Graph LSTM for Automated Theorem Proving Search Space Reduction
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Solution Overview
Problem
Current automated theorem provers face challenges when dealing with large collections of axioms, as they tend to get lost in irrelevant regions of the proof search-space, leading to inefficiencies and prolonged CPU time.
Innovation Solution
The method involves generating a directed acyclic graph from a logical formula, assigning initial embeddings, computing node states using feed-forward networks, and performing layer-by-layer embedding updates to narrow the search space, allowing for more focused automatic theorem proving.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If automated theorem provers use large collections of axioms to improve proof completeness, then the ability to prove more theorems is enhanced, but the system gets lost in irrelevant regions of proof search-space leading to increased CPU time and reduced efficiency
Solution Approach 1:
The patent segments the proof search process into distinct phases: (1) generating a directed acyclic graph from the logical formula to represent the proof structure, (2) assigning initial embeddings to nodes and edges to capture local structural information, (3) computing layer-by-layer embedding updates to propagate information through the graph, and (4) using the final embedding to guide the theorem prover. This segmentation allows the system to process large axiom collections systematically rather than exhaustively searching all combinations.
Solution Approach 2:
The patent introduces graph embeddings as an intermediary representation between the logical formula and the theorem proving process. The embedding process acts as a mediator that transforms the structured logical formula into a continuous vector representation that captures global structure and relationships, enabling the theorem prover to efficiently identify relevant axioms without exhaustive search.
2Reliability
If automated theorem provers perform exhaustive search to ensure complete exploration of proof space, then proof completeness is improved, but CPU time increases significantly
Solution Approach 1:
The patent performs preliminary actions by pre-processing the logical formula into a directed acyclic graph structure and computing embedding representations before the actual theorem proving search begins. This pre-computation of structural embeddings provides the theorem prover with prior knowledge about the proof space topology, enabling it to navigate more efficiently without exhaustive exploration.
Solution Approach 2:
The patent replaces the mechanical exhaustive search mechanism with a learning-based embedding approach. Instead of systematically exploring all possible proof paths through mechanical search, the system uses neural network-based graph embeddings to capture global structural patterns and guide the search process, substituting computational brute force with intelligent pattern recognition.
Data Source
AI summary
Generate, from a logical formula, a directed acyclic graph having a plurality of nodes and a plurality of edges. Assign an initial embedding to each mode and edge, to one of a plurality of layers. Compute a plurality of initial node states by using feed-forward networks, and construct cross-dependent embeddings between conjecture node embeddings and premise node embeddings. Topologically sort the DAG with the initial embeddings and node states. Beginning from a lowest rank, compute layer-by-layer embedding updates for each of the plurality of layers until a root is reached. Assign the embedding update for the root node as a final embedding for the DAG. Provide the final embedding for the DAG as input to a machine learning system, and carry out the automatic theorem proving with same.