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 ScienceNational Chiao Tung University

Language

Entscheidungsproblem

What solves a problem?

What is a problem

Do you love me?

What is a problem

{01}* y/n

What is a problem

{01}* y/n

Is it a good model?

Does it converge?

1 1 1 1{ } { } {1, , , ,...}2 3 4ns

n= =

Introduction

Given a sequence of real numbers

Does it converge?

1 1 1 1{ } { } {1, , , ,...}2 3 4ns

n= =

Introduction

Given a sequence of real numbers

Ratio test, root test, divergence test, monotonic convergence theorem… etc

Introduction

Now given a sequence of languages

1 2 3{ } { , , ,....}nL L L L=

Does it converge?

Introduction

Now given a sequence of language

1 2 3{ } { , , ,....}nL L L L=

Does it converge? ?

1 1 1 1{ } { } {1, , , ,...}2 3 4ns

n= =

Introduction

Let’s review how calculus proves1lim 0

n n→∞=

Prove that 1lim lim 0nn n

Sn→∞ →∞

= =

10 R such that | 0 |

1

1

1

1 1| 0 |

k n kn

let k

n k

n

n n

ε ε

ε

ε

ε

ε

∀ > ∃ ∈ ∀ > − <

⇒ =

⇒∀ > =

⇒ >

⇒ − = <

Introduction

For example, given

{0 1 | 0}n nL n= ≥

Introduction

A little observation shows that

0

{0 1 | 0} lim

01

n nnn

ni i

ni

L n L

where L

→∞

=

= ≥ =

=

Introduction

0

{ } { 01 }

{{ },{ ,01},{ ,01,0011},{ ,01,0011,000111},...}

ni i

ni

L

ε ε ε ε=

=

=

Each is regular but lim is context-freen nnL L

→∞

Introduction

These discussions depend on two keys:

1. The property of a field2. The notion of distance, called “metric”

Metric spaceConsider a set X, if

1. 2.3.

Then we call X is a metric spaceAny function with these 3 properties is called a distance function, or a metric.

( , ) 0 if ; otherwise ( , ) 0 d 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∀ ∈

Metric space

e.g:Euclidean space is an important example of metric space,whose metric is defined as

kR

( , ) | | where , kd x y x y x y R= − ∈

2

1(For , | | ( ))

kk

ii

x R x x=

∈ = ∑

Summary

{01}* y/n

We did not alter the model

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!

Any Question ?

Thank you for listening