Problem
Research questions and friction points this paper is trying to address.
random functions
word reconstruction
function composition
total variation distance
auto-correlation
Composition of random functions and word reconstruction
🤖 AI Summary
This work addresses the identifiability of an unknown word \( w \) from a single sample of the random function it induces. Specifically, it investigates whether \( w \) can be partially recovered with high probability when the observed function arises from the composition of two independent uniform random functions according to the pattern prescribed by \( w \). The authors introduce a novel analytical framework based on total variation distance and define a word-dependent constant \( c(w) \) that quantifies the distinguishability of functions induced by different words. Assuming Schanuel’s conjecture, they prove that non-isomorphic words yield distinct constants \( c(w) \). Their main contributions include high-probability recovery of the length of \( w \) and its maximal exponent, as well as an effective separation guarantee: if \( c(w_1) \neq c(w_2) \), the corresponding induced functions are distinguishable under total variation distance.
📝 Abstract
Given two functions $\mathbf{a}\!:\! [n] \rightarrow [n]$ and $\mathbf{b}\!:\! [n] \rightarrow [n]$ chosen uniformly at random, any word $w=w_1w_2\dots w_k\in \{a,b\}^k$ induces a random function $\mathbf{w}\!:\! [n] \rightarrow [n]$ by composition, i.e. $\mathbf{w}=φ_{w_k}\circ \dots \circ φ_{w_1}$ with $φ_a=\mathbf{a}$ and $φ_b=\mathbf{b}$. We study the following question: assuming $w$ is fixed but unknown, and $n$ goes to infinity, does one sample of $\mathbf{w}$ carry enough information to (partially) recover the word $w$ with good enough probability?
We show that the length of $w$, and its exponent (largest $d$ such that $w={u}^d$ for some word ${u}$) can be recovered with high probability. We also prove that the random functions stemming from two different words are separated in total variation distance, provided that certain ``auto-correlation'' word-depending constant $c(w)$ is different for each of them. We give an explicit expression for $c(w)$ and conjecture that non-isomorphic words have different constants. We prove that this is the case assuming a major conjecture in transcendental number theory, Schanuel's conjecture.
Innovation
Methods, ideas, or system contributions that make the work stand out.
random function composition
word reconstruction
total variation distance
auto-correlation constant
Schanuel's conjecture
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