A Theory of Interactive Computation Jan van Leeuwen, Jiri Widermann Presented by Choi, Chang-Beom KAIST

A Theory of Interactive Computation

Jan van Leeuwen, Jiri Widermann

Presented by Choi, Chang-BeomKAIST

A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 2

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

Hacker

Req

uest R

esp

on

d

Atta

ck

Peer Server

Sensor

Info

rm

Action

Ub

iqu

itou

s E

nvir

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men

t

Human

Reactio

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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



A Model of Interactive Computation

Component (C)

Environment (E)

alp

hab

et

Alphabet Σ = {0, 1, τ, #}


Definitions C : Component E : Environment Alphabet : Σ = {0, 1, τ, #}

0, 1 : actual symbols τ : silent or empty symbol # : fault or error symbol

Interactive input streams e = e0e1 … et …

Interactive output streams c = c0c1 … ct …

(if C’s output is c then C is interactive component )

τ


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 (Con’t) 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

C’s output at time t is depend on e0e1…et-1 and c0c1

…ct-1

ē : e with out τ ċ : c with out τ

τ


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’ > t and vice versa

Non-sile

nt

silent

t

silent

silent

t+1

silent

silent

t+2

silent

Non-sile

nt

t’ = t+3


Definition 1

An interaction pair of C and E is any pair (e,c) such that e = e0e1 … et … and c = c0

c1 … 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 fini

te time


Component Memory space of C is always finite but potentiall

y unbounded C can build up an infinite database of knowledge

Algorithmicity Program evolves over time and which answers wheth

er Et-1(ct-1) ∋ et or not Regardless of E’s actual behavior, there is an algorith

mic way to verify afterwards that a sequence could have been generated by E


Interactive Transduction

E Ce c

ω-transducer on infinite sequence


Definition 2 & 3 The behavior of C with respect to E is the set

TC = {(e, ċ)|(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) = ċ and say that ċ 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 0’s

(including empty sequence) 1* : set of finite sequences of 1’s {0,1}* : set of all finite sequences over {0,1} {0,1}ω : set of infinite sequences or streams ove

r {0,1}


Environment fools the Component

There is no C can exist that transduces input streams of the from 1α1β1γ to output 1β1α1 with α, β ∈ 0* and γ ∈ {0,1}ω

Suppose C can transduce 1α1β1γ to 1β1α1 C must response to an input from E (100…) First symbol of c will be 1 If second symbol of c is 0 then E’s input will be 1α11γ If second symbol of c is 1 then E’s input will be 1α101γ If second symbol of c is # then it is not fault-free


Content



Interactively Computable Relations

Interactive computations can be view as classical, monotonic computations taken to infinity


Definition for Interactively Computable Relations

y ∈ {0,1}ω and t ≥ 0 preft(y) be length–t prefix of y

x is a finite and strict prefix of y


Theorem 1

Proof Think about Turing Machine (Mg) which represents g w

ith 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 => : Thm 1 <= Design a component C


Theorem 3 Interactiveness is recursively undecidable

Proof Cantor’s Diagonal argument


Content



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

Peer Server

SensorIn

form

Action

Ub

iqu

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En

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Reactio

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Interactive Translations Interactive components perform the online translation of infinite stre

ams 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 co

mputable


Content



Conclusion

It requires knowledge of Basic Automata Theory Omega Language Theory

Future works How about nonuniformly evolving of interactiv

e systems and programs?

Documents

A Theory of Interactive Computation Jan van Leeuwen, Jiri Widermann Presented by Choi, Chang-Beom KAIST