Motivated by creating physical theories, formal languages $S$ with variables are considered and a kind of distance between elements of the languages is defined by the formula $d(x,y)= \ell(x \nabla y) - \ell(x) \wedge \ell(y)$, where $\ell$ is a length function and $x \nabla y$ means the united theory of $x$ and $y$. Actually we mainly consider abstract abelian idempotent monoids $(S,\nabla)$ provided with length functions $\ell$. The set of length functions can be projected to another set of length functions such that the distance $d$ is actually a pseudometric and satisfies $d(x\nabla a,y\nabla b) \le d(x,y) + d(a,b)$. We also propose a "signed measure" on the set of Boolean expressions of elements in $S$, and a Banach-Mazur-like distance between abelian, idempotent monoids with length functions, or formal languages.

翻译：受创建物理理论的启发，本文考虑带有变量的形式语言$S$，并通过公式$d(x,y)= \ell(x \nabla y) - \ell(x) \wedge \ell(y)$定义了语言元素间的一种距离，其中$\ell$是长度函数，$x \nabla y$表示$x$与$y$的联合理论。实际上，我们主要研究配备长度函数$\ell$的抽象阿贝尔幂等幺半群$(S,\nabla)$。长度函数集合可投影到另一长度函数集合，使得距离$d$实际上是一个伪度量，并满足$d(x\nabla a,y\nabla b) \le d(x,y) + d(a,b)$。我们还提出了$S$中元素布尔表达式集合上的一个“符号测度”，以及一个类似于巴拿赫-马祖尔距离的概念，用于衡量带有长度函数的阿贝尔幂等幺半群或形式语言之间的距离。