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：我们提出了一种将一阶统一化推广到项代数的通用方法，其中变量索引是对象语言的一部分。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，值域为可能包含序列中变量的项。对于给定项$t$，通过迭代应用$\sigma$可生成一个无限项序列。给定统一化问题$U$和映射$\sigma$，**图解统一化问题**询问是否所有统一化问题$U$、$\sigma(U)$、$\sigma(\sigma(U))$、$\dots$均可统一化。我们提供了一种可终止且可靠的算法。若进一步限制于所谓的$\infty$-稳定问题，该算法是**完备的**。我们推测此额外要求对于完备性并非必要。图解统一化与基于归结的归纳证明变换及归纳推理方法相关。