Information Flow Complexity Theory: A Comprehensive Mathematical Framework
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2025
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| author | Kilpatrick, Christian |
| author_facet | Kilpatrick, Christian |
| contents | <p>We introduce <strong>Information Flow Complexity</strong> as a new field within computational complexity theory, establishing rigorous mathematical foundations and proving fundamental theorems. This framework measures the <em>dynamic flow of information</em> through algorithms during execution, providing a fundamentally different perspective from traditional static complexity measures.</p> <p>Our key contribution is proving that Information Flow bypasses all three known barriers to complexity class separation: <strong>Natural Proofs</strong> (Razborov–Rudich 1997), <strong>Relativization</strong> (Baker–Gill–Solovay 1975), and <strong>Algebraization</strong> (Aaronson–Wigderson 2009). This establishes Information Flow as the first framework proven suitable for fundamental separation results.</p> <p>We define Information Flow as <strong>conditional Shannon entropy</strong>:</p> <p><span><span><span>Flow(M,x,t)=H(Statet∣Statet−1)\text{Flow}(M,x,t) = H(\text{State}_t \mid \text{State}_{t-1})</span><span><span><span><span>Flow</span></span><span>(</span><span>M</span><span>,</span><span>x</span><span>,</span><span>t</span><span>)</span><span>=</span></span><span><span>H</span><span>(</span><span><span>State</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>∣</span></span><span><span><span>State</span><span><span><span><span><span><span><span>t</span><span>−</span>1</span></span></span><span></span></span></span></span></span><span>)</span></span></span></span></span></p> <p>measuring information gained per computational step. <strong>Total Flow</strong> aggregates across all steps. We prove barrier bypass theorems with complete proofs, apply Information Flow to prove <strong>P ≠ NP</strong> conditionally (assuming pseudorandom generators), and analyze concrete algorithms.</p> <p>Applications include information-theoretic lower bounds, algorithm design principles, and potential approaches to other major separations (<strong>P vs PSPACE</strong>, <strong>NP vs coNP</strong>).</p> <p>This work establishes <strong>Information Flow Complexity</strong> as a rigorous mathematical field with proven theoretical advantages and practical applications.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17373031 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Information Flow Complexity Theory: A Comprehensive Mathematical Framework Kilpatrick, Christian math proof new math <p>We introduce <strong>Information Flow Complexity</strong> as a new field within computational complexity theory, establishing rigorous mathematical foundations and proving fundamental theorems. This framework measures the <em>dynamic flow of information</em> through algorithms during execution, providing a fundamentally different perspective from traditional static complexity measures.</p> <p>Our key contribution is proving that Information Flow bypasses all three known barriers to complexity class separation: <strong>Natural Proofs</strong> (Razborov–Rudich 1997), <strong>Relativization</strong> (Baker–Gill–Solovay 1975), and <strong>Algebraization</strong> (Aaronson–Wigderson 2009). This establishes Information Flow as the first framework proven suitable for fundamental separation results.</p> <p>We define Information Flow as <strong>conditional Shannon entropy</strong>:</p> <p><span><span><span>Flow(M,x,t)=H(Statet∣Statet−1)\text{Flow}(M,x,t) = H(\text{State}_t \mid \text{State}_{t-1})</span><span><span><span><span>Flow</span></span><span>(</span><span>M</span><span>,</span><span>x</span><span>,</span><span>t</span><span>)</span><span>=</span></span><span><span>H</span><span>(</span><span><span>State</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>∣</span></span><span><span><span>State</span><span><span><span><span><span><span><span>t</span><span>−</span>1</span></span></span><span></span></span></span></span></span><span>)</span></span></span></span></span></p> <p>measuring information gained per computational step. <strong>Total Flow</strong> aggregates across all steps. We prove barrier bypass theorems with complete proofs, apply Information Flow to prove <strong>P ≠ NP</strong> conditionally (assuming pseudorandom generators), and analyze concrete algorithms.</p> <p>Applications include information-theoretic lower bounds, algorithm design principles, and potential approaches to other major separations (<strong>P vs PSPACE</strong>, <strong>NP vs coNP</strong>).</p> <p>This work establishes <strong>Information Flow Complexity</strong> as a rigorous mathematical field with proven theoretical advantages and practical applications.</p> |
| title | Information Flow Complexity Theory: A Comprehensive Mathematical Framework |
| topic | math proof new math |
| url | https://doi.org/10.5281/zenodo.17373031 |