← 研踪 PaperTrailarXiv:2611.02560 · v1
Circuit Lower Bounds for Extended Gate Bases from Subexponential Counting
首次提交 2026-11-02 13:38
版本更新 2026-11-02 13:38
雷达观测 2026-11-04 23:42
摘要
We study circuit lower bounds for extended gate bases using exact CNF counting and an extension-ring form of the contraction theorem of OpenAI. The deterministic counting theorem of Claude (Anthropic) yields an $\mathsf{E}^{\mathsf{NP}}$ language requiring $Ω(m\sqrt{\log m}/\log\log m)$ gates over unbounded AND/OR with binary XOR, infinitely often. Using that contraction theorem, a bounded binary base with a linear budget of arbitrary XOR and symmetric gates requires $Ω(m\sqrt{\log m})$ gates infinitely often. These results extend the binary-gate lower bound of OpenAI. Signed long-clause selectors reduce the first counting problem to the difference of two ordinary 3-CNF counts. Affine characters and unit-separated interpolation give the second counting interface, with logarithmic cost per symmetric-gate arity. A quantitative local-reduction transfer accounts for auxiliary variables, branch counts, and integer recovery.
作者备注：16 pages
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v1 · 2026-11-02 13:38 · 最新观测版本
Circuit Lower Bounds for Extended Gate Bases from Subexponential Counting
We study circuit lower bounds for extended gate bases using exact CNF counting and an extension-ring form of the contraction theorem of OpenAI. The deterministic counting theorem of Claude (Anthropic) yields an $\mathsf{E}^{\mathsf{NP}}$ language requiring $Ω(m\sqrt{\log m}/\log\log m)$ gates over unbounded AND/OR with binary XOR, infinitely often. Using that contraction theorem, a bounded binary base with a linear budget of arbitrary XOR and symmetric gates requires $Ω(m\sqrt{\log m})$ gates infinitely often. These results extend the binary-gate lower bound of OpenAI. Signed long-clause selectors reduce the first counting problem to the difference of two ordinary 3-CNF counts. Affine characters and unit-separated interpolation give the second counting interface, with logarithmic cost per symmetric-gate arity. A quantitative local-reduction transfer accounts for auxiliary variables, branch counts, and integer recovery.
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