关于有界基数条件独立蕴含问题的复杂性
On the Complexity of the Conditional Independence Implication Problem With Bounded Cardinalities
AI总结：
本文证明有界基数的条件独立蕴含问题即使在二元变量下仍为 co-NEXPTIME 难，并结合已知 EXPSPACE 上界刻画其复杂性。
摘要：
我们证明了有界基数的条件独立（CI）蕴含问题是 co-NEXPTIME 难的，该问题询问给定的 CI 蕴含是否对所有具有给定基数的离散随机变量成立。即使所有变量都是二元的，该问题仍然是 co-NEXPTIME 难的。该归约来自铺砌问题的一个变体，并基于 Cheuk Ting Li 先前使用的构造，该构造用于证明一个相关问题的不可判定性，其中某些变量的基数保持无界。众所周知，有界基数的 CI 蕴含问题属于 EXPSPACE，因为其否定可以表述为一个关于实数的存在性一阶逻辑公式，其大小相对于输入大小是指数级的。
英文摘要：
We show that the conditional independence (CI) implication problem with bounded cardinalities, which asks whether a given CI implication holds for all discrete random variables with given cardinalities, is co-NEXPTIME-hard. The problem remains co-NEXPTIME-hard if all variables are binary. The reduction goes from a variant of the tiling problem and is based on a prior construction used by Cheuk Ting Li to show the undecidability of a related problem where the cardinality of some variables remains unbounded. The CI implication problem with bounded cardinalities is known to be in EXPSPACE, as its negation can be stated as an existential first-order logic formula over the reals of size exponential with regard to the size of the input.
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