Abstract
Information technologies for analysis and processing heterogeneous data often face the necessity to unify representation of such data. To solve this problem, it seems reasonable to search for a universal structure that would allow for reducing different formats of data and knowledge to a single mathematical model with unitized manipulation methods. The concept of relation looks very prospective in this sense. So, with a view to developing a general theory of relations, the authors propose n-tuple algebra (NTA) developed as a theoretical generalization of structures and methods applicable in intelligence systems. NTA allows for formalizing a wide set of logical problems (deductive, abductive and modified reasoning, modeling uncertainties and so on).
TopN-Tuple Algebra: Basics And Features
N-tuple algebra is a mathematical system to deal with arbitrary n-ary relations. In NTA, such relations can be expressed as four types of structures called NTA objects. Every NTA object is immersed into a certain space of attributes. Names of NTA objects contain an identifier followed by a sequence of attributes names in square brackets; these attributes determine the relation diagram in which the NTA object is defined. For example, R[XYZ] denotes an NTA object defined within the space of attributes
Key Terms in this Chapter
D-System: A set of homotypic D - n -tuples equal to the intersection of these D - n -tuples.
C-n-Tuple: An n -tuple of components defined in a certain relation diagram; domain of each component is a subset of the domain of the corresponding attribute.
Collisions: Situations occurring during defeasible reasoning when some new knowledge (hypothesis) is inputted. Such situations can be recognized as violations of some formally expressed rules and/or limitations which control consistency and meaning content of a logical system.
D-n-Tuple: An n -tuple of components enclosed in reversed square brackets and equal to a diagonal C -system with the same diagonal components.
N-Tuple Algebra: An algebraic system whose support is an arbitrary set of n -ary relations expressed by specific structures, namely, C - n -tuple, C -system, D - n -tuple, and D -system. These structures provide a compact expression for sets of elementary n -tuples.
C-System: A set of homotypic C - n -tuples that are denoted as a matrix in square brackets and equal to the intersection of these C - n -tuples. The rows of this matrix are C - n -tuples.
Generalized Operations and Relations: They differ from similar operations and relations of algebra of sets by the only feature: NTA objects (operands) are reduced to the same relation diagram before executing these operations or checking the relations.