We Built a Language Model 14,000,000x Smaller Than GPT3 and Formally Verified It

3 months ago 3

Extremely lightweight universal grammar implementation with provable recursion

A mathematically rigorous, recursively complete language model that fits in under 50kB with zero runtime dependencies. Built on Chomsky's Minimalist Grammar theory with formal verification and empirical validation.

📊 Mind-Blowing Size Comparison:

Our Model: 0.05 MB (50 KB)
GPT-3: 700,000 MB
Ratio: We're 14,000,000x smaller!

Yet we still provide provable recursion, next-token prediction, and formal verification. See how we did it →

✨ What Makes This Special

🧮 Mathematically Proven: Formal proofs of recursive capability using Coq
⚡ Ultra-Lightweight: Complete implementation under 50kB binary
🔬 Scientifically Validated: Tested with standard linguistic benchmark suites
🏗️ Universal Grammar: Based on Chomsky's Minimalist Grammar theory
♾️ Provably Recursive: Generates a^n b^n patterns, proving non-regularity
🤖 Probabilistic Language Model: Next-token prediction with formal guarantees

Rust 1.70+ (for compilation)
Git (for cloning)
Python 3.8+ (optional, for probabilistic language model)

# Clone the repository git clone https://github.com/user/atomic-lang-model.git cd atomic-lang-model/atomic-lang-model # Run the demo (shows recursive generation + parsing) cargo run --release # Run mathematical proof tests cargo test --release test_complete_recursive_proof # Run full benchmark suite cargo test --release --features bench # NEW: Try the probabilistic language model cd ../python && python tiny_lm.py

🧬 Atomic Language Model - Recursive Grammar Demo ============================================================ 📐 Mathematical Proof: aⁿbⁿ Generation ---------------------------------------- n=0: ε (empty) n=1: a b n=2: a a b b n=3: a a a b b b n=4: a a a a b b b b n=5: a a a a a b b b b b 🔍 Parsing Test: Recursive Structures ---------------------------------------- ✅ 'the student left' → the student left Category: S, Complete: true ♾️ Unbounded Recursion Demonstration ---------------------------------------- ✅ Can generate a^6b^6 (length: 12) ✅ Can generate a^7b^7 (length: 14) ✅ Can generate a^8b^8 (length: 16) 🎯 Theoretical Capacity: INFINITE 🔬 Practical Limit: Memory bounded

1. 🎯 Start Here: Understanding Recursion

Read First: Recursive Language Overview

What is recursion in language?
Why does it matter?
How does this implementation work?

2. 🧮 The Mathematical Foundation

Next: Chomsky's Mathematical Proofs

The 1956 proof that changed linguistics
How finite-state grammars fail
Why recursion is mathematically necessary

3. ⚙️ Technical Deep Dive

Then: Formal Language Theory

Grammar hierarchies and automata
Minimalist Grammar operations
Complexity theory and parsing

4. 💻 Implementation Details

Implementation: Atomic Language Model

5. 🧪 Testing and Validation

Validation: NLP Verification Methods

Agreement test suites
Colorless green tests
Performance benchmarking

Advanced: Machine Verification

Coq proof development
Mechanized theorem proving
Mathematical rigor

This project demonstrates the full journey from mathematical theory to practical implementation:

📖 The Recursive Story - The complete narrative connecting all pieces

🆕 Probabilistic Language Model Extension

We've extended the atomic language model with probabilistic next-token prediction capabilities while maintaining all formal guarantees:

🎲 Weighted Grammar Rules: Each production has learned probabilities
🔮 Next-Token Prediction: Monte Carlo sampling for language modeling
🔄 Hybrid Architecture: Combines Rust validation with Python inference
🌐 REST API: Flask server for easy integration
📦 Still Ultra-Light: <100kB total with all features

# Run the probabilistic language model cd atomic-lang-model/python python tiny_lm.py # Start the API server python api_server.py # Try the interactive demo cd ../examples python language_model_demo.py

# Predict next token curl localhost:5000/predict?prefix=the+student # Generate sentences curl localhost:5000/generate?count=5 # Validate syntax curl -X POST localhost:5000/validate \ -H "Content-Type: application/json" \ -d '{"sentences": ["the student left", "student the left"]}'

This extension bridges formal grammar theory with practical NLP applications, creating the world's smallest formally verified language model.

