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

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 a → ab, b → a.^[1][4]
• The tribonacci word is generated over {a,b,c} by the endomorphism a → ab, b → ac, c → a.^[5]
• The Rudin–Shapiro sequence is obtained from the fixed point of the 2-uniform morphism a → ab, b → ac, c → db, d → dc 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 a → ab, b → cb, c → ad, d → cd 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 = lim_n→∞ 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

• 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. 

Further reading

• Cassaigne, Julien; Karhumäki, Juhani (1997). "Toeplitz words, generalized periodicity and periodically iterated morphisms". European Journal of Combinatorics 18: 497–510. doi:10.1006/eujc.1996.0110. Zbl 0881.68065.