Zdeněk Frolík

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] 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.[3][4]
• The Fibonacci word is generated over {a,b} by the endomorphism aab, ba.[1][3]
• The tribonacci word is generated over {a,b,c} by the endomorphism aab, bac, ca.[4]
• 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.[4]
• 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.[5]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
1. Lothaire (2005) p.524
2. Lothaire (2011) p. 10
3. Lothaire (2011) p. 11
4. Lothaire (2005) p.525
5. Lothaire (2005) p.526