Information content in formal languages

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 $d(x,y)=\mathrm{\ell }(x\mathrm{\nabla }y)-\mathrm{\ell }(x)\wedge \mathrm{\ell }(y)$, where $\mathrm{\ell }$ is a length function and $x\mathrm{\nabla }y$ means the united theory of $x$ and $y$. Actually we mainly consider abstract abelian idempotent monoids $(S,\mathrm{\nabla })$ provided with length functions $\mathrm{\ell }$. 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 $d(x\mathrm{\nabla }a,y\mathrm{\nabla }b)\le d(x,y)+d(a,b)$. 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.