Imperative Knowledge Representation (PyIRK)

• Imperative Representation of Knowledge (PyIRK) is a Python-centric framework that formalizes domain knowledge using higher-order logic and imperative constructs.
• It integrates human-readable LaTeX and natural language with automated RDF graph generation, enabling interactive semantic querying and modular knowledge bases.
• The framework advances control engineering by merging LLM-assisted natural language processing with explicit, Python-based graph construction and semantic rules.

The Imperative Representation of Knowledge (PyIRK) is a framework for representing domain knowledge in a way that is simultaneously human-readable, machine-interpretable, and highly expressive, especially in contexts where higher-order logical structures and imperative constructs are required. Its design leverages Python as a host language and targets knowledge-intensive domains such as control engineering, facilitating the transformation of natural-language and mathematical descriptions (often in LaTeX) into a formalized RDF knowledge graph that supports both semantic querying and interactive user interfaces (Fiedler et al., 4 Nov 2025).

1. Motivation and Background

The rapid expansion of research output in highly formalized fields such as control engineering has exposed limitations in traditional knowledge representation (KR) approaches. Descriptive logic (including OWL/DL) excels at stating static facts but struggles with representing higher-order logical structures (such as theorems with premises and assertions), interactive processes, and procedural knowledge. While alternatives like Computability Logic (CL) expand expressiveness to include accomplishable tasks and interactive processes (Kwon et al., 2013), they often require bespoke languages with steep learning curves.

PyIRK addresses these gaps by providing:

• A unified, expressive formalism for engineering knowledge compatible with both human editors and automated tools;
• An imperative, Python-centric API, reducing the barrier to entry for engineers already proficient in Python;
• Native support for concepts exceeding classical ontology expressiveness, including higher-order logic and procedural constructs;
• Seamless integration with LLM-driven semiautomated pipelines for bootstrapping knowledge bases directly from LaTeX sources (Fiedler et al., 4 Nov 2025).

2. Formal Structure and Syntax

2.1 Core Constructs

PyIRK programs are written as standard Python scripts using context managers and helper constructors:

• item(uri, label, type=...): Declares a new knowledge graph node with a globally unique URI, human-readable label, and type.
• statement(): Opens a block for specifying RDF-style triples via subject(...), predicate(...), and object(...).
• theorem(uri, label): Bundles higher-order structures (“setup–premise–assertion packages”) via subblocks: premise(), assertion(), proof().

The language is intentionally imperative; statements are evaluated eagerly at runtime, constructing an in-memory RDF graph.

Example

with item("ex:PIDController", label="PID Controller", type="ControlMethod") as pid: pass with statement() as st: st.subject(pid) st.predicate("hasGain") st.object(k_p)

2.2 Semantic Rules

PyIRK’s formal semantics are modeled as follows:

• Item introduction

$$\frac{}{\text{item}(u,\ell,t) \Longrightarrow u \in 𝕀,\ \texttt{rdfs:label}(u,\ell),\ \texttt{rdf:type}(u,t)}$$

where $𝕀$ is the set of item URIs.

• Triple insertion

$$\frac{}{\text{statement}(s,p,o) \Longrightarrow \langle s, p, o \rangle \in G}$$

where $G \subseteq 𝕀 \times ℛ \times (𝕀 \cup 𝕃)$ is the knowledge graph.

• Theorem schema

$$\text{theorem}(τ) = \{\,\text{premise}(τ),\; \text{assertion}(τ)\, \}$$

This design captures the imperative, procedural encoding of both factual and higher-order knowledge.

3. LLM-Supported Knowledge Generation Pipeline

The PyIRK methodology integrates LLMs in a semiautomated pipeline for converting LaTeX mathematical text into formal knowledge graphs (Fiedler et al., 4 Nov 2025):

Pipeline Steps

Step	Description	Tool/API
0	Preprocess LaTeX, inject snippet delimiters around logical chunks	Custom scripts
1a	Extract Formal Natural Language (FNL) using LLMs (e.g., Google Gemini)	LLM + Markdown prompt
1b	Manual review/correction (10–20% of lines edited)	Human-in-the-loop
2	Algorithmic conversion: FNL lines parsed into PyIRK Python code	Parser/Emitter
3a	Execute PyIRK code: populate RDF graph in memory	PyIRK API
3b	Serialize graph (JSON-LD, Turtle), load into SPARQL endpoint	RDF tools
3c	Inject semantic tooltips into HTML documents	JavaScript/HTML

FNL extraction via the LLM follows an explicit prompt grammar to ensure bullet-point, subject–predicate–object statements, which are then parsed programmatically.

Illustrative Example

Given a LaTeX definition:

\begin{definition}[Orthogonal Complement] Let U be a subspace of a Hilbert space H. The orthogonal complement %%%%5%%%% \end{definition}

Corresponding FNL:

• Definition “OrthogonalComplement” has argument U.
• OrthogonalComplement is subset of H.
• OrthogonalComplement definedBy “x∈H | ∀u∈U: ⟨x,u⟩=0”.

