Formalized Knowledge Representations

Updated 7 September 2025

Formalized knowledge representations are mathematically and logically grounded frameworks that encode and structure knowledge for unambiguous inference.
They underpin applications in legal reasoning, digital forensics, robotics, and engineering by unifying symbolic inference with probabilistic decision-making.
Recent theoretical advances demonstrate that universal KR formalisms are recursively isomorphic, ensuring consistent query answering across diverse systems.

Formalized knowledge representations (KR) are precise, mathematically and logically grounded methods for encoding, structuring, and manipulating knowledge in a form suitable for unambiguous reasoning and automated processing. They provide not only the substrate for symbolic inference and decision-making but also serve as the backbone for semantic interoperability across diverse systems and domains. This article surveys key theoretical frameworks, formal methodologies, and practical applications of formalized KR, highlighting mechanisms for context sensitivity, expressiveness, integration of uncertainty, and the rigorous treatment of representation heterogeneity.

1. Foundational Principles and Formalisms

Formalized KR frameworks are typically defined by their syntactic primitives, semantic commitments, and the nature of the inferences they support. The expressiveness, computational properties, and interoperability of a KR formalism are governed by its foundational choices.

Set-theoretic approaches (Zhou, 2016) take naive set theory as a universal substrate: knowledge is modeled by a triple $\langle \mathcal{I}, \mathcal{C}, \mathcal{O} \rangle$ , with $\mathcal{I}$ as individuals, $\mathcal{C} \subseteq 2^{\mathcal{I}}$ as concepts (interpreted as sets), and $\mathcal{O}$ as operators (functions). All assertions reduce to equality between terms, e.g., $a = b$ , with richer constructs (membership, multi-assertion schemas, logical operators, quantifiers) derived directly from set-theoretic operations: $\lnot(a = b) ::= \{a\} \cap \{b\} = \emptyset, \qquad \forall(C,\,A(C)) ::= C|A(C) = C$ This yields a uniform architecture allowing additivity (through definitions) and expressiveness up to, and including, full first-order logic, but at the cost of potential undecidability.

Network-based and categorical approaches provide alternative structural frameworks. In the context-sensitive network formalism (Leong, 2013), each "concept" is a contextual tuple $(a\#b)$ , enabling representation of knowledge that is valid only in specific contexts. Three relation types—categorical (inheritance, part-of, etc.), uncertain (causal/probabilistic), and context (CXT)—organize semantics. Chaining is associative, e.g.,

$(Color\#Royal~Elephant\#Thailand)$

with efficient inference modeled as path problems in labeled directed graphs.

Relational ologs (Patterson, 2017) leverage bicategories of relations to encode knowledge. Concepts correspond to objects (types), and relations to morphisms, with a deeply integrated type system: $\Cl(\Lang(\mathcal{B})) \simeq \mathcal{B}$ This equivalence of the classifying category of the theory and the bicategory of relations ensures all diagrammatic, algebraic, and logical properties can be unified.

2. Ontologies, Semantic Models, and Expressive Scope

Ontologies formalize shared conceptualizations, providing structures for domain knowledge via logic-based or graph-based taxonomies, constraints, and semantic relations.

In mathematical knowledge management (Elizarov et al., 2014), ontologies are formally stated as logical theories $O_K$ whose models coincide with the intended models under a specified conceptualization. Key structures include:

Extensional structure: $S = (D, R)$ , with $D$ the domain and $R$ the set of relations.
Intensional (conceptual) structure: $\rho^n : W \rightarrow 2^{D^n}$ , mapping world states $W$ to subsets of $D^n$ .
Ontology: a logical theory characterized by the set of "intended models" matching ontological commitments.

OntoMath $^{PRO}$ (as in (Elizarov et al., 2014)) operationalizes these abstractions for mathematics, supporting multilingual, hierarchically-structured coverage and semantic search, with applications to formula search and educational assessment.

Stratified approaches (Bagchi, 14 Apr 2025) explicitly layer representation across the concept, language, knowledge, and data levels. The Universal Knowledge Core (UKC) architecture separates supralingual concepts, natural language lexicons (with synsets and glosses), and domain-specific namespaces. Language teleontologies provide rigorous mappings from vocabulary to unique concept identifiers, while knowledge teleontologies axiomatize types, relations, and properties, driving semantic integration in data-centric projects (e.g., DataScientia, JIDEP).

3. Integration of Symbolic and Uncertain Knowledge

Many real-world applications demand that deterministic, taxonomic knowledge be integrated with models of uncertainty, context, and temporal or causal dependencies.

