AI 中文总结
该研究探讨加权树自动机初始代数语义的生成能力，证明特定强双半群的性质，得出含高秩符号时加权树自动机可生成强双半群全部元素等结论，与加权串自动机形成对比。
AI 中文摘要
我们研究在强双半群（因此也包括半环）上的加权树自动机的初始代数语义的生成能力，以及加权树自动机仅能生成有限个值的条件问题。我们证明存在一个右分配强双半群，它是双局部有限的但不是局部有限的。我们还证明，如果秩字母表包含一个秩至少为2的符号，那么对于任何有限生成的强双半群，加权树自动机可通过其初始代数语义生成该强双半群的所有元素。作为这些结果的推论，对于双局部有限且非局部有限的右分配强双半群，只要输入秩字母表包含一个秩至少为2的符号，加权树自动机就能生成无限多个值，这与加权串自动机仅能生成有限个值的情况形成鲜明对比。进一步的推论是，对于任何有限生成的半环，存在一个加权树自动机，可通过其运行语义生成该半环的所有元素。
英文摘要
We consider the generating power of the initial algebra semantics of weighted tree automata over strong bimonoids (hence also over semirings) and the question under which conditions the weighted tree automata can produce only finitely many values. We show that there exists a right-distributive strong bimonoid which is bi-locally finite but not locally finite. We also show that if the ranked alphabet contains a symbol with rank at least two, then for any finitely generated strong bimonoid, weighted tree automata can generate, via their initial algebra semantics, all elements of the strong bimonoid. As a consequence of these results, for bi-locally finite right-distributive strong bimonoids which are not locally finite, weighted tree automata can generate infinitely many values, provided that the input ranked alphabet contains a symbol with rank at least two. This is in sharp contrast to the setting of weighted string automata, which can generate only finitely many values. As a further consequence, for any finitely generated semiring, there exists a weighted tree automaton which generates, via its run semantics, all elements of the semiring.
CommentsIn Proceedings AFL 2026, arXiv:2608.23071
Journal refEPTCS 451, 2026, pp. 123-139
