Information content in formal languages

TL;DR

This paper explores the structure of formal languages with a focus on defining a distance measure between elements, introducing new mathematical tools like length functions, pseudometrics, and a Banach-Mazur-like distance to analyze their properties.

Contribution

It introduces a novel framework for measuring distances in formal languages using length functions and proposes a Banach-Mazur-like metric for comparing algebraic structures.

Findings

Defined a distance between elements of formal languages using length functions

Established conditions under which the distance becomes a pseudometric

Proposed a Banach-Mazur-like distance for comparing algebraic structures

Abstract

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.