The network formalism (Leong, 2013) integrates categorical (structural) and uncertain (behavioral) knowledge using two orthogonal relation types. Categorical relations (type hierarchies, part-whole, equivalence) enable inheritance and exception handling; uncertain relations encode probabilistic/temporal interactions (with values like "cause", "inhibit", "co-occurrence"). The context (CXT) operator delimits where and when structural knowledge holds, enabling representations such as "Royal elephants in Thailand are white" vs. "Elephants are gray in general".

Probabilistic graphical models and logic-based rule systems are unified in technical systems (Scharei et al., 2020), where frameworks such as Bayesian networks, Markov Logic Networks (MLNs), and deep neural networks coexist with logic- and ontology-based components. This enables the handling of uncertainty and learning in robotics, planning, and complex perception tasks.

4. Knowledge Patterns, Modularization, and Reuse

Reusable theory schemata—knowledge patterns (Clark et al., 2020)—are abstract, modular first-order theories encapsulating recurring modeling structures. Rather than inheritance, these patterns are instantiated via explicit morphisms: mappings from the abstract pattern's vocabulary to domain-specific predicate and constant symbols. For example, a generic "container" pattern is instantiated for both RAM and expansion slots via separate symbol mappings.

A knowledge representation ( $\Gamma$ ) maps theories to knowledge bases, and patterns can be morphed into different domains with syntactic substitution. Computationally, this yields efficient reuse, modular design, and clarification of modeling decisions, with the caveat that runtime dynamic selection and conflict handling are manual.

5. Representation Heterogeneity and Unification

Complex technical systems and data integration scenarios confront heterogeneity at the level of concepts, language, properties, and data formats. Stratified methodologies (Bagchi, 14 Apr 2025) and modular ontological frameworks address this by:

Factoring out language- and concept-level variation via unique identifiers (UKC GIDs), synsets, and controlled vocabularies.
Mitigating schema mismatches through language and knowledge teleontologies that formally declare entity types, properties, domains, and ranges.
Enforcing iterative reuse, standardization, and cross-domain harmonization via metadata catalogs (as exemplified by the LiveKnowledge infrastructure).

This modularization supports scalable construction of semantic resources, multilingual interoperability, and iterative model evolution.

6. Computational Equivalence and Theoretical Unification

A prominent theoretical result (Zhang et al., 16 Dec 2024) is that, under general assumptions, all universal knowledge representation formalisms are recursively isomorphic. That is, for the canonical mapping

$\Gamma = \Gamma_0 \circ p$

for some recursive bijection $p$ , expressive formalisms (declarative, procedural, symbolic, connectionist) can represent and answer the same queries up to efficient offline compilation. Universal KRFs (knowledge representation formalisms) map all recursively enumerable knowledge bases and support uniform query answering. This theoretical unification has several implications:

The "declarative vs. procedural" and "symbolic vs. connectionist" disputes are, at the formal level, questions of representation and practical engineering, not of inherent expressiveness.
As long as padding and effective intertranslatability are present, different KRFs offer identical theoretical power for reasoning and knowledge management, subject to computability and resource constraints.

7. Domain-Specific Applications and Case Studies

Formalized KR has profound domain-specific impacts:

In legal reasoning, modular, multi-agent frameworks (Sadowski et al., 31 Aug 2025) decompose statutes into ontological schemas (TBox) and apply rule-based inference on case facts (ABox), enabling verifiable, explainable legal decision-making with performance benefits over end-to-end neural solutions.
Digital forensics (Chabot et al., 2019) employs formal models integrating events, entities, and temporal logic (e.g., Allen’s algebra) to reconstruct timelines and correlate evidence, prioritizing reproducibility and semantic rigor for legal admissibility.
Design knowledge in engineering (Buzon et al., 2018) is captured with standards-based formalisms (UML, XML, ontologies), enabling machine-interpretable, context-aware exchange and validation, as illustrated by structured models of collaborative design objects.
Robotics and AI (Scharei et al., 2020) leverage a taxonomy of KR techniques for perception, planning, reasoning, and learning, integrating logic, probabilistic models, deep learning, and semantic graphs for flexible and adaptive technical systems.

In conclusion, formalized knowledge representations span a diverse set of logical, set-theoretical, categorical, and ontological frameworks. These frameworks support powerful, expressive, and interoperable encoding of knowledge, accommodating context, uncertainty, and domain heterogeneity. Recent theoretical advances establish deep equivalences among universal formalisms, while practical methodologies address modularization, reuse, and semantic integration, making formalized KR essential to domains requiring explanation, verification, and sophisticated inference.