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From P ≟ NP to Practice: Description Complexity and Certificate-First Algorithm Discovery for Hard Problems.

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Title: From P ≟ NP to Practice: Description Complexity and Certificate-First Algorithm Discovery for Hard Problems.
Authors: Abela, John, Cachia, Ernest, Layfield, Colin
Source: Mathematics (2227-7390); Jan2026, Vol. 14 Issue 1, p41, 33p
Subject Terms: KOLMOGOROV complexity, COMPUTATIONAL complexity, COMPLEXITY (Philosophy), REPLICATION (Experimental design), HEURISTIC, HEURISTIC algorithms
Abstract: The celebrated question of whether P = N P continues to define the boundary between the feasible and the intractable in computer science. In this paper, we revisit the problem from two complementary angles: Time-Relative Description Complexity and automated discovery, adopting an epistemic rather than ontological perspective. Even if polynomial-time algorithms for NP-complete problems do exist, their minimal descriptions may have very high Kolmogorov complexity. This creates what we call an epistemic barrier, making such algorithms effectively undiscoverable by unaided human reasoning. A series of structural results—relativization, Natural Proofs, and the Probabilistically Checkable Proofs (PCPs) theorem—already indicate that classical proof techniques are unlikely to resolve the question, which motivates a more pragmatic shift in emphasis. We therefore ask a different, more practical question: what can systematic computational search achieve within these limits? We propose a certificate-first workflow for algorithmic discovery, in which candidate algorithms are considered scientifically credible only when accompanied by machine-checkable evidence. Examples include Deletion/Resolution Asymmetric Tautology (DRAT)/Flexible RAT (FRAT) proof logs for SAT, Linear Programming (LP)/Semidefinite Programming (SDP) dual bounds for optimization, and other forms of independently verifiable certificates. Within this framework, high-capacity search and learning systems can explore algorithmic spaces far beyond manual (human) design, yet still produce artifacts that are auditable and reproducible. Empirical motivation comes from large language models and other scalable learning systems, where increasing capacity often yields new emergent behaviors even though internal representations remain opaque. This paper is best described as a position and expository essay that synthesizes insights from complexity theory, Kolmogorov complexity, and automated algorithm discovery, using Time-Relative Description Complexity as an organising lens and outlining a pragmatic research direction grounded in verifiable computation. We argue for a shift in emphasis from the elusive search for polynomial-time solutions to the constructive pursuit of high-performance heuristics and approximation methods grounded in verifiable evidence. The overarching message is that capacity plus certification offers a principled path toward better algorithms and clearer scientific limits without presuming a final resolution of P = ? N P . [ABSTRACT FROM AUTHOR]
Copyright of Mathematics (2227-7390) is the property of MDPI and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The celebrated question of whether P = N P continues to define the boundary between the feasible and the intractable in computer science. In this paper, we revisit the problem from two complementary angles: Time-Relative Description Complexity and automated discovery, adopting an epistemic rather than ontological perspective. Even if polynomial-time algorithms for NP-complete problems do exist, their minimal descriptions may have very high Kolmogorov complexity. This creates what we call an epistemic barrier, making such algorithms effectively undiscoverable by unaided human reasoning. A series of structural results—relativization, Natural Proofs, and the Probabilistically Checkable Proofs (PCPs) theorem—already indicate that classical proof techniques are unlikely to resolve the question, which motivates a more pragmatic shift in emphasis. We therefore ask a different, more practical question: what can systematic computational search achieve within these limits? We propose a certificate-first workflow for algorithmic discovery, in which candidate algorithms are considered scientifically credible only when accompanied by machine-checkable evidence. Examples include Deletion/Resolution Asymmetric Tautology (DRAT)/Flexible RAT (FRAT) proof logs for SAT, Linear Programming (LP)/Semidefinite Programming (SDP) dual bounds for optimization, and other forms of independently verifiable certificates. Within this framework, high-capacity search and learning systems can explore algorithmic spaces far beyond manual (human) design, yet still produce artifacts that are auditable and reproducible. Empirical motivation comes from large language models and other scalable learning systems, where increasing capacity often yields new emergent behaviors even though internal representations remain opaque. This paper is best described as a position and expository essay that synthesizes insights from complexity theory, Kolmogorov complexity, and automated algorithm discovery, using Time-Relative Description Complexity as an organising lens and outlining a pragmatic research direction grounded in verifiable computation. We argue for a shift in emphasis from the elusive search for polynomial-time solutions to the constructive pursuit of high-performance heuristics and approximation methods grounded in verifiable evidence. The overarching message is that capacity plus certification offers a principled path toward better algorithms and clearer scientific limits without presuming a final resolution of P = ? N P . [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematics (2227-7390) is the property of MDPI and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.3390/math14010041
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        Text: English
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      – SubjectFull: COMPUTATIONAL complexity
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      – SubjectFull: COMPLEXITY (Philosophy)
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      – SubjectFull: HEURISTIC
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              Text: Jan2026
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