Tuesday 18 June 2013

Boundary of Words

Topological Group Language Theory

Preliminary Note 3
Boundary of Words

Distance space     (X, d)
Gromov product (y|z)x
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).

Hyperbolicity for general distance space is defined by Gromov product.
Details are below.
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.
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
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.

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).
From the condition limi→∞(xi|yi), spherical surface is considered as boundary at infinity by the comparison with Poincaré model.
Spherical surface is considered as the unit of language. 
Language's distance and warp is also considered under hyperbolic space.
References are below.

#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

No comments:

Post a Comment