描述复杂性理论中的随机化与去随机化
Randomisation and Derandomisation in Descriptive Complexity Theory
摘要：
我们从逻辑的角度研究概率复杂性类和去随机化问题。对于每一个逻辑L，我们引入一个新的逻辑BPL，有界错误概率L，这是从L中定义的复杂性类BPP，有界错误概率多项式时间，是从PTIME中定义的类似的方式。我们的主要重点在于问题的去随机化，我们证明，有一个查询是可定义的BPFO，概率版本的一阶逻辑，但不是在Cinf，有限变量无穷逻辑计数。这意味着有限模型理论的许多标准逻辑，如传递闭包逻辑和不动点逻辑，无论有无计数，都不能去随机化。同样，我们提出了一个查询的有序结构，这是在BPFO定义的，但不是在一元二阶逻辑，和一个查询的加法结构，这是在BPFO定义的，但不是在FO。这些查询中的后一个表明，AC0（有界深度多项式大小的电路）的某些统一变体不能被去随机化。这些结果与大多数标准复杂性类可以去随机化的普遍看法相反。最后，我们注意到，BPIFP+C，概率版本的定点逻辑计数，捕获的复杂性类BPP，即使在无序结构。
英文摘要：
We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.
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