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：以物理理论为动力,考虑正式语言中带有变量的美元,并且根据公式 $d(x,y) =\ ell(x\ nabla y) -\ ell(x)\wedge\ ell(y) $,美元是一个长函数,$x\ nabla y y 美元是指美元和美元的统一理论。实际上,我们主要考虑以长度函数 $(S,\nabla) 提供的抽象ABelian 一元(S,\nabla) 和美元为单位。一套长度函数可以预测成另一组长函数,这样一组长函数的距离实际上是一个假的(x\ nabla a,y\ nabla b)\ le d(x,y) + d(a,b) 美元。我们还提议对以 $S $ 提供的元素的布利安语表达式表示“ 签名” 。
相关内容
微信扫码咨询专知VIP会员