88 lines
4.0 KiB
Markdown
88 lines
4.0 KiB
Markdown
# The Ubiquity of Space-Time Simulation in Modern Computing: From Theory to Practice
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This repository contains the academic paper exploring how Ryan Williams' 2025 theoretical result, TIME[t] ⊆ SPACE[√(t log t)], manifests in real-world computing systems.
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## Paper
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**Title**: The Ubiquity of Space-Time Simulation in Modern Computing: From Theory to Practice
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**Author**: David H. Friedel Jr., Founder, MarketAlly LLC (USA) & MarketAlly Pte. Ltd. (Singapore)
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**Status**: Ongoing research: Version 2 released with additional clarity, preliminary results, and planned further experiments and refinements.
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## Abstract
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Ryan Williams' 2025 result demonstrates that any time-bounded algorithm can be simulated using only O(√(t log t)) space, establishing a fundamental limit on the space-time relationship in computation. This paper bridges the gap between this theoretical breakthrough and practical computing systems. Through controlled experiments and analysis of production systems, we show that space-time tradeoffs following the √n pattern are ubiquitous across databases, machine learning frameworks, and distributed systems. However, we find that practical constant factors range from 100× to 10,000×, primarily due to memory hierarchies and I/O overhead.
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## Related Repositories
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- **[Experiments & Code](https://github.com/sqrtspace/sqrtspace-experiments)**: Full implementation, experiments, and interactive dashboard
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- **[Interactive Dashboard](https://github.com/sqrtspace/sqrtspace-experiments/tree/main/dashboard)**: Streamlit app for exploring space-time tradeoffs
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## Key Findings
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Note: All findings in this version are subject to refinement as additional experiments and cross-hardware validation are conducted.
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1. **Experimental validation**:
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- Checkpointed sorting: 375-627× slowdown for √n space reduction
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- Real LLM inference (Ollama): 18.3× slowdown for √n context chunking
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- Stream processing: 30× speedup with sliding windows (less memory = faster!)
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- SQLite: Counterintuitive faster performance with smaller caches on modern SSDs
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2. **Production system analysis**:
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- SQLite (billions of deployments): Buffer pools sized at √(database_size)
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- Flash Attention (GPT-4, etc.): O(n²) → O(n) memory enabling 10× longer contexts
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- PostgreSQL & Apache Spark: √n patterns in buffer pools and shuffle operations
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3. **Theory vs practice gap**:
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- Williams predicts √n slowdown, we observe 100-10,000× due to memory hierarchies
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- L1 cache: ~1ns, RAM: ~100ns, SSD: ~100μs, HDD: ~10ms
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- Modern hardware can invert predictions (bandwidth bottlenecks)
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4. **Practical framework**:
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- When beneficial: Streaming data, sequential access, distributed systems
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- When harmful: Interactive apps, random access, small datasets
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- Interactive dashboard and tools for practitioners
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## Building the Paper
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```bash
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# Compile the paper
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pdflatex main.tex
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bibtex main
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pdflatex main.tex
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pdflatex main.tex
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# Compile two-page summary
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pdflatex two_page_summary.tex
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```
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## Citation
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Once published on arXiv:
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```bibtex
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@article{friedel2025ubiquity,
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title={The Ubiquity of Space-Time Simulation in Modern Computing: From Theory to Practice},
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author={Friedel Jr., David H.},
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journal={arXiv preprint arXiv:25XX.XXXXX},
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year={2025}
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}
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```
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## Reading Order
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1. **Quick Overview**: Read `executive_summary.md` (2 pages)
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2. **Technical Summary**: Read `two_page_summary.tex` (2 pages, compile to PDF)
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3. **Full Paper**: Read `main.tex` (26 pages, compile to PDF)
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4. **Try It Yourself**: Visit the [experiments repository](https://github.com/sqrtspace/sqrtspace-experiments)
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## Contact
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- **Email**: dfriedel@marketally.ai
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- **Organization**: [MarketAlly LLC](https://marketally.com)
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## License
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This paper is licensed under CC BY 4.0. You may share and adapt the material with proper attribution.
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## Acknowledgments
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This work was carried out independently as part of early-stage R&D at MarketAlly LLC and MarketAlly Pte. Ltd. We acknowledge the use of large language models for drafting assistance. |