Nominal syntax is a generalisation of first-order syntax that includes names, a notion of name binding and an elegant axiomatisation of alpha-equivalence, based on nominal set theory. However, it does not take into account non-capturing atom substitution, which is not a primitive notion in nominal syntax. We consider an extension of nominal syntax with non-capturing atom substitutions and show that matching is decidable and finitary but unification is undecidable in general. The proof of undecidability of unification is obtained by reducing Hilbert's tenth problem to unification of extended nominal terms. We provide a general matching algorithm and characterise a class of problems for which matching is unitary, giving rise to expressive and efficient rewriting systems.

标称语法是基于标称集合论的一阶语法的概括，包括名称，名称绑定的概念和alpha等价的优雅公理化。但是，它没有考虑非捕获原子替换，这不是名义语法中的原始概念。我们考虑使用不捕获原子的替换来扩展名义语法，并表明匹配是可确定的和确定的，但统一通常是不可确定的。通过将希尔伯特的第十个问题简化为扩展名义条款的统一，可以获得统一的不可判定性的证明。我们提供了一种通用的匹配算法，并描述了一类匹配是单一的问题，从而产生了表达高效的重写系统。