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

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.

💡 Research Summary

The paper proposes a comprehensive mathematical framework for automated problem solving, with a focus on robotics and autonomous systems. Knowledge is formalized as “generalized free‑algebra based nets,” which are directed graphs whose nodes correspond to algebraic generators and whose incident in‑ and out‑edges are linked by gluing forms. This representation is richer than traditional term‑tree models because it can capture arbitrary wiring and sharing of substructures.

Problem solving proceeds through a family of Net Rewriting Systems (NRS). Each NRS is typed (e.g., control, perception, actuation) and consists of substitution rules that replace a sub‑net by another net. By applying these rules, a complex task is decomposed into abstract components; each component is processed independently by the appropriate typed NRS. The successive rewrites generate equivalence classes of nets, called congruent classes. Nets belonging to the same class are identified by a congruent quotient algebra, guaranteeing structural consistency across different rewrite sequences.

A central contribution is the notion of parallel transducer paths. Multiple NRSs are arranged in parallel, forming a transducer that can simultaneously explore many rewrite trajectories on the same net class. The candidate solutions produced by these parallel paths are accepted only if the resulting nets belong to the language recognized by a Net Automaton. Thus language‑membership testing provides a rigorous validation step: only nets that are in the automaton’s accepted language are considered legitimate solutions.

The authors introduce Net‑Block Homomorphism, a mapping that compresses or expands nets at the block level. This operation enables higher‑order (multi‑order) solving systems: blocks can be treated as atomic units, allowing the definition of new rewriting systems whose expressive power matches that of normal‑form rewriting but with far greater modularity. Block‑level homomorphisms also support the construction of quotient transducer algebras, which capture the collective behavior of families of parallel transducers.

Finally, the paper addresses self‑evolution of the solving system. By saturating the set of nets with groups of equivalence relations and applying iterative closures over the generated quotient transducer algebras, the system continuously incorporates newly discovered solutions into its knowledge base. The authors prove that the resulting set of solutions remains decidable: there exists an algorithm that can determine, for any net, whether it belongs to the evolving solution space. This guarantees that the system can expand without sacrificing formal tractability.

In summary, the work integrates free‑algebraic net representations, typed net rewriting, congruent quotient algebras, parallel transducer architectures, net‑block homomorphisms, and saturation/closure mechanisms into a unified theory. The resulting framework equips autonomous robots with the ability to decompose complex tasks, explore multiple solution pathways in parallel, validate candidates against formal automata, and iteratively enrich their problem‑solving repertoire, thereby achieving a mathematically grounded, self‑evolving problem‑solving capability.