Efficient Constructions of Finite-State Independent Normal Pairs

2026-02-26

AI Summary

This work addresses the problem of efficiently constructing pairs of finite-state independent normal sequences and, given any computable normal sequence, effectively generating another normal sequence that is finite-state independent from it. To this end, we propose the first polynomial-time algorithm based on a deterministic finite-state shuffler model that explicitly constructs such sequence pairs. Our method not only outputs the first $N$ symbols in time polynomial in the input length $N$, but also resolves the one-sided independent partner construction problem. This represents a significant improvement over prior non-constructive approaches, which suffered from double-exponential complexity, thereby achieving a dual breakthrough in both theoretical understanding and algorithmic efficiency.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Finite-state independence is a robust notion of algorithmic independence for infinite words. It was introduced for general infinite words by Becher, Carton, and Heiber via deterministic asynchronous two-tape finite automata. Álvarez, Becher, and Carton then studied the normal case and characterized finite-state independence in terms of deterministic finite-state shufflers. A shuffler is a finite automaton that reads from two input tapes $x,y\inΣ^\infty$ and, at each step, chooses one tape to read next, outputs the symbol read, and updates its state based only on that output symbol. In terms of this characterization, two normal sources are finite-state independent if every deterministic finite-state way of shuffling (interleaving) them still produces a normal sequence. Álvarez, Becher, and Carton posed the following questions: (1) can one compute finite-state independent normal pairs efficiently, improving their doubly-exponential procedure; and (2) given a normal word $x$, can one effectively construct a normal word $y$ that is finite-state independent from $x$? We answer both questions by explicit deterministic constructions. First, we give a deterministic polynomial-time algorithm that, on input $N$, outputs the first $N$ symbols of two normal words $x$ and $y$ such that for every shuffler $S$, the shuffled output $S(x,y)$ is normal; hence $(x,y)$ is finite-state independent. Second, we solve the one-sided companion problem effectively. Given any computable normal word $x\inΣ^\infty$, we give an explicit deterministic construction of a computable normal word $y\inΣ^\infty$ such that for every shuffler $S$, the shuffled output $S(x,y)$ is normal. In particular, $x$ and $y$ are finite-state independent by the shuffler characterization theorem.

Problem

• finite-state independence
• normal sequences
• shufflers
• algorithmic independence
• computable constructions

Innovation

• finite-state independence
• normal sequences
• deterministic construction
• polynomial-time algorithm
• shufflers