This document discusses using the concept of a metric space to determine if a sequence of languages converges. It introduces the idea of a metric space as a set where a distance function is defined that satisfies specific properties. As an example, it shows that a sequence of languages where each language is a finite union of regular languages converges in the limit to a context-free language. Determining if sequences of languages converge depends on understanding the properties of the spaces involved and how to define a suitable distance metric between languages.

1. A Metric for
Language and Problem
Project members:
郭安哲 Kuo, An Che
陳敬之 Chen, Ching Tzu
Instructor:
陳穎平 Ying-ping Chen
Department of Computer Science
National Chiao Tung University

2. Language

3. Entscheidungsproblem

4. What solves a problem?

5. What is a problem
Do you love me?

6. What is a problem
{01}* y/n

7. What is a problem
{01}* y/n
Is it a good model?

8. Does it converge?
1 1 1 1
{ } { } {1, , , ,...}
2 3 4
ns
n
= =
Introduction
Given a sequence of real numbers

9. Does it converge?
1 1 1 1
{ } { } {1, , , ,...}
2 3 4
ns
n
= =
Introduction
Given a sequence of real numbers
Ratio test, root test, divergence test,
monotonic convergence theorem… etc

10. Introduction
Now given a sequence of languages
1 2 3{ } { , , ,....}nL L L L=
Does it converge?

11. Introduction
Now given a sequence of language
1 2 3{ } { , , ,....}nL L L L=
Does it converge?
?

12. 1 1 1 1
{ } { } {1, , , ,...}
2 3 4
ns
n
= =
Introduction
Let’s review how calculus proves
1
lim 0
n n→∞
=

13. Prove that
1
lim lim 0n
n n
S
n→∞ →∞
= =
1
0 R such that | 0 |
1
1
1
1 1
| 0 |
k n k
n
let k
n k
n
n n
ε ε
ε
ε
ε
ε
∀ > ∃ ∈ ∀ > − <
⇒ =
⇒ ∀ > =
⇒ >
⇒ − = <

14. Introduction
For example, given
{0 1 | 0}n n
L n= ≥

15. Introduction
A little observation shows that
0
{0 1 | 0} lim
01
n n
n
n
n
i i
n
i
L n L
where L
→∞
=
= ≥=
= 

16. Introduction
0
{ } { 01 }
{{ },{ ,01},{ ,01,0011},{ ,01,0011,000111},...}
n
i i
n
i
L
ε ε ε ε
=
=
=

Each is regular but lim is context-freen n
n
L L
→∞

17. Introduction
These discussions depend on two keys:
1. The property of a field
2. The notion of distance, called “metric”

18. Metric space
Consider a set X, if
1.
2.
3.
Then we call X is a metric space
Any function with these 3 properties is called a distance
function, or a metric.
( , ) 0 if ; otherwise ( , ) 0d p q p q d p q> ≠ =
( , ) ( , )d p q d q p=
( , ) ( , ) ( , )d p q d p r d r q r X≤ + ∀ ∈
,p q X∀ ∈

19. Metric space
e.g:
Euclidean space is an important example of metric space,
whose metric is defined as
k
R
( , ) | | where , k
d x y x y x y R=− ∈
2
1
(For , | | ( ))
k
k
i
i
x R x x
=
∈ =∑

20. Summary
{01}* y/n
We did not alter the model

21. What is a problem
{01}*
+ - < ||
y/n
We add properties to the model,
and hope it will help us understand
the nature of problem solving
better!

22. Any Question ?

23. Thank you for listening 