We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables ($X_0,\ X_1,\ X_2,\dots$) with a mapping $\sigma$ whose domain is the variable sequence and whose range consist of terms that may contain variables from the sequence. From a given term $t$, an infinite sequence of terms may be produced by iterative application of $\sigma$. Given a unification problem $U$ and mapping $\sigma$, the \textit{schematic unification problem} asks whether all unification problems $U$, $\sigma(U)$, $\sigma(\sigma(U))$, $\dots$ are unifiable. We provide a terminating and sound algorithm. Our algorithm is \textit{complete} if we further restrict ourselves to so-called $\infty$-stable problems. We conjecture that this additional requirement is unnecessary for completeness. Schematic unification is related to methods of inductive proof transformation by resolution and inductive reasoning.

翻译：我们提出了一种面向术语代数的一阶统一的推广，其中变量索引作为对象语言的一部分。我们通过将某些变量序列（$X_0,\ X_1,\ X_2,\dots$）与一个映射$\sigma$关联来利用变量索引，该映射的定义域为变量序列，值域为可包含该序列中变量的项。从给定项$t$出发，通过迭代应用$\sigma$可生成无限项序列。给定统一问题$U$和映射$\sigma$，\textit{图解统一问题}询问所有统一问题$U$、$\sigma(U)$、$\sigma(\sigma(U))$、$\dots$是否均可统一。我们提供了一个可终止且可靠的算法。若进一步约束至所谓$\infty$-稳定问题，该算法是\textit{完备的}。我们猜想该附加条件对完备性并非必要。图解统一与基于消解和归纳推理的归纳证明变换方法相关。