Composition of random functions and word reconstruction

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.

Figures (8)

• Figure 1: Samples of the random function $\mathbf{w}$ for $n=1000$, for $w=ab$ (left) and $w=aaabaa$ (right). Since the two words have different length one can distinguish them by looking at the proportion of leaves (Theorem \ref{['thm:main_leaves']}).
• Figure 2: Plot of the constant $c(w)$ for the 511 words of length at most $9$ (each cross is a word, the ordinate is the length and the abscissa is $c(w)$). Most words seem to concentrate towards lower values when the length increases, which is consistent with the fact that "typical words have small auto-correlation". Although this cannot be checked visually, there are $2^{k-1}$ distinct values at height $k$, and they are altogether distinct, in accordance with Conjecture \ref{['conj:cwseparate']}. The rightmost point on the $k$-th height corresponds to the word $w=a^{k}$.
• Figure 3: The $w$-in-ball and universal $w$-in-cover of a vertex $x$ for $w=aaba$. On the left figure, the ball is a directed tree so the cover is isomorphic to the ball. On the middle figure, the ball is not a tree. The in-cover of the middle example is displayed on the right figure, with vertices arbirarily labeled. The following pairs of vertices in the right figure correspond to the same vertex in the $w$-in-ball: $(6,x)$, $(1,8)$, $(2,9)$ and $(3,10)$. Note that the right example coincide, as a tree, with the left one, in particular the left and middle examples have the same $w$-in-cover.
• Figure 4: Direct interpretation of the formula in \ref{['eq:branch-simple']} for $\mathbb P_{ BGW}^{\langle i\rangle}(Z^{\langle i\rangle}_k=0)$. The marked vertex at generation $i$ (among $Z^{\langle i\rangle}_i$) is squared. The number in each subtrees indicate a strict upper bound on the height of its offspring under the event $Z^{\langle i\rangle}_k=0$, each contributing a factor $\eta_{k-t}$ in \ref{['eq:branch-simple']}.
• Figure 5: Left: A branching process evolving according to the law $\mathbb P^{i,j,\ell}_{\mathbf{X}}$ where the two marked vertices are not related: two vertices (boxed), respectively at heights $i$ and $j$, which are not ancestor one of the other, are marked and conditioned to the fact that the subtrees hanging from these marked vertices coincide for $\ell$ generations. Here the event $X_k=0$ is considered, which forces respectively $(Z_i^{(1)}-2),(Z_{j-i}^{(2)}-1),Z_\ell^{(3)}$ and $Z_\ell^{(3)}$ independent subtrees to have heights less than $(k-i),(k-j),(k-i-\ell)$, and $(k-j-\ell)$, which directly gives \ref{['eq:monster_int1']}. Right: The "spine" representation of this event. Numbers in each small subtree indicate a strict upper bound on their height under the event $X_k=0$. This directly leads to the expression in Proposition \ref{['prop:little_monster']}.

Theorems & Definitions (97)

• Conjecture I
• Theorem 1: Number of leaves
• Corollary 2: Guessing the length
• Theorem 3: Weighted cycle-count
• Example 1: Cycles of length at most $L$
• Corollary 4: Guessing the exponent
• Remark 2
• Theorem 5: Variance and second moment for the number of leaves
• Corollary 6: TV-separation
• Conjecture II