Boundary of Words
q.v. Zoho Paper. SIL
Topological Group Language Theory
TANAKA Akio
Preliminary Note 3
Boundary of Words
[Theorem]
1
Distance space ( X , d)
Gromov product ( y | z ) x
2
Hyperbolic space X
point sequence { x i 
X }
X }
Set of all the { x i 
X } S ∞ ( X )
X } S ∞ ( X )
Point sequence of X { y i }
When { y i } is lim i →∞ ( x i | y i ) = 0, { y i } 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 x 0 ( X, d )
Arbitrary 3 points of X x, y, z
When ( X, d ) satisfies the next, it is called δ -hyperbolic.
( x | y ) x 0 
min{( x | z)x 0 , ( y | z ) x 0 }- δ
min{( x | z)x 0 , ( y | z ) x 0 }- 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.
, 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 ) } +2 δ
max{ d ( x, z ) + d ( y, w ), d ( x, w ) + d ( y, z ) } +2 δ
4
Distance space X
Arc of X α : [0, λ ] → X
Arbitrary s, t 
[0, λ ]
[0, λ ]
d ( α ( s ), α (t)) = | s-t |
Arc α is called geodesic segment.
Geodesic segment from x to y that is x,y 
X is expressed by 
.
X 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.
xyz = is called geodesic triangle.
[Impression]
1
Word is identified with point sequence { x i 
X }.
X }.
Language is identified with S ∞ ( X ).
New generated word is identified Point sequence of X , { y i } that has condition lim i →∞ ( x i | y i ) .
The new generated word is also in language, that is to say, S ∞ ( X ).
2
From the condition lim i →∞ ( x i | y i ), 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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