有限状态 $\epsilon$-机器的有限摘要可辨识性

中文摘要 AI 辅助

我们研究多少可观测的有限维信息足以识别一个有限状态生成器 $\epsilon$-机器。可观测摘要是由给定有限历史和连续块定律的条件概率组成的标记表。对于平稳遍历的有限字母过程，我们首先证明具有过去视界 $L$ 和未来视界 $R$ 的表等价于长度为 $L+R$ 的块定律。然后我们回顾经典的线性实现界。具有 $r$ 和 $s$ 个状态的两个隐马尔可夫表示，如果它们在长度至多 $r+s-1$ 的词上一致，则它们在整个过程定律上一致。我们的主要结果给出了一个适用于精确有限状态生成器 $\epsilon$-机器的结构同步界。如果它们的目标同步半径分别为 $a$ 和 $b$，且最大预测分离半径为 $d$，则长度为 $a+b+d+1$ 的一个块定律的相等性意味着同构。证明将两个机器的状态与同时同步两个生成器的词对齐。对于具有成对不同参数的 $m$ 阶全支持二元上下文子族，结构同步界为 $2m+2$，而经典线性实现界的状态计数和秩维形式为 $2^{m+1}-1$。一个构造表明需要 $m$ 阶的视界。当 $m$ 阶上下文结构事先已知时，精确视界为 $m+1$ 个符号。因此，原始的态计数和秩维公式在该子族上可能呈指数级悲观。比较是在一般充分界之间进行的，而非与最优实现视界比较。

英文摘要

We study how much observable finite-dimensional information is sufficient to identify a finite-state generator $ε$-machine. The observable summaries are labeled tables of conditional probabilities given finite histories and contiguous block laws. For stationary ergodic finite-alphabet processes, we first prove that a table with past horizon $L$ and future horizon $R$ is equivalent to the block law of length $L+R$. We then recall the classical linear-realization bound. Two hidden Markov presentations with $r$ and $s$ states agree on the entire process law if they agree on words through length $r+s-1$. Our main result gives a structural synchronization bound adapted to exact finite-state generator $ε$-machines. If their target synchronization radii are $a$ and $b$, and their maximum predictive separation radius is $d$, then equality of one block law of length $a+b+d+1$ implies isomorphism. The proof aligns the states of both machines with words that synchronize the two generators simultaneously. For the full-support binary context subfamily of order $m$ with pairwise distinct parameters, the structural synchronization bound is $2m+2$, whereas the state-count and rank-dimension forms of the classical linear-realization bound are $2^{m+1}-1$. A construction shows that a horizon of order $m$ is necessary. When the order-$m$ context structure is known in advance, the sharp horizon is $m+1$ symbols. Thus the raw state-count and rank-dimension formulas can be exponentially pessimistic on this subfamily. The comparison is between general sufficient bounds, not with an optimal realization horizon.

发表机构

• Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）

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