Dedicated Geometry Solver

• Dedicated geometry solvers are computational systems that use formal languages and controlled natural language interfaces to unambiguously specify and solve geometric problems.
• They utilize graph-based representations such as ontological graphs and dependency lattices to model geometric entities and their interrelations for robust querying.
• These systems integrate algorithmic reasoning with dynamic geometry platforms to support educational, proof discovery, and automated theorem proving applications.

A dedicated geometry solver refers to a computational system or suite of algorithms explicitly designed to process, reason about, and solve geometric problems using formal representations, symbolic deduction, constraint solving, or a combination of algorithmic and declarative approaches. These solvers are distinguished by their capability to handle geometric entities (points, lines, circles, etc.) and their interrelations directly, often supporting querying, construction, proof, and retrieval tasks at scale. They play a central role in geometric knowledge management, educational tools, computational geometry, and mathematical software infrastructures.

1. Formal Query Languages and Declarative Interfaces

Dedicated geometry solvers often employ a controlled natural language (CNL) or domain-specific language (DSL) to articulate queries and constructions in a manner that is syntactically precise yet close to mathematical parlance. The rationale is to constrain the expressiveness just enough to avoid ambiguity, making semantic parsing and formal translation tractable.

A compelling example is provided by the controlled language described in (Haralambous et al., 2014): queries can specify geometric constructions such as

“D, E, F are midpoints of A–C, A–B, B–C ; C–E intersects B–D at G ; draw A–C, A–F, A–B, B–C, B–D, C–E.”

Such formulations are interpreted unambiguously by the solver and compiled into an internal formalism. This systematic formalization, supported by grammars defined in Backus–Naur form, enables automated reasoning engines to accept complex geometric queries from human users or external systems (such as dynamic geometry systems, DGS) and acts as a critical frontend for geometric knowledge management.

2. Ontological Graphs and Dependency Lattices

An essential feature of modern dedicated geometry solvers is the utilization of graph-based internal representations for geometric figures and constructions. In the approach of (Haralambous et al., 2014), geometric entities—points, segments, lines, circles, angles—are instantiated as nodes, and their relationships (belongingness, parallelism, incidence, ratios) are represented as labeled edges in an ontological graph.

Procedural constructions, on the other hand, are encoded as dependency lattices, where each construction step induces dependencies (e.g., the definition of a midpoint creates edges from the defining points to the new midpoint, labeled accordingly). The complete dependency lattice forms a directed acyclic graph with a global source and global sink node, reflecting construction flows and supporting systematic reduction strategies.

Graphical representations—used both for reasoning and for visualization—explicitly factor out construction order and idiosyncratic naming, focusing instead on invariant relationships crucial for geometric matching and search.

3. Algorithms for Querying and Search

The core computational engine of a dedicated geometry solver comprises reasoning, matching, and search algorithms operating on these formal representations. The process typically includes:

1. Parsing Input: Accepting input from a DGS or CNL, building the ontological graph.
2. Inference: Applying geometric inference rules (e.g., transitivity of parallelism) to enrich the representation.
3. Graph Conversion: Translating the query representation into a database or search engine language (e.g., Cypher for Neo4j).
4. Subgraph Matching: Performing invariant subgraph search to detect all figures in the corpus sharing the query’s geometric relationships.
5. Dependency-Based Reduction: If no exact match is found, employing dependency lattice reduction by iteratively removing terminal parts of the construction, thereby weakening the query constraints until a match in the repository is located.

The following pseudocode excerpt (condensed from Algorithm 1 in (Haralambous et al., 2014)) summarizes the search loop:

Input: Corpus of ontological graphs, Query (from DGS or CNL) Process: Build ontological graph of Query Apply inference enrichments Convert to Cypher query; query Neo4j If no match: Build dependency lattice Repeat: Remove bottom node/edge from lattice Convert reduced graph to Cypher Query database Until match found or lattice exhausted

This approach enables handling both strict and partial matches and is robust to variations in construction order or redundant definitions.

4. Applications and Integration in Dynamic Geometry

Dedicated geometry solvers have significant implications for educational software, DGS platforms, and future mechanized mathematics environments:

• Educational Integration: Teachers and students can query repositories of geometric constructions (e.g., TGTP, Intergeo) not just by keywords but by specifying geometric properties or partial constructions, enabling discovery and exploration far beyond static digital libraries.
• Assisted Proof Discovery: Mechanized mathematics assistants can leverage these solvers to check, complete, or supply alternative constructions and proofs, filtering results for relevance based on invariant geometric patterns.
• Automatic Theorem Proving: The solver’s formal backbone supports integration with geometry automated theorem provers, providing a consistent base for automated deduction and verification tasks.
• Figure Search: Unlike string matching or metadata searching, ontological subgraph queries retrieve all figures sharing essential properties, regardless of superficial variation or auxiliary construction details.

5. Comparison to Prior and Contemporary Systems

Legacy systems such as TGTP and Intergeo provide valuable geometric repositories but lack sophisticated geometric querying mechanisms. They focus on cataloguing and display, with rudimentary metadata-based retrieval that cannot exploit the deep geometric structure.

By contrast, the mechanisms advanced in (Haralambous et al., 2014) provide:

• Semantic Search: Relying on invariant graph structure rather than string patterns or construction order.
• Flexible, User-Friendly Input: Via CNLs, supporting both non-expert and expert users.
• Robustness to Redundancy: With query reduction through dependency lattices, ensuring that extraneous definitions do not prevent locating relevant constructions.
• Extensibility: Easy integration with corpus extension, DGS frontends, and proof assistants.

A summary of distinguishing features is provided below:

System	Query Method	Invariant Subgraph Search	Query Reduction
TGTP, Intergeo	Keyword/Metadata	No	No
Dedicated Solver (Haralambous et al., 2014)	CNL/DGS to Ontological Graph	Yes	Yes

6. Broader Implications and Extensions

Dedicated geometry solvers exemplify a broader shift toward relation-based modeling and retrieval in mathematical knowledge management. They enable:

• Corpus-Scale Geometric Data Mining: High-level mining for geometric patterns, conjecture suggestion, and educational content curation.
• Unified Mathematical Assistants: Integration of controlled language query, formal knowledge representation, and scalable search as core capabilities of next-generation mathematical software.
• Research in Representation Theory: These formalisms motivate further exploration into canonical representations (e.g., graph canonization relative to geometric invariance) and efficient subgraph isomorphism algorithms in mathematical domains.

Although current approaches still rely on predefined grammars and corpus-specific graph schemes, they set a paradigm for combining symbolic computation, database technology, and controlled language processing to create versatile, rigorously grounded geometry solvers. The extension to full integration with proof engines, as well as handling a wider range of geometric relations and higher-dimensional constructs, remains an open direction for future research.