Reframing P Vs. NP via Structural Descent: SBI, Alankar Chains and the Emergence of Temporal Complexity Theory
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| Language: | English |
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2026
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| author | Khara, Alankar Sukhdev Singh |
| author_facet | Khara, Alankar Sukhdev Singh |
| contents | <p>This paper introduces Temporal Complexity Theory (TCT), a generalization of classical computational complexity in which problem difficulty is modeled as a function of evolving knowledge states rather than fixed representations. Building on the Structural Descent Framework (SDF), we formalize problem solving as a trajectory through a dynamically evolving theory-space, where admissible representations, invariants, and transformations are updated via Structural Bayesian Inference (SBI) and operationalized through Alankar chains. Within this framework, we define three complementary notions of complexity:</p> <p>absolute complexity, corresponding to classical worst-case cost; knowledge-conditioned complexity, which depends on the current knowledge state; and retrospective complexity, capturing the minimal cost of solution from a structurally optimal representation. This decomposition explains the pervasive empirical phenomenon of hindsight simplicity, wherein historically difficult problems become trivial once appropriate structural representations are discovered. A central contribution is the introduction of the coordination invariant, a representation-dependent measure of irreducible coupling among problem constraints, formalized via graph-theoretic, communication, and circuit complexity perspectives. We show that this invariant lower-bounds computational complexity and governs the emergence of epistemic plateaus (resolvable through structural updates) and ontic plateaus (potentially irreducible). From this perspective, the P versus NP problem is reframed as a question of global coordination compressibility: whether all instances admit representations in which constraint interactions can be reduced to polynomial structure. While this reformulation does not resolve the problem, it isolates the structural hypothesis underlying computational hardness and provides a unified account linking complexity theory, learning dynamics, and scientific discovery. More broadly, TCT establishes a principled framework in which computation, representation, and knowledge evolution are treated as a single coupled process, shifting the study of complexity from static analysis to dynamic structural accessibility.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19154136 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Reframing P Vs. NP via Structural Descent: SBI, Alankar Chains and the Emergence of Temporal Complexity Theory Khara, Alankar Sukhdev Singh Structural Descent Framework Temporal Complexity Theory Structural Bayesian Inference P Vs NP Alankar Chains <p>This paper introduces Temporal Complexity Theory (TCT), a generalization of classical computational complexity in which problem difficulty is modeled as a function of evolving knowledge states rather than fixed representations. Building on the Structural Descent Framework (SDF), we formalize problem solving as a trajectory through a dynamically evolving theory-space, where admissible representations, invariants, and transformations are updated via Structural Bayesian Inference (SBI) and operationalized through Alankar chains. Within this framework, we define three complementary notions of complexity:</p> <p>absolute complexity, corresponding to classical worst-case cost; knowledge-conditioned complexity, which depends on the current knowledge state; and retrospective complexity, capturing the minimal cost of solution from a structurally optimal representation. This decomposition explains the pervasive empirical phenomenon of hindsight simplicity, wherein historically difficult problems become trivial once appropriate structural representations are discovered. A central contribution is the introduction of the coordination invariant, a representation-dependent measure of irreducible coupling among problem constraints, formalized via graph-theoretic, communication, and circuit complexity perspectives. We show that this invariant lower-bounds computational complexity and governs the emergence of epistemic plateaus (resolvable through structural updates) and ontic plateaus (potentially irreducible). From this perspective, the P versus NP problem is reframed as a question of global coordination compressibility: whether all instances admit representations in which constraint interactions can be reduced to polynomial structure. While this reformulation does not resolve the problem, it isolates the structural hypothesis underlying computational hardness and provides a unified account linking complexity theory, learning dynamics, and scientific discovery. More broadly, TCT establishes a principled framework in which computation, representation, and knowledge evolution are treated as a single coupled process, shifting the study of complexity from static analysis to dynamic structural accessibility.</p> |
| title | Reframing P Vs. NP via Structural Descent: SBI, Alankar Chains and the Emergence of Temporal Complexity Theory |
| topic | Structural Descent Framework Temporal Complexity Theory Structural Bayesian Inference P Vs NP Alankar Chains |
| url | https://doi.org/10.5281/zenodo.19154136 |