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A Theory of Interactive Computation

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A Theory of Interactive Computation. Jan van Leeuwen, Jiri Widermann ... Why 'Interactive System' ... Traditional Turing Machine is not adequate to Interactive System ... – PowerPoint PPT presentation

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Title: A Theory of Interactive Computation


1
A Theory of Interactive Computation
  • Jan van Leeuwen, Jiri Widermann
  • 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

3
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

4
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?
5
Purpose of Interactive System
  • Not to compute some finial result
  • React to environment or Interact with environment
  • Maintain a well-defined action-reaction behavior

6
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

7
Examples of Inactive Systems
Server
Request
Respond
Attack
Hacker
8
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

9
Content
  • Introduction
  • A Model of Interactive Computation
  • Interactively Computable Relations
  • Interactive Recognitions
  • Interactive Generations
  • Interactive Translations
  • Conclusion and Future works

10
A Model of Interactive Computation
Component (C)
alphabet
Environment (E)
Alphabet S 0, 1, t,
11
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
12
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)

13
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
14
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
15
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

16
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

17
Interactive Transduction
E
C
e
c
?-transducer on infinite sequence
18
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

19
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

20
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

21
Content
  • Introduction
  • A Model of Interactive Computation
  • Interactively Computable Relations
  • Interactive Recognitions
  • Interactive Generations
  • Interactive Translations
  • Conclusion and Future works

22
Interactively Computable Relations
  • Interactive computations can be view as
    classical, monotonic computations taken to
    infinity

23
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

24
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

25
Theorem 2
  • Proof
  • gt Thm 1
  • lt Design a component C

26
Theorem 3
  • Interactiveness is recursively undecidable
  • Proof
  • Cantors Diagonal argument

27
Content
  • Introduction
  • A Model of Interactive Computation
  • Interactively Computable Relations
  • Interactive Recognitions
  • Interactive Generations
  • Interactive Translations
  • Conclusion and Future works

28
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.

29
Definitions
30
Lemma
31
Interactive Generations
  • Proves that interactive generation and
    interactive recognition is dual

Ubiquitous Environment
Peer Server
Inform
Reaction
Action
Sensor
Human
32
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

33
Content
  • Introduction
  • A Model of Interactive Computation
  • Interactively Computable Relations
  • Interactive Recognitions
  • Interactive Generations
  • Interactive Translations
  • Conclusion and Future works

34
Conclusion
  • It requires knowledge of
  • Basic Automata Theory
  • Omega Language Theory
  • Future works
  • How about nonuniformly evolving of interactive
    systems and programs?
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