We consider the complexity class ACC1 and related families of arithmetic circuits. We prove a variety of collapse results, showing several settings in which no loss of computational power results if fan-in of gates is severely restricted, as well as presenting a natural class of arithmetic circuits in which no expressive power is lost by severely restricting the algebraic degree of the circuits. We draw attention to the strong connections that exist between ACC1 and VP, via connections to the classes CC1[m] for various m. These results tend to support a conjecture regarding the computational power of the complexity class VP over finite algebras, and they also highlight the significance of a class of arithmetic circuits that is in some sense dual to VP. In particular, these dual-VP classes provide new characterizations of ACC1 and TC1 in terms of circuits of semiunbounded fan-in. As a corollary, we show that ACCi = CCi for all $${i \geq 1}$$i≥1.

中文翻译：

---

双 VP 课程

我们考虑复杂度等级 ACC1 和相关的算术电路系列。我们证明了各种崩溃结果，显示了几种设置，其中如果严格限制门的扇入，则不会导致计算能力损失，并展示了一类自然的算术电路，其中通过严格限制，不会损失表达能力电路的代数次数。我们提请注意 ACC1 和 VP 之间存在的强连接，通过连接到不同 m 的类 CC1[m]。这些结果倾向于支持关于复杂性类 VP 在有限代数上的计算能力的猜想，并且它们也突出了一类在某种意义上与 VP 对偶的算术电路的重要性。特别是，这些双 VP 类在半无界扇入电路方面提供了 ACC1 和 TC1 的新特征。作为推论，我们证明对于所有 $${i \geq 1}$$i≥1，ACCi = CCi。