1. Languages and Codes
Chapter 15
第 15 章

2. P0L Schemes
A 0L scheme is an ordered pair (A, P), where A is a finite alphabet and P, called the set of productions, is a finite non-empty subset of A A* s.t. for aA  at least one u A* s.t. (a,u)  P.
A language which can be generated by a 0L scheme is called a 0L language.
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3. 0L Schemes  Substitutions
A mapping h : A 2B * is said to be a substitution of A into 2B * for finite alphabets A and B if h (a)  for every a A.
If (A, P) is a 0L scheme, then the mapping h defined by h (a) = {u | (a,u)  P}, a A, determines a substitution over A.
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4. Substitutions  0L Schemes
Every substitution h over A defines a 0L scheme (A, P), where P = {(a,u) | a A, uh (a)}.
A 0L scheme can be defined either by (A, P) or (A, h ).
A 0L scheme (A, P) is termed propagating, called a P0L scheme, if (a,1)  P for every
a A.
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5. Strongly Injective
A substitution h : A 2B * is strongly injective (shortly s-injective) if for  wh(A*),
! u A* s.t. wh(u).
Note that 1h(a) for each a A whenever h is s-injective.
A substitution h is non-erasing if none of h(a), a A, contains the empty word.
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6. P0L Languages
A 0L system is a triple (A, P,w), where (A, P) is a 0L scheme and w A*, called the axiom of (A, P,w).
For a substitution h over A, let h0(w) = {w} and h i(w) = h (h i–1(w )) for w A* and i 1.
The language L(A, h,w) =  i  0 h i(w) is called the 0L language generated by (A, h,w).
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7. Properties
Prop. Let h be an s-injective substitution over A. Then for each w1, w2 A*,
L(A, h,w1)  L(A, h,w2)  if and only if
either L(A, h,w1)  L(A, h,w2) or vice versa.
Prop. If h be an s-injective substitution over A, then every P0L language L with the scheme (A,h) is contained in a unique maximal P0L language with the same scheme.
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8. Property Preserving Substitutions
For an alphabet A and a property P of languages, let PA denote the family of languages with the property P over A.
A substitution h : 2A *  2B * is said to be
P-preserving if h(L)PB for every LPA .
If h : A*  2A * is a P-preserving substitution, then we said that the 0L scheme (A,h) is
P-preserving.
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9. D-Primitivity-Preserving Homomorphisms
Th. Let h : A*  B* be a homomorphism. Then the following statements are equivalent:
(1) |h ( A)| = | A| and h ( A) is a d-code;
(2) h is D(n)-preserving for every n 1;
(3) h is D(n)-preserving for some n 1;
(4) h(a), h(ab) D(1) for any two distinct
letters a,b A.
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10. D-Primitivity-Preserving Substitutions
Prop. Let h : A*  2B * be a substitution s.t. h(a) is an infix code for  a A.
If h (ab), h (a2b), h (ab2)  D(1) for  a  bA, then h (D(1))  D(1) h (A).
Prop. Let h : A*  2B * be a substitution s.t. h(a)  h(b) = for  a  b A and h (A) is a d-code. Then h is D(n)-preserving for some n  2 iff
h is a homomorphism.
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11. Pure Languages
A language L  A+ is called pure if for any uL+,  (u)L+.
Prop. Let h : A*  2B * be an s-injective substitution. Then h preserves pure language if and only if h (A) is a pure language.
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12. Primitivity-Preserving Substitutions
Prop. Let h : A*  2B * be an s-injective substitution. Then h (A) being a pure language implies that h is primitivity-preserving.
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13. Other Language-Preserving Substitutions
Prop. Let h : A*  2B * be a substitution. Then h (A) containing a maximal code over B implies that h is dense-preserving.
Prop. Let h : A*  2B * be a substitution s.t. h(A) is a thin code over B. Then h is dense-preserving if and only if h (A) is a maximal code over B .
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14. A Special Case
Prop. Let h : A*  A* be a homomorphism. Then the following statements are equivalent:
(1) h (A) = A;
(2) h is dense-preserving;
(3) h is disjunctive-preserving.
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15. D(n)-Generating 0L Schemes
For any monomorphism h over A s.t. h (A) is a d-code, a word wD(n) iff the M0L language L(A,h,w)  D(n) for any n 1.
Thus the M0L scheme (A,h) is
D(n)-generating for each n 1.
Prop. No D0L scheme (A,h) is dense-generating, where h is a homomorphism over A and | A|  2.
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16. Dense-Generating 0L Schemes
Prop. Let A={a1,a2,,am} and v = a1a2  am. If h is a non-erasing substitution over A s.t. h (A) contains a maximal prefix code C with lg(C)  2, then the P0L language L(A,h,v) is dense, i.e., (A,h) is dense-gtenerating.
