随机有根树上的二阶逻辑
Second order logic on random rooted trees
摘要：
我们在随机有根树的背景下解决逻辑和可表达性问题。有根树的无限性不能表示为一阶句，而可以表示为存在一元二阶句。另一方面，有限性不能用EMSO来表达。对于一类广泛的随机树模型，包括高尔顿-沃森树的后代分布，有充分的支持，我们证明了更强的声明，有限性不同意任何EMSO的零集。我们构造了一个有限树和一个非零集的无限树，不能区分彼此的任何EMSO的给定参数。这一点可以通过集合卵石埃克塞特福赫特游戏（其中初始着色轮之后是给定数量的卵石轮）来证明。
英文摘要：
We address questions of logic and expressibility in the context of random rooted trees. Infiniteness of a rooted tree is not expressible as a first order sentence, but is expressible as an existential monadic second order sentence (EMSO). On the other hand, finiteness is not expressible as an EMSO. For a broad class of random tree models, including Galton-Watson trees with offspring distributions that have full support, we prove the stronger statement that finiteness does not agree up to a null set with any EMSO. We construct a finite tree and a non-null set of infinite trees that cannot be distinguished from each other by any EMSO of given parameters. This is proved via set-pebble Ehrenfeucht games (where an initial colouring round is followed by a given number of pebble rounds).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
