# 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

${\displaystyle asf(s)f(f(s))\cdots f^{(n)}(s)\cdots \ }$

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