The Mathematical Abstraction Theory, The Fundamentals for Knowledge Representation and Self-Evolving Autonomous Problem Solving Systems

TL;DR

This paper develops a mathematical framework using generalized free algebra nets for automated problem solving in robotics, enabling self-evolving systems with expanded solving capabilities and formal solution generation methods.

Contribution

It introduces a novel algebraic approach with net rewriting systems, quotient algebras, and homomorphisms for universal abstraction and self-evolution in problem solving systems.

Findings

Construction of solution algorithms parallel to candidate solutions.

Establishment of quotient algebras for class-based problem solving.

Expansion of solving power through new solutions and self-evolution mechanisms.

Abstract

The intention of the present study is to establish the mathematical fundamentals for automated problem solving essentially targeted for robotics by approaching the task universal algebraically introducing knowledge as realizations of generalized free algebra based nets, graphs with gluing forms connecting in- and out-edges to nodes. Nets are caused to undergo transformations in conceptual level by type wise differentiated intervening net rewriting systems dispersing problems to abstract parts, matching being determined by substitution relations. Achieved sets of conceptual nets constitute congruent classes. New results are obtained within construction of problem solving systems where solution algorithms are derived parallel with other candidates applied to the same net classes. By applying parallel transducer paths consisting of net rewriting systems to net classes congruent quotient algebras are established and the manifested class rewriting comprises all solution candidates whenever produced nets are in anticipated languages liable to acceptance of net automata. Furthermore new solutions will be added to the set of already known ones thus expanding the solving power in the forthcoming. Moreover special attention is set on universal abstraction, thereof generation by net block homomorphism, consequently multiple order solving systems and the overall decidability of the set of the solutions. By overlapping presentation of nets new abstraction relation among nets is formulated alongside with consequent alphabetical net block renetting system proportional to normal forms of renetting systems regarding the operational power. A new structure in self-evolving problem solving is established via saturation by groups of equivalence relations and iterative closures of generated quotient transducer algebras over the whole evolution.