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Podrobná bibliografie
Hlavní autor:	Li, Y.Y.N.
Médium:	Recurso digital
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Vydáno:	Zenodo 2025
On-line přístup:	https://doi.org/10.5281/zenodo.17656331
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We propose a fundamental variational law of intelligent computation, the \emph{Principle of Least Structural Action} (PLSA). Given a reasoning trajectory $\psi(t)$ produced by a solver, algorithm, or large language model (LLM), we define a structural Lagrangian \[ L(\psi,\dot\psi) = \alpha K(\psi) + \beta S(\psi) + \gamma C(\psi), \] where $K$ is structural incompressibility (a computable Kolmogorov-style measure), $S$ is structural curvature, and $C$ is task-consistency (correctness potential).   PLSA states that intelligent systems select trajectories minimizing the structural action \[ \mathcal A[\psi] = \int_0^T L(\psi,\dot\psi)\,dt. \] At the macroscopic level this yields a Structure--Time Law of the form \[ T(x)=\Theta\!\left(N(x)^{\beta_T}\,2^{\alpha_T h(x)}\right), \] where $N(x)$ is a size parameter and $h(x)$ is a normalized structural hardness index derived from the trace (for instance, a monotone transform of a conditional incompressibility measure). Equivalently, \[ \log_2 T(x)\approx \alpha_T\,h(x) + \beta_T\,\log_2 N(x) + \gamma_T, \qquad \alpha_T,\beta_T>0. \] We show that this scaling law can be derived as the time-integral of the microscopic PLSA (Theorem~3). We instantiate these ideas empirically on random 3-SAT using a CDCL solver and a small number of LLM chain-of-thought (CoT) runs. On 120 SAT instances, a gzip-based structural overhead $\lambda_K(x)$ of the solver trace explains a substantial fraction of runtime variance under a linear model \[ \log_2 T(x)\approx a\,\lambda_K(x) + b\,\log_2 n + c \quad(R^2\approx 0.70), \] with fitted coefficients \[ a\approx -6.80,\quad b\approx 1.89,\quad c\approx -20.49. \] In our normalization, \emph{harder instances empirically exhibit smaller $\lambda_K(x)$}, so that the fitted coefficient $a<0$ is naturally interpreted after a change of variables to a hardness index $h(x):=1-\lambda_K(x)$. In that parameterization the same regression takes the form \[ \log_2 T(x)\approx \alpha_T\,h(x)+\beta_T\,\log_2 n + \gamma_T, \] with \[ \alpha_T=-a\approx 6.80>0,\quad \beta_T=b\approx 1.89,\quad \gamma_T = a + c \approx -27.29. \] For SAT instances solved by an LLM via CoT, we observe a dramatic increase in trace overhead (roughly $5$--$7\times$ larger $\lambda_K$) and several orders of magnitude increase in effective runtime relative to the CDCL solver on the same problems, consistent with the view that explicit reasoning corresponds to an expansion of structural residuals and a higher structural action. We conjecture that PLSA is a universal law of intelligence across artificial, biological, and physical systems.

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