Artificial Intelligence Systems Based on NTA

This chapter describes NTA capabilities in modeling various kinds of intelligence systems, namely discrete automata, systems for Formal Concept Analysis and for solving Constraint Satisfaction Problems, as well as for unified representation and processing of knowledge expressed in different conventional structures: productions, semantic networks, frames, etc.

NTA Techniques to Decrease Complexity

Matrix properties of NTA objects allow to decrease computational complexity of intellectual procedures as well as to efficiently parallel logical inference algorithms. New structural and statistical classes of CNFs with polynomially identifiable satisfiability properties were discovered in NTA. Consequently, many classes of problems, whose complexity evaluation is theoretically high, e.g., exponential, can in practice be solved in polynomial time, on the average.

Deductive Analysis in NTA

This chapter mostly describes implementation of logical inference by means of NTA. Besides the known logical calculus methods, NTA logical inference procedures can include new algebraic methods for checking correctness of a consequence or for finding corollaries to a given axiom system. Above feasibility of certain substitutions, these inference methods consider inner structure of knowledge to be processed thus providing faster solving of standard logical analysis tasks.

Data Management in NTA Structures

As for making databases more intelligent, NTA can be considered an extension of relational algebra to knowledge processing. Besides, we propose an approach to development of search engines, in particular, question-and-answer teaching systems based on controlled languages and algebraic models for representation and processing of question-and-answer texts.

Theoretical Basics of NTA

The chapter introduces theoretical features of n-tuple algebra developed by the authors as a theoretical generalization of structures and methods applied in intelligence systems. NTA supports formalization of a wide set of logical problems (abductive and modified conclusions, modeling of graphs, semantic networks, expert rules, etc.). This chapter contains main definitions and theorems of NTA. Unlike relational algebra and theory of binary relations, NTA uses Cartesian product of sets rather than sequences of elements (elementary n-tuples) as a basic structure and implements a general theory of n-ary relations. Novelty of our approach is that we developed some new mathematical structures allowing to implement many techniques of semantic and logical analyses; these methods have no analogies in conventional theories.

Probabilistic Modeling

Besides using logic-probabilistic analysis for probabilistic estimation itself, it is often necessary to consider uncertain or underdetermined information of a different nature expressed in the following forms: interval estimates, lists of possible options, hypothetical or possibilistic statements, etc. This chapter discusses methods of logic and logic-probabilistic analysis of uncertain data and knowledge by means of NTA. Using NTA, we can solve direct and inverse problems of probabilistic analysis of logical systems in multidimensional spaces with no limitations on classes of probability distributions.

Analysis of Defeasible Reasoning

This chapter describes implementation of abductive and modified conclusions by means of NTA. The algorithm and rules to form hypotheses for abductive conclusions are proposed. They can be applied not only to NTA objects expressing formulas of propositional calculus, but also to a more general case when attribute domains contain more than two values. Within a specific knowledge system, choosing variables and their values depends on criteria determined by the content of the system. The techniques that we developed simplify generating abductive conclusions for given limitations, for instance, in composition and number of variables. A distinctive feature of the proposed methods is that they are based on the classical foundations of logic, that is, they do not use non-monotonic logic, the logic of defaults, etc., which allowed some violations of laws of Boolean algebra and algebra of sets.

Basic Mathematical Structures

This chapter presents some conventional means of logical analysis. It is necessary to show the scope and features of our approach. In particular, we refer to algebra of sets and algebra of logic (propositional algebra), which belong to the class of Boolean algebras, as well as partially ordered sets and the theory of relations.