AI 中文总结
研究可逆逻辑综合中变换规则的完备性，提出无辅助比特和垃圾输出的完备变换规则集$\mathcal{RC}^{r}$，在此基础上给出有辅助比特和垃圾输出的扩展完备规则集$\mathcal{RC}^+$，为电路相关工作奠定理论基础，助力自动化设计工具开发。
AI 中文摘要
变换规则在可逆电路优化、基于模板的重写和等价性检查中起着核心作用，确定其完备性是可逆逻辑综合中的一个基本问题。本文研究了有无辅助比特和垃圾输出的可逆电路变换规则的完备性。对于无辅助比特和垃圾输出的可逆电路，引入了一个精炼且完备的变换规则集$\mathcal{RC}^{r}$，通过用一个更简单且广泛采用的变换规则替换先前工作中提出的规则集$\mathcal{RC}$中的一条规则得到。基于此结果，进一步提出了扩展规则集$\mathcal{RC}^+$，并证明了其对于有辅助比特和垃圾输出的可逆电路的完备性。这是首个针对任意可逆电路的完备变换规则集，为电路优化等奠定了理论基础，有助于可逆和量子电路自动化设计工具的开发。
英文摘要
Transformation rules play a central role in reversible circuit optimization, template-based rewriting, and equivalence checking, and establishing their completeness is a fundamental problem in reversible logic synthesis. In this work, we investigate the completeness of transformation rules for reversible circuits both with and without ancillary bits and garbage outputs. For reversible circuits without ancillary bits and garbage outputs, we introduce a refined and complete transformation rule set $\mathcal{RC}^{r}$, obtained by replacing one rule in the rule set $\mathcal{RC}$ proposed in the previous work (TCAD, 45, pp. 3711--3724, 2026) with a simpler and widely adopted transformation rule. Since all rules in $\mathcal{RC}^{r}$ are commonly used in reversible logic synthesis, this result reveals that practically adopted transformation rules are already sufficient to establish a complete rewriting framework. Based on this result, we further propose an extended rule set, denoted by $\mathcal{RC}^+$, and prove its completeness for reversible circuits with ancillary bits and garbage outputs. To the best of our knowledge, this work presents the first complete transformation rule set for arbitrary reversible circuits, regardless of whether ancillary bits or garbage outputs are employed. The proposed framework establishes a theoretical foundation for circuit optimization, template generation, and equivalence checking, and may facilitate the development of automated design tools for reversible and quantum circuits.
