Tuesday, 18 June 2013

Boundary of Words

Topological Group Language Theory

   
Preliminary Note 3
Boundary of Words


[Theorem]
1
Distance space     (X, d)
Gromov product (y|z)x
2
Hyperbolic space     X
point sequence     {xiX}
Set of all the {xiX}    S(X)
Point sequence of      {yi}    
When {yi} is limi→∞(xi|yi) = 0, {yi} is in S(X).


[Explanation]
1
Hyperbolicity for general distance space is defined by Gromov product.
Details are below.
2
Distance space is defied by the next.
Distance space that has basic point x0       (X, d)
Arbitrary 3 points of X       x, y, z
When (X, d) satisfies the next, it is called δ-hyperbolic.
(x|y)x0 min{(x|z)x0, (y|z)x0}-δ
When distance space (X, d) is δ-hyperbolic for arbitrary base point, X is called δ-hyperbolic.
Here for a certain δ-hyperbolic space is abbreviatedly called hyperbolic.
3
The next condition is equivalent with what (X, d) is δ-hyperbolic.
d(x, y) + d(z, w max{ d(x, z) + d(y, w), d(x, w) + d(y, z) } +2δ
4
Distance space       X
Arc of X     α : [0, λ] → X
Arbitrary s, t [0, λ]
d(α(s), α(t)) = |s-t|
Arc α is called geodesic segment.
Geodesic segment from x to y that is x,yX is expressed by .
When arbitrary 2 points of distance space (X, d) are combinable by geodesic segment, X is called geodesic space.xyz =  is called geodesic triangle.


[Impression]
1
Word is identified with point sequence {xiX}.
Language is identified with S(X).
New generated word is identified Point sequence of X, {yi} that has condition limi→∞(xi|yi).
The new generated word is also in language, that is to say, S(X).
2
From the condition limi→∞(xi|yi), spherical surface is considered as boundary at infinity by the comparison with Poincaré model.
3
Spherical surface is considered as the unit of language. 
Language's distance and warp is also considered under hyperbolic space.
References are below.


[References]
#1 Quantum Theorey for Language
#2 Distance Theory
#3 Warp Theory
The upper papers and the related papers with the themes are seen at Sekinan Linguistuic Field.


To be continued
Tokyo February 12, 2009

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