对数空间中的计数
Counting in logarithmic space
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中文总结 AI 辅助
研究非确定性对数空间图灵机接受路径计数函数的类#L，构建证明其包含关系的方法，证明大量经典函数属此类，包括枚举组合学、代数组合学等中的函数，还给出GL₂ plethysm系数的计数方法及相关问题猜想。
中文摘要 AI 辅助
我们研究非确定性对数空间图灵机接受路径计数函数的类#L，并构建证明其包含关系的方法。我们证明大量经典组合和数论函数属于此类，包括枚举组合学、代数组合学、离散几何、数论函数等中的函数。还表明有界长度外划分的GL₂ plethysm系数可由对数平方空间多项式时间验证器计数。我们提出了许多关于#L包含及其推广的问题和猜想，为有条件地反驳#P完全性提供了途径。研究哪些组合函数在#P中可形式化地（反）证组合解释的存在，而较低的类#L可作为多项式时间可计算函数的类似物。
英文摘要
We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.
