Resolution of P ̸= NP: Compression Barriers, Non-Relativizing Arguments, and Entropy Formalization
Fuente:
Zenodo
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Recurso digital |
| Veröffentlicht: |
Zenodo
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866902308357406720 |
|---|---|
| author | Fisher, Christopher |
| author_facet | Fisher, Christopher |
| contents | <p>This paper presents a complete and rigorous proof that P is not equal to NP, resolving one of the most fundamental open problems in theoretical computer science. The proof integrates three independent approaches: an entropy-based decision tree complexity barrier, a compression impossibility theorem grounded in Kolmogorov complexity, and a non-relativizing circuit complexity separation. Together, these demonstrate that NP-complete problems cannot be solved in polynomial time by any deterministic algorithm.</p> <p>The entropy argument shows that distinguishing among all possible solutions requires an exponential number of steps. The compression theorem proves that no polynomial-time function can reduce an NP-complete instance to a smaller equivalent form while preserving decidability. The circuit complexity result establishes that solving NP-complete problems requires circuits of exponential size, beyond the scope of polynomial-time computation.</p> <p>These results hold universally, even under oracle and quantum models, making the separation between P and NP robust across computational paradigms. The proof draws from foundational principles in complexity theory, including Shannon entropy, Kolmogorov complexity, and structural bounds from circuit lower bounds literature.</p> <p>This work has significant implications for cryptography, optimization, computational hardness, and our broader understanding of algorithmic limits.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15088639 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Resolution of P ̸= NP: Compression Barriers, Non-Relativizing Arguments, and Entropy Formalization Fisher, Christopher P ≠ NP proof theoretical computer science entropy barrier compression impossibility non-relativizing arguments circuit complexity Kolmogorov complexity NP-complete problems decision tree depth cryptography implications Clay Millennium Problem computational complexity algorithmic barriers structural complexity quantum complexity <p>This paper presents a complete and rigorous proof that P is not equal to NP, resolving one of the most fundamental open problems in theoretical computer science. The proof integrates three independent approaches: an entropy-based decision tree complexity barrier, a compression impossibility theorem grounded in Kolmogorov complexity, and a non-relativizing circuit complexity separation. Together, these demonstrate that NP-complete problems cannot be solved in polynomial time by any deterministic algorithm.</p> <p>The entropy argument shows that distinguishing among all possible solutions requires an exponential number of steps. The compression theorem proves that no polynomial-time function can reduce an NP-complete instance to a smaller equivalent form while preserving decidability. The circuit complexity result establishes that solving NP-complete problems requires circuits of exponential size, beyond the scope of polynomial-time computation.</p> <p>These results hold universally, even under oracle and quantum models, making the separation between P and NP robust across computational paradigms. The proof draws from foundational principles in complexity theory, including Shannon entropy, Kolmogorov complexity, and structural bounds from circuit lower bounds literature.</p> <p>This work has significant implications for cryptography, optimization, computational hardness, and our broader understanding of algorithmic limits.</p> |
| title | Resolution of P ̸= NP: Compression Barriers, Non-Relativizing Arguments, and Entropy Formalization |
| topic | P ≠ NP proof theoretical computer science entropy barrier compression impossibility non-relativizing arguments circuit complexity Kolmogorov complexity NP-complete problems decision tree depth cryptography implications Clay Millennium Problem computational complexity algorithmic barriers structural complexity quantum complexity |
| url | https://doi.org/10.5281/zenodo.15088639 |