# Morphic word

In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid.

Every automatic sequence is morphic.[1]

## Definition

Let f be an endomorphism of the free monoid A on an alphabet A with the property that there is a letter a such that f(a) = as for a non-empty string s: we say that f is prolongable at a. The word

$a s f(s) f(f(s)) \cdots f^{(n)}(s) \cdots \$

is a pure morphic or pure substitutive word. It is clearly a fixed point of the endomorphism f: the unique such sequence beginning with the letter a.[2][3] In general, a morphic word is the image of a pure morphic word under a coding.[1]

If a morphic word is constructed as the fixed point of a prolongable k-uniform morphism on A then the word is k-automatic. The n-th term in such a sequence can be produced by a finite state automaton reading the digits of n in base k.[1]

## Examples

• The Thue–Morse sequence is generated over {0,1} by the 2-uniform endomorphism 0 → 01, 1 → 10.[4][5]
• The Fibonacci word is generated over {a,b} by the endomorphism aab, ba.[1][4]
• The tribonacci word is generated over {a,b,c} by the endomorphism aab, bac, ca.[5]
• The Rudin–Shapiro sequence is obtained from the fixed point of the 2-uniform morphism aab, bac, cdb, ddc followed by the coding a,b → 0, c,d → 1.[5]
• The regular paperfolding sequence is obtained from the fixed point of the 2-uniform morphism aab, bcb, cad, dcd followed by the coding a,b → 0, c,d → 1.[6]

## D0L system

A D0L system (deterministic context-free Lindenmayer system) is given by a word w of the free monoid A on an alphabet A together with a morphism σ prolongable at w. The system generates the infinite D0L word ω = limn→∞ σn(w). Purely morphic words are D0L words but not conversely. However if ω = uν is an infinite D0L word with an initial segment u of length |u| ≥ |w|, then zν is a purely morphic word, where z is a letter not in A.[7]

## References

1. ^ a b c d Lothaire (2005) p.524
2. ^ Lothaire (2011) p. 10
3. ^ Honkala (2010) p.505
4. ^ a b Lothaire (2011) p. 11
5. ^ a b c Lothaire (2005) p.525
6. ^ Lothaire (2005) p.526
7. ^ Honalka (2010) p.506
• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Honkala, Juha (2010). "The equality problem for purely substitutive words". In Berthé, Valérie; Rigo, Michel. Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications 135. Cambridge: Cambridge University Press. pp. 505–529. ISBN 978-0-521-51597-9. Zbl 1216.68209.
• Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 0-521-84802-4. Zbl 1133.68067.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.