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Efficient Constructions of Finite-State Independent Normal Pairs

Subin Pulari

TL;DR

A deterministic polynomial-time algorithm is given that outputs the first $N$ symbols of two normal words x andy such that for every shuffler $S$ the shuffled output $S(x,y)$ is normal; hence x,y is finite-state independent.

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.

Efficient Constructions of Finite-State Independent Normal Pairs

TL;DR

A deterministic polynomial-time algorithm is given that outputs the first symbols of two normal words x andy such that for every shuffler the shuffled output is normal; hence x,y is finite-state independent.

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 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 , can one effectively construct a normal word that is finite-state independent from ? We answer both questions by explicit deterministic constructions. First, we give a deterministic polynomial-time algorithm that, on input , outputs the first symbols of two normal words and such that for every shuffler , the shuffled output is normal; hence is finite-state independent. Second, we solve the one-sided companion problem effectively. Given any computable normal word , we give an explicit deterministic construction of a computable normal word such that for every shuffler , the shuffled output is normal. In particular, and are finite-state independent by the shuffler characterization theorem.
Paper Structure (16 sections, 27 theorems, 146 equations, 1 algorithm)

This paper contains 16 sections, 27 theorems, 146 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Organization.
  4. Preliminaries
  5. Polynomial-time construction of a finite-state independent normal pair
  6. Proof organization.
  7. Aligned block counts and the $F_n$ constraints
  8. Conditional tail bounds under short prefix conditioning
  9. Aggregating conditional failure probabilities via a rolling potential function
  10. The algorithm and correctness
  11. Polynomial-time implementation
  12. A computable finite-state independent word for a fixed computable normal input
  13. Proof outline.
  14. Finite test sets and a computable extraction lemma
  15. Almost-sure normality for a fixed shuffler
  16. ...and 1 more sections

Key Result

theorem 1

Let $x,y\in\Sigma^\infty$ be normal. Then $x$ and $y$ are finite-state independent in the sense of Becher, Carton, and Heiber Becher2018 if and only if for every shuffler $S$ over $\Sigma$, the output $S(x,y)$ is normal.

Theorems & Definitions (60)

  • Definition 1: Normality Borel1909
  • Definition 2: Computable infinite word
  • Definition 3: Deterministic shuffler ABC
  • Definition 4: Finite-state independence for normal words
  • theorem 1: Shuffler characterization ABC
  • theorem 2
  • Definition 5
  • Definition 6: $F_n$
  • Lemma 1
  • Lemma 2
  • ...and 50 more