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.

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

$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. In general, a morphic word is the image of a pure morphic word under a coding.

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.

Examples

• The Thue–Morse sequence is generated over {0,1} by the 2-uniform endomorphism 0 → 01, 1 → 10.
• The Fibonacci word is generated over {a,b} by the endomorphism a → ab, b → a.
• The tribonacci word is generated over {a,b,c} by the endomorphism a → ab, b → ac, c → a.
• 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.
• 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.

See also

• Lindenmayer system

References

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• 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