function of some interval *I* of space *M,* and this interval is defined as a direct product of non-empty subsets *α _{i}*, taken by one from every

That means that an elementary conjunction *k* is defined as a conjunction of several one-argument predicates *x _{i}*

The multiplicands, for which *α _{i}*

Note, that in the simplest case, when all arguments become two-valued, this definition coincides with the definition of elementary conjunction in Boolean algebra.

Similarly, we shall define an *elementary disjunction d* as a disjunction of one-argument predicates distinct from true:

If *α _{i}*

The disjunctive and conjunctive normal forms are defined as usual: DNF is a disjunction of elementary conjunctions, and CNF is a conjunction of elementary disjunctions.

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