Ontological Mapping of AI Algorithms

• Ontological mapping of AI algorithms is a formal framework that systematically organizes AI models, cognitive biases, and architectures for enhanced explainability and integration.
• It employs modular, multi-layered abstractions—ranging from physical components to agency—to structure AI systems using methods like UML, category theory, and vector ontologies.
• Evaluation taxonomies and lifecycle mappings within this framework ensure robust benchmarking, system reliability, and accountable AI development.

Ontological mapping of AI algorithms refers to the systematic representation and organization of algorithms, models, architectures, and related concepts within a formal or semi-formal framework, permitting rigorous reasoning, interoperation, and extension across both theoretical and practical domains. Such mapping is crucial for enabling explainability, composability, benchmarking, semantic annotation, and ultimately, the evolution of artificial intelligence systems in research and deployment.

1. Foundational Mapping Principles and Cognitive Bias Integration

Ontological mapping in the context of algorithmic intelligence begins with the tension between universal models (e.g., AIXI based on Solomonoff induction) and non-universal, pragmatic cognitive architectures. The synthesis proposed in "Cognitive Bias for Universal Algorithmic Intelligence" (Potapov et al., 2012) considers cognitive functions (perception, planning, memory, knowledge representation, theory of mind, language) as formal metaheuristics—mathematically encoded biases (priors and search heuristics) that steer universal induction towards tractable, environment-typical regularities.

Formally, the universal prior of Solomonoff induction,

$P(x) = \sum_{p} 2^{-l(p)}$

can be extended with a bias weight $H(p)$: $P_{bias}(x) = \sum_{p} 2^{-l(p)} \cdot H(p)$ where $H(p)$ encodes heuristic structure such as perceptual regularities or planning shortcuts, improving efficiency whilst maintaining universality. This illustrates the ontological stratification—cognitive algorithms as primitives or "bias modules" augment the base universal model, defining a continuum from purely universal (inefficient) agents to cognitive, bias-enhanced (efficient) agents.

2. Formal Models, UML-Based Mapping, and Modular Architecture

Ontological mapping in applied systems is exemplified by formal software engineering methodologies for intelligent ontological processing systems. As seen in the Instrumental Complex for Ontological Engineering Purpose (Palagin et al., 2018), module-level mathematical models and functional–component UML breakdowns provide the skeleton:

• Module sum: $\text{System} = \sum_{i=1}^{n} P_{i}$ (where $P_{i}$ are programming modules)
• Integration mapping: $$S_{\text{System}} : S_{\text{modules}} \to F_{\text{overall}}$$

Objects (entities, terms, concepts, relationships) extracted from unstructured data are mapped via dynamic/static/physical/component models and integrity predicates. Three-tier architectures (presentation, logic, data) separate concerns, enabling scalable and secure integration of AI-driven ontological mapping workflows—linguistic analysis, extraction, ontology construction, validation, and database storage.

3. Algebraic and Category-Theoretic Frameworks

Mathematical ontologies for mapping algorithms rely on algebraic and categorical structures. Class algebra and calculus (Buehrer, 2018) encode classes with intent (logical description) and extent (set of instances), organized by IS-A hierarchies and evaluated via Galois connections:

• Eval–eval$^{-1}$ connection links Boolean algebras of (super/sub)classes and biclique relationships.
• Residual operators, e.g., $$x (R \setminus S) y = \bigvee_{z \in X} x R z > z S y$$ capture binary and causal relations.

Categorical approaches (Guo, 3 Feb 2025) treat machine learning systems as objects in categories with morphisms (algebraic operations, binary relations) and transformations preserving system structure:

• Adjunctions $(F, G, \varphi)$ define optimal transformation loops, facilitating problem-solving via universal properties.
• Yoneda embedding provides full and faithful mapping from elements to their interaction sets.

These frameworks support both explicit symbolic mappings and the rigorous comparison, transformation, and unification of diverse algorithmic structures.

4. Layered Abstraction and Multi-Level Ontologies

Ontological mapping spans abstraction hierarchies. The five-layer framework (Serb et al., 2019):

1. Physical: elementary phenomena (electrons, transistors, memristors)
2. Functional: circuits, logic gates, artificial neurons
3. Computational: signal processing, neural networks, learning procedures
4. Semantic: symbol manipulation, representation, planning, reasoning
5. Agency: goal-seeking, behavior, ethical assurance

Mapping AI algorithms involves tracking the flow and conversion of representations and uncertainties across these layers, with each level negotiating complexity-performance and control-complexity trade-offs. This multidimensional stratification enables targeted innovation, safety assurance, and modularity.

5. Conceptual Modeling, Taxonomies, and Organizational Structures

Systematic mapping between conceptual modeling languages (DSML, UML, Petri nets, ontologies) and AI techniques (ML, NLP, multi-agent systems) is essential for mutual explainability and automation (Bork et al., 2023). Taxonomies classify modeling purposes—representation, analysis, (re-)design, code generation—and cross-reference these with AI domains (reasoning, planning, learning, perception, communication, integration/interaction).

LaTeX-formalized search queries and tables, e.g.,

$$Q = \Bigl(\bigvee_{i} CM_{i}\Bigr) \land \Bigl(\bigvee_{j} AI_{j}\Bigr)$$

serve to specify boundaries and intersections in mapping studies. Such intertwined taxonomies build the ontological graph needed for scalable knowledge representation and practical system integration.

6. Vector Spaces as Truly Formal Ontologies

A rigorous formalism rooted in Husserl’s a priori criteria is embodied by vector ontologies (Rothenfusser, 20 May 2025). Here, basis vectors index quality dimensions; objects exist as linear combinations, and their interrelations—pattern of existence, causality (linear dependence), convex regions (mereology)—are mathematically explicit:

• Axioms (commutativity, associativity, identity, inverse, distributivity) are chosen a priori

$\forall u,v \in V_{ont},\ u+v=v+u$

• Object membership, relationships, and reconstruction distances support existence and ontology mapping

This formalism is manifest in the internal hidden-state representations of neural networks, suggesting that advanced AI architectures already deploy implicit vector ontologies for entity mapping, reconstruction, and interpretation.

7. Evaluation Ontologies and Lifecycle Mapping

Comprehensive framework mapping harmonizes evaluation terminology, taxonomies, and lifecycle stages (Xia et al., 8 Apr 2024):

• Terminology ensures interoperability (accuracy, correctness, benchmarking, system evaluation)
• Taxonomy separates component-level (data, model, non-AI) and system-level (usability, reliability, capability) assessments
• Lifecycle mapping links evaluation phases to stakeholder responsibilities via function $E: S \times R \to \text{Evaluation}_\text{Types}$

Such mapping ensures robust, transparent, and accountable development, deployment, and ongoing operation of AI systems.

8. Synthesis and Future Directions

Ontological mapping of AI algorithms is inherently multidisciplinary, bridging universal mathematical frameworks, cognitive bias modules, formal and modular software architectures, conceptual modeling taxonomies, layered abstractions, vector-space formalism, and evaluation supply chains. The convergence of these approaches supports explainability, composability, interoperability, and principled development, facilitating the evolution from narrow to general intelligence and ensuring methodological robustness in both research and application. Future research is oriented toward expanding tensorial ontologies, extending category-theoretic mappings, integrating multi-layered architectures, and empirically aligning machine and human conceptual worlds.