Letter-to-letter transducers are a standard formalism for modeling reactive systems. Often, two transducers that model similar systems differ locally from one another, by behaving similarly, up to permutations of the input and output letters within "rounds". In this work, we introduce and study notions of simulation by rounds and equivalence by rounds of transducers. In our setting, words are partitioned to consecutive subwords of a fixed length $k$, called rounds. Then, a transducer $\mathcal{T}_1$ is $k$-round simulated by transducer $\mathcal{T}_2$ if, intuitively, for every input word $x$, we can permute the letters within each round in $x$, such that the output of $\mathcal{T}_2$ on the permuted word is itself a permutation of the output of $\mathcal{T}_1$ on $x$. Finally, two transducers are $k$-round equivalent if they simulate each other. We solve two main decision problems, namely whether $\mathcal{T}_2$ $k$-round simulates $\mathcal{T}_1$ (1) when $k$ is given as input, and (2) for an existentially quantified $k$. We demonstrate the usefulness of the definitions by applying them to process symmetry: a setting in which a permutation in the identities of processes in a multi-process system naturally gives rise to two transducers, whose $k$-round equivalence corresponds to stability against such permutations.

翻译：字母到字母转换器是建模反应式系统的标准形式化工具。通常，两个建模相似系统的转换器在局部行为上存在差异，表现为在“轮次”内对输入和输出字母进行排列后具有相似行为。本文引入并研究了转换器的轮次模拟与轮次等价概念。在我们的设定中，单词被划分为固定长度$k$的连续子词，称为轮次。直观而言，若对于每个输入词$x$，可对$x$的每个轮次内的字母进行排列，使得转换器$\mathcal{T}_2$在排列后单词上的输出本身即为$\mathcal{T}_1$在$x$上输出的一个排列，则称转换器$\mathcal{T}_1$被转换器$\mathcal{T}_2$进行$k$轮模拟。最后，若两个转换器相互模拟，则称它们为$k$轮等价。我们解决了两个主要判定问题：即(1)当$k$作为输入给定时，以及(2)针对存在量化的$k$，判定$\mathcal{T}_2$是否$k$轮模拟$\mathcal{T}_1$。通过将定义应用于过程对称性（即多过程系统中过程标识的排列自然产生两个转换器，其$k$轮等价对应于针对此类排列的稳定性），我们展示了定义的有效性。