A Theory of Interactive
Computation
Jan van Leeuwen, Jiri Widermann
Presented by Choi, Chang-Beom
KAIST
Content
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





Introduction
A Model of Interactive Computation
Interactively Computable Relations
Interactive Recognitions
Interactive Generations
Interactive Translations
Conclusion and Future works
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
2
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
3
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!
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
Why?
4
Purpose of Interactive System

Not to compute some finial result

React to environment or Interact with
environment
Maintain a well-defined action-reaction
behavