Elsevier

Theoretical Computer Science

Volume 806, 2 February 2020, Pages 28-41
Theoretical Computer Science

Embedding a θ-invariant code into a complete one

https://doi.org/10.1016/j.tcs.2018.08.022Get rights and content
Under an Elsevier user license
open archive

Highlights

  • Given an alphabet A, and an (anti-)automorphism θ, a set LA is θ-invariant if θ(L)=L.

  • In the framework of θ-invariant sets, a defect effect is highlighted.

  • Several non-trivial examples of finite complete θ-invariant codes are presented.

  • Over a finite alphabet, any regular non complete θ-invariant code can be embedded into a complete one.

  • Given a thin θ-invariant code, being a maximal code, or being maximal in the family of θ-invariant codes, or being complete are equivalent properties.

Abstract

Let A be an arbitrary alphabet and let θ be an (anti-)automorphism of A (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ (θ-invariant for short) that is, languages L satisfying θ(L)L. We establish an extension of the famous defect theorem. With regard to the so-called notion of completeness, we provide a series of examples of finite complete θ-invariant codes. Moreover, we establish a formula which allows to embed any non-complete θ-invariant code into a complete one. As a consequence, in the family of the so-called thin θ-invariant codes, maximality and completeness are two equivalent notions.

Keywords

Antimorphism
Anti-automorphism
Automorphism
(Anti-)automorphism
Bernoulli distribution
Bifix
Code
Complete
Context-free
Defect
Equation
Finite
Invariant
Involutive
Label
Maximal
Morphism
Order
Overlap
Overlapping-free
Prefix
Regular
Suffix
Thin
Tree
θ-Invariant
θ-Code
Uniform
Variable-length code
Word

Cited by (0)