Defining Logic Structures In Functional Spaces
Author(s)
D. Sitnikov, N. Matski & O. Ryabov
Abstract
Many practical problems of geometric information representation and pattern
recognition require specific methods and tools for describing complex functions
and geometric objects in a way that allows us to use such descriptions for
making effective mathematical transformations and logic inferences. Since in
real world situations we rarely encounter ideal objects that can be described with
the help of one elementary function it is important to develop methods and
models for integrating geometric information. There are some original
approaches to describing complicated geometric structures, one of which is a
method based on so-called R-functions. These functions allow complicated
geometric objects to be described analytically, which gives an opportunity to
operate with complex objects with the help of a single real function. The concept
of R-functions is quite simple but the range of practical applications of this
theory is very broad. In this paper we have suggested a generalization of
R-functions, which we call Rp-functions. Any R-function can be associated with
a Boolean function and it is always possible to construct an R-function
corresponding to a given Boolean one. Similarly, any Rp-function can be
associated with a finite predicate and it is always possible to construct an
Rp-function corresponding to a given predicate. If R-functions allow us to use
Boolean logic for describing complex objects, Rp-functions provide a possibility
of using predicate logic, which is more rich and general than Boolean one. Like
R-functions, Rp-functions allow the use of knowledge on what logic rules have
been used for constructing a complex object to build a function which is positive
inside the given domain and negative outside it. Rp-functions also seem to be
interesting by themselves as they allow discovery of logic properties of quite a
large class of real functions.
Keywords
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