PyIRK code:

with item("ex:OrthogonalComplement", label="Orthogonal Complement", type="Definition") as oc: pass with statement() as s1: s1.subject(oc) s1.predicate("hasArgument") s1.object(U) with statement() as s2: s2.subject(oc) s2.predicate("isSubsetOf") s2.object(H) with statement() as s3: s3.subject(oc) s3.predicate("definedBy") s3.object("x∈H | ∀u∈U: ⟨x,u⟩=0")

4. Expressiveness: Imperative and Higher-Order Knowledge Representation

PyIRK captures higher-order structures that are not directly expressible in classical description logics. The explicit imperative design (using Python context managers and runtime objects) supports:

• Structured representation of theorems as packages of premise–assertion–proof;
• Parameterized and modular definitions;
• Immediate evaluation and graph materialization upon execution.

Compared to classical fact-based schemes, this approach is analogous to the expressiveness gains obtained by Computability Logic in general knowledge representation (Kwon et al., 2013). In CL, a fact is a trivial task and a capability is an accomplishable task, a paradigm that PyIRK implicitly supports via higher-order graph encodings and imperative semantics.

A plausible implication is that, in PyIRK, an engineer can represent not only static ontological fact bases but also executable specifications, including procedures and interactive protocols, encoded as Python routines constructing graph nodes and higher-order relations at runtime.

5. Interactive Semantic Layer and User Experience

PyIRK supports the generation of an “interactive semantic layer” for enhanced document-based knowledge transfer (Fiedler et al., 4 Nov 2025):

1. Source LaTeX material is converted to HTML (e.g., via pandoc).
2. Each annotated token in mathematical content is wrapped in an HTML <span> with a data-uri attribute that links to the corresponding RDF graph node.
3. On hover, a JavaScript component reads the URI, queries a precomputed JSON store, and displays definition snippets, relationships, or “See also” links (triggering SPARQL endpoint queries or redirecting to ontology browsers).
4. The user interface supports toggling the semantic mode or interacting with tooltips to minimize reading disruption and promote rapid access to core definitions.

In practice, this approach led to ≈700 tooltips being generated over eight pages of a control-engineering monograph, with an average of ~15% manual correction effort in FNL extraction (Fiedler et al., 4 Nov 2025). Reading speed was subjectively improved due to faster definition lookup and reduced need for context switching.

6. Evaluation, Limitations, and Future Prospects

Significant reported advantages include:

• Version-controlled, explicit knowledge bases with stable URIs and direct traceability to source text;
• Expressive power for representing higher-order constructs and procedural knowledge (theorems, interactive definitions) not possible in OWL/DL;
• Efficient semantic querying via SPARQL as opposed to unstructured LLM prompts;
• Seamless integration into established LaTeX-based academic workflows.

Current limitations identified are:

• Manual correction of LLM-extracted FNL statements is a persistent bottleneck (10–20% rate);
• The pipeline requires LaTeX source—PDF-only workflows without embedded LaTeX remain unsupported;
• User studies quantifying learning benefits of the interactive semantic layer have not yet been conducted.

Proposed directions include increased LLM supervision in FNL extraction to reduce the need for human correction, robust PDF-to-FNL conversion via OCR and parsing, and comprehensive user studies. A full-featured interactive knowledge assistant—capable of dereferencing URIs, tracing inference chains, and answering control-theoretic queries—is projected as a future development.

7. Connections to Task-Based Knowledge and Computability Logic

PyIRK’s imperative representation aligns with the broader shift in KR from static facts to representations of accomplished and accomplishable tasks (Kwon et al., 2013). In Computability Logic, knowledge is interpreted via games/tasks, and “truth” corresponds to an agent's ability to execute a winning strategy. The PyIRK paradigm—executable, compositional, and higher-order—allows knowledge bases to mirror computational capabilities, analogous to CL formulas.

This suggests that PyIRK can serve both as a pragmatic framework for semantic knowledge capture in engineering domains and as a concrete realization of the paradigm shift advocated by task logic and Computability Logic. The main trade-off, as with CL-based systems, lies in the complexity and performance overhead of operationally interpreting such expressive representations in real-world settings. Robust tool support and Python-level interpreters will be key to the practical adoption of this approach.

---

In summary, the Imperative Representation of Knowledge (PyIRK) provides a Python-driven, highly expressive framework for formalizing knowledge in domains characterized by complex mathematical and procedural content. It supports LLM-assisted construction of semantic layers, higher-order logic, and seamless integration with both human-centric workflows and machine-oriented querying, positioning itself as a versatile tool for next-generation knowledge bases in control engineering and related fields (Fiedler et al., 4 Nov 2025, Kwon et al., 2013).