Prop. Let A={a,b} and h a substitution over A defined by h (a) = {b} and h (b) = {ab,aa}. Then the P0L language L(A,h,a) is dense.
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17. -Words
A homomorphism h can be extended to -words by setting h () = h (a1) h (a2)h (an) for each -word  = a1a2 an .
For a homomorphism h over A, h i is defined by h 1 = h and h i(u) = h (h i–1(u)) for any uA* and i  2. Let wA+ be s.t exists. Then we denote this limit by h (w).
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18. Strongly Cube-Free Words
A word or an -word over A is termed square-free (resp. cube-free) if it contains no subword of the form u2 (resp. u3), u 1.
A word or an -word w is said to be strongly cube-free if wA*(au)2a(A*A) for any aA and uA*.
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19. Strong-Cube-Free-Preserving Homomorphisms
Prop Let A and B be two non-empty finite alphabets. Let h : A* A  B* B  be a homomorphism s.t. h (A) consists of strongly cube-free words and that h (A) is a non-empty subset of aB* for some aB.
If  an -word   A s.t. h ( ) is strongly cube-free, then  is square-free.
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20. Example 1 The Thue-Morse -Word
Let A={a,b} and let  be a homomorphism over A by  (a) = ab and  (b) = ba. The Thue-Morse -word is the -word   (a).
Let A={a,b,c}, B={a,b} and h : A*  B* be a homomorphism defined by h (a) = a,
h (b) = ab and h (c) = abb. Let h ( ) be the Thue-Morse -word, i.e.,
h ( ) =  (a) = abbabaabbaababbabaaba.
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21. Example 2 The Square-Free -Word 
The Thue-Morse -word h ( ) =  (a) is strongly cube-free.
By Prop ,  = h –1(  (a)) is square- free.
 = cbacabcbabcacba.
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22. Square-Freeness-Preserving Homomorphisms 1
Prop. Let h : A*  B* be a homomorphism with h (A)  {1} s.t.
(1) h (u) is square-free for
 square-free word u with lg(u)  3,
(2) No h (a) is a proper factor of an h (b)
(a,b A).
Then h preserves square-free words.
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23. Square-Freeness-Preserving Homomorphisms 2
Theorem Let h : A*  B* be a non-erasing homomorphism and
m = max{k | h (A)  B*h (Ak )B*  }.
Then h preserves square-free words if and only if h (w) is square-free for each square-free word w with lg(w)  max{| A|, m +2}.
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24. Square-Freeness-Preserving Homomorphisms 3
Prop Let h : A*  B* be a homomorphism. Let M (h) = max{lg(h (a)) | a A} and m (h) = min{lg(h (a)) | a A}.
Then h is square-freeness-preserving if and only if h (w) is square-free for any square-free word w with
lg(w)  max{3, (M (h) – 3)/ m (h)}.
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25. Square-Freeness-Preserving Example 1
Ex. Let A={a1, a2,  , an} and h : A*  A* a homomorphism. Let k : A*  A* be a homomorphism defined by k (ai) = hk (ai).
Consider h(a1) = uv, where u,v A+ with v p u and u s v. Let  : (A{an+1})*  (A{an+1})* be a homomorphism defined by
 (a1) = uan+1v,  (ai) = h(ai), i = 2,  , n, and
 (an+1) = an+1.
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26. Square-Freeness-Preserving Example 2
If h is square-freeness-presserving, then k and  are square-freeness-presserving for any k  2.
One can use this procedure to construct a square-freeness-preserving homomorphism s.t. M (h) – m (h) is large. In this case Th performs better than Prop
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27. Square-Freeness-Preserving v.s. Primitivity-preserving
Prop. Each non-erasing square-freeness-preserving homomorphism is primitivity-preserving.
Prim.-preserv.
S.-F.-preserv.
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28. A Construction of Primitivity-Preserving Homomorphisms
Cor. Let A={a,b}, B a non-empty finite alphabet and h : A*  B* a non-erasing injective homomorphism with
lg(h(a))  lg(h(b)).
Let m = max{k | h (A)  B*h (Ak )B*  }.
If h (w) is primitive for each primitive word w with lg(w)  m +2, then for any primitive word w A*a3A*, h (w) is primitive.
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29. Primitivity-Generating 0L Schemes Example
Ex. Let A={a,b} and h : A*  A* a homomorphism defined by h(a) = ba3 and h(b) = b. Since m = max{k | h (A)  B*h (Ak )B*  }=1, and h(a), h(b), h(ab), h(ab2) and h(a2b) are primitive words, h i(a3b) is primitive for any i 1.
That is, ( A,h) is a primitivity-generating 0L scheme.
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