We introduce semiring-weighted regular tree grammars with storage where the semiring is complete. We show that the class of weighted tree languages generated by them is a principal abstract family of weighted tree languages provided that the semiring is commutative, the rank of symbols occurring in trees is bounded by a global parameter, and that the storage is finitely encoded and contains a reset instruction. Moreover, we prove the same statement for the iterated pushdown (instead of finitely encoded storage containing a reset instruction).

中文翻译：

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加权树语言的主要抽象族

我们引入了半环加权的正则树文法，其中半环是完整的。我们表明，由它们生成的加权树语言类是加权树语言的主要抽象族，前提是半环是可交换的，树中出现的符号的等级受全局参数的限制，并且存储是有限编码的，并且包含复位指令。此外，我们证明了迭代下推的相同语句（而不是包含重置指令的有限编码存储）。