The unification problem in a normal modal logic is to determine, given a formula φ, whether there exists a substitution σ such that σ(φ) is in that logic. In that case, σ is a unifier of φ. We shall say that a set of unifiers of a unifiable formula φ is minimal complete if for all unifiers σ of φ, there exists a unifier τ of φ in that set such that τ is more general than σ and for all σ,τ in that set, σ≠τ, neither σ is more general than τ, nor τ is more general than σ. When a unifiable formula has no minimal complete set of unifiers, the formula is nullary. We usually distinguish between elementary unification and unification with parameters. In elementary unification, all variables are likely to be replaced by formulas when one applies a substitution. In unification with parameters, some variables—called parameters—remain unchanged. In this paper, we prove that normal modal logics KB, KDB and KTB as well as infinitely many normal modal logics between KDB and KTB possess nullary formulas for unification with parameters.

中文翻译：

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关于$$\mathbf {KB}$$KB 和$$\mathbf {KTB}$$KTB 之间模态逻辑的合一类型

正常模态逻辑中的统一问题是在给定公式 φ 的情况下确定是否存在替代 σ，使得 σ(φ) 在该逻辑中。在这种情况下，σ 是 φ 的统一符。如果对于 φ 的所有统一子 σ，在该集合中存在一个 φ 的统一子 τ，使得 τ 比 σ 更一般，并且对于所有 σ,τ ，则我们会说可统一公式 φ 的一组统一子是最小完备的set, σ≠τ, σ 既不比 τ 更一般， τ 也不比 σ 更一般。当可统一公式没有最小完整统一词集时，该公式是无效的。我们通常区分基本统一和参数统一。在基本统一中，当应用替代时，所有变量都可能被公式替代。在与参数的统一中，一些变量——称为参数——保持不变。在本文中，