Theory of Computing Systems

, Volume 62, Issue 7, pp 1555–1572

# Finite-State Independence

• Verónica Becher
• Olivier Carton
• Pablo Ariel Heiber
Article

## Abstract

In this work we introduce a notion of independence based on finite-state automata: two infinite words are independent if no one helps to compress the other using one-to-one finite-state transducers with auxiliary input. We prove that, as expected, the set of independent pairs of infinite words has Lebesgue measure 1. We show that the join of two independent normal words is normal. However, the independence of two normal words is not guaranteed if we just require that their join is normal. To prove this we construct a normal word x 1 x 2 x 3… where x 2n = x n for every n. This construction has its own interest.

## Keywords

Finite-state automata Infinite sequences Normal sequences Independence

## Notes

### Acknowledgements

The authors acknowledge Alexander Shen for many fruitful discussions. The authors are members of the Laboratoire International Associé INFINIS, CONICET/Universidad de Buenos Aires–CNRS/Université Paris Diderot. Becher is supported by the University of Buenos Aires and CONICET.

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## Authors and Affiliations

• Verónica Becher
• 1
• Olivier Carton
• 2
• Pablo Ariel Heiber
• 3
1. 1.Departamento de Computación, Facultad de Ciencias Exactas y Naturales & ICCUniversidad de Buenos Aires & CONICETBuenos AiresArgentina
2. 2.Institut de Recherche en Informatique FondamentaleUniversité Paris DiderotParisFrance
3. 3.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos Aires & CONICETBuenos AiresArgentina