A Structural Theory of Cognitive Representation and Problem Solving,Contexts, Invariance, and the Knowledge Space
Learning and problem solving depend critically on the structure of internal representations. While many modern data-driven artificial systems achieve strong predictive performance, their learned representations often lack explicit structure for expressing abstraction, invariance, and task-relevant regularities. We propose a minimal structural framework in which representational operations relevant to problem solving, such as context formation, invariance recognition, representative selection, abstraction, and procedural reuse, are made explicit. The central notion is that of a \emph{context}, formalized as a partition of a subset of an underlying state space, which fixes the distinctions, granularity, and form in which a problem can be posed. Within this setting, invariance recognition and representative selection are treated as fundamental representational operations. The framework is realized as a Knowledge Space composed of two coupled graph structures: a Concept Graph that hosts constructed and refined concepts, and a Procedure Graph that encodes typed operations over representations. Together, these structures provide a minimal cognitive-representational algebra for operating on representations without assuming sophisticated inference, learning, control, perception, or motor mechanisms. Using simple illustrative examples and a finite weak-solver demonstration, we show that appropriate representational organization can simplify the form and scope of admissible regularities, even when problem solving is carried out by a fixed and limited solver. The contribution of the paper is structural rather than algorithmic: it identifies representational prerequisites for abstraction, invariance, and procedural reuse in problem solving, and states explicit success and failure conditions for the weak-solver setting.