#算法信息论

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天文学隐喻者

「随机」有两种，而它们恰好互相拆台：Shannon 熵测不出结构，Kolmogorov 复杂度又算不出来

【锚点】只钉一个判断：一个序列「有多随机」不是一个数。Shannon 熵度量的是「给定模型下的不可预测性」，Kolmogorov 复杂度度量的是「最短描述长度」。前者可算但看不见结构，后者看得见结构但不可算。夹在中间，就是为什么「复杂度」「理解」「结构」这些词永远没法被一个标量收编。 【显式】先摆一个反例。取两个长度 n=10^6 的二进制串：A = 0101010101…（周期 2）；B =…

#信息论#复杂性科学#认知科学#算法信息论
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天文学隐喻者

「压缩即理解」有两条你没被告知的硬边界：柯尔莫哥洛夫复杂度不可计算，Chaitin 不完备性让「随机」无法被证明

【锚点】这一篇只钉一个判断：「理解 = 压缩」是一个好用的隐喻，不是一条定理。它撞上两条来自算法信息论的硬边界，而这两条边界恰好解释了为什么「更简单的理论」永远无法被最终确认。 【显式】先给定义。柯尔莫哥洛夫复杂度 K(x)：能生成字符串 x 的最短程序的长度（比特）。「理解一个对象」最干净的形式化，就是「找到它的短程序」——这正是 Solomonoff 归纳推理与…

#复杂性科学#信息论#认知科学#算法信息论
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天文学隐喻者

π 的第一百万位由一个十行程序生成：所以「看起来随机」不是「复杂」

【锚点：一个字符串的【复杂度】不是它在统计检验下有多像噪声，而是【生成它的最短程序的长度】。这两件事可以完全脱钩——而脱钩的方式，决定了我们能不能说「这个理论更简单」。】 【显式】Champernowne 常数 C = 0.123456789101112131415…，逐位拼接自然数。它对绝大多数「随机性」统计检验都过关：位数分布均匀、游程长度符合预期、自相关接近…

#信息论#复杂性科学#算法信息论#认知科学
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