Loading...

PPT – A Theory of Interactive Computation PowerPoint presentation | free to download - id: 4ea7f-MDJlN

The Adobe Flash plugin is needed to view this content

A Theory of Interactive Computation

- Jan van Leeuwen, Jiri Widermann
- Presented by Choi, Chang-Beom
- KAIST

Content

- Introduction
- A Model of Interactive Computation
- Interactively Computable Relations
- Interactive Recognitions
- Interactive Generations
- Interactive Translations
- Conclusion and Future works

Preliminary

- On-line Algorithm
- online algorithm is one that can process its

input piece-by-piece, without having the entire

input available from the start - Example Stock estimation
- Off-line Algorithm
- offline algorithm is given the whole problem data

from the beginning and is required to output an

answer which solves the problem - Example Summation of 1 100

Introduction

- Why Interactive System?
- Modern computer systems are built from components

that communicate and compute, while interacting

with their environment. - Web Server Client (Server/Client Model)
- Ubiquitous computing
- Traditional Model is incomplete!

Why?

Purpose of Interactive System

- Not to compute some finial result
- React to environment or Interact with environment

- Maintain a well-defined action-reaction behavior

Why Traditional Model is Incomplete to Capture

Interactive Properties

- Input is unpredictable
- Input is not specified in advance
- Interactive system never terminate (unless a

fault occurs) - Interactive system may change over time
- It is concurrent processes and continuing

interaction

Examples of Inactive Systems

Server

Request

Respond

Attack

Hacker

Difference Between Interactive System and

Traditional System

- Traditional system
- There is no interaction between input and output
- Accepting input on initiation
- Producing output on termination
- Turing Machine with fixed input
- Interactive System
- Interaction between input and output
- Inputs can depend on intermediate outputs
- Traditional Turing Machine is not adequate to

Interactive System

Content

- Introduction
- A Model of Interactive Computation
- Interactively Computable Relations
- Interactive Recognitions
- Interactive Generations
- Interactive Translations
- Conclusion and Future works

A Model of Interactive Computation

Component (C)

alphabet

Environment (E)

Alphabet S 0, 1, t,

Definitions

- C Component
- E Environment
- Alphabet S 0, 1, t,
- 0, 1 actual symbols
- t silent or empty symbol
- fault or error symbol
- Interactive input streams
- e e0e1 et
- Interactive output streams
- c c0c1
ct
(if Cs output is c then C is

interactive component )

t

Faults

- Fault Rules
- If C receives a symbol from E, then C will

output a within a finite amount of time after

this as well (and vice versa) - If no s are exchanged, the interaction between

E and C is called fault-free (error-free)

Definitions (Cont)

- Assumptions
- E(C) sends a signal to C(E) during time t then

C(E) knows this signal from next-time moments

onward - E is totally nondeterministic and unpredictable

in generating its next signal Et-1(ct-1) ? et - Cs output at time t is depend on e0e1
et-1 and

c0c1 ct-1 - e e with out t
- c c with out t

t

Interactiveness

- For all times t, when E sends a non-silent signal

to C at time t, then C sends a non-silent signal

to E at some time t with t gt t and vice versa

t

Non-silent

silent

Definition 1

- An interaction pair of C and E is any pair (e,c)

such that e e0e1 et and c c0c1 ct

represent an interactive computation of C in

response to E - Full environmental activity
- At all time t, E sends a non-silent signal to C
- Only for E, C can emit silent signal but for

finite time

Component

- Memory space of C is always finite but

potentially unbounded - C can build up an infinite database of knowledge
- Algorithmicity
- Program evolves over time and which answers

whether Et-1(ct-1) ? et or not - Regardless of Es actual behavior, there is an

algorithmic way to verify afterwards that a

sequence could have been generated by E

Interactive Transduction

E

C

e

c

?-transducer on infinite sequence

Definition 2 3

- The behavior of C with respect to E is the set

TC (e, c)(e,c) is an interaction pair of C

and E. If (e,c) is an interaction pair of C and

E, then we also write TC(e) c and say that c is

the interactive transduction of e by C - A relation T on infinite sequences is called

interactively computable iff there is an

interactive component C such that T TC

Example

- 0 set of finite sequences of 0s (including

empty sequence) - 1 set of finite sequences of 1s
- 0,1 set of all finite sequences over 0,1
- 0,1? set of infinite sequences or streams

over 0,1

Environment fools the Component

- There is no C can exist that transduces input

streams of the from 1a1ß1? to output 1ß1a1 with

a, ß ? 0 and ? ? 0,1? - Suppose C can transduce 1a1ß1? to 1ß1a1
- C must response to an input from E (100 )
- First symbol of c will be 1
- If second symbol of c is 0 then Es input will be

1a11? - If second symbol of c is 1 then Es input will be

1a101? - If second symbol of c is then it is not

fault-free

Content

- Introduction
- A Model of Interactive Computation
- Interactively Computable Relations
- Interactive Recognitions
- Interactive Generations
- Interactive Translations
- Conclusion and Future works

Interactively Computable Relations

- Interactive computations can be view as

classical, monotonic computations taken to

infinity

Definition for Interactively Computable Relations

- y ? 0,1? and t 0 preft(y) be lengtht prefix

of y - x is a finite and strict prefix of y

Theorem 1

- Proof
- Think about Turing Machine (Mg) which represents

g with finite input stream - x preft(u)
- Mg simulates C
- Output of c is a signal 0 or 1 Mg writes

corresponding symbol - Output of c is a silent symbol Mg writes nothing
- Output of c is , Mg is sent to indefinite loop

Theorem 2

- Proof
- gt Thm 1
- lt Design a component C

Theorem 3

- Interactiveness is recursively undecidable
- Proof
- Cantors Diagonal argument

Content

- Introduction
- A Model of Interactive Computation
- Interactively Computable Relations
- Interactive Recognitions
- Interactive Generations
- Interactive Translations
- Conclusion and Future works

Interactive Recognition

- Interactive systems perform tasks in monitoring
- Recognition of patterns in infinite streams of

signals from environment (ex. intrusion

detection system) - Interactive system cannot detect that automaton

(Component) passing an infinite number of times

through one or more accepting states during the

processing of the infinite input sequence - In Interactive systems there is a specification

which environment has to follow and component has

to observe that this specification is adhere to.

Definitions

Lemma

Interactive Generations

- Proves that interactive generation and

interactive recognition is dual

Ubiquitous Environment

Peer Server

Inform

Reaction

Action

Sensor

Human

Interactive Translations

- Interactive components perform the online

translation of infinite streams into other

infinite streams of signal - Related notion of omega-transduction
- Function f is interactively computable iff f is

limit-continuous - If f and g are interactively computable, then so

is f g - Let f be interactively computable and 1-1. Then

f-1 is interactively computable

Content

- Introduction
- A Model of Interactive Computation
- Interactively Computable Relations
- Interactive Recognitions
- Interactive Generations
- Interactive Translations
- Conclusion and Future works

Conclusion

- It requires knowledge of
- Basic Automata Theory
- Omega Language Theory
- Future works
- How about nonuniformly evolving of interactive

systems and programs?