✅ Formal Specification: Complete mathematical definition of operations
✅ Coq Proofs: Machine-verified theorems about recursive properties
✅ Non-regularity Proof: Constructive demonstration via a^n b^n generation
✅ Complexity Bounds: Polynomial parsing with exponential generation capacity

Universal Grammar Implementation

✅ Merge Operation: Merge(α:=_X β, X:γ) = ⟨X, [], [α, γ]⟩
✅ Move Operation: Implements wh-movement and feature checking
✅ Feature System: Categories, selectors, and movement features
✅ Minimalist Compliance: Based on Chomsky's Minimalist Program

✅ Zero Dependencies: No runtime requirements
✅ Size Optimized: ~35kB binary with full functionality
✅ Memory Efficient: <256kB peak usage for 20-word sentences
✅ Fast Parsing: Polynomial-time complexity O(n³)

✅ Agreement Tests: Subject-verb agreement across center-embedding (Linzen et al. 2016)
✅ Colorless Green: Syntactic processing with semantic anomalies (Gulordava et al. 2018)
✅ Recursive Capability: Demonstrates unbounded center-embedding
✅ Performance Metrics: Comprehensive benchmarking framework

atomic-lang-model/ ├── docs/ # 📚 Complete documentation suite │ ├── recursive-language-overview.md # Start here! │ ├── chomsky-mathematical-proofs.md # The math foundation │ ├── formal-language-theory.md # Technical theory │ ├── computational-processing.md # How machines handle recursion │ ├── nlp-verification-methods.md # Testing approaches │ ├── machine-verification.md # Formal proofs │ └── the-recursive-story.md # Complete narrative ├── atomic-lang-model/ # 🧬 Core implementation │ ├── src/lib.rs # Main implementation (~3k lines) │ ├── tests/recursion.rs # Mathematical proof tests │ ├── bench/ # NLP evaluation suites │ ├── Coq/Minimalist.v # Machine-verified proofs │ ├── spec.md # Formal specification │ ├── REPORT.md # Implementation analysis │ ├── python/ # 🤖 Probabilistic LM extension │ │ ├── tiny_lm.py # Core probabilistic grammar │ │ ├── hybrid_model.py # Rust-Python bridge │ │ └── api_server.py # REST API server │ └── examples/ # 🎮 Demo applications │ ├── language_model_demo.py # Interactive demo │ └── quick_example.py # Simple usage example └── flow/ # 🌊 Claude-flow integration └── claude-flow/ # AI orchestration platform

# Generate recursive patterns cargo run --release -- generate an_bn 5 # Output: a a a a a b b b b b # Parse sentences cargo run --release -- parse "the student who left smiled" # Shows parse tree and derivation steps # Run mathematical tests cargo test test_an_bn_generation cargo test test_recursive_capability

# Run linguistic evaluation suites cargo test --release agreement_suite cargo test --release colorless_green_suite # Performance benchmarking cargo test --release run_complete_benchmark # Formal verification (requires Coq) cd Coq && coqc Minimalist.v

This implementation demonstrates that:

Recursive universal grammar is implementable in extremely constrained environments
Mathematical theory and practical engineering can be unified effectively
Formal verification and empirical testing provide complementary validation
Chomsky's insights about recursion remain relevant for modern AI systems

We welcome contributions! See Contributing Guidelines

Great starting points:

📖 Improve documentation and examples
🧪 Add more linguistic test cases
⚡ Optimize performance and memory usage
🔬 Extend Coq formalization
🌍 Test on additional languages

If you use this work in research, please cite:

@software{atomic_language_model, title={Atomic Language Model: Recursive Universal Grammar in 50kB}, author={Atomic Language Model Team}, year={2024}, url={https://github.com/user/atomic-lang-model} }

MIT License - see LICENSE for details.

Built on the mathematical foundations of:

Noam Chomsky: Recursive language theory and Minimalist Grammar
Edward Stabler: Formal implementation of Minimalist Grammars
Linzen et al.: Agreement test methodology
Gulordava et al.: Colorless green evaluation framework

Glossary of Concepts → A comprehensive A-Z guide to all key terms and concepts in the documentation

Built with mathematical rigor. Validated through empirical testing. Optimized for practical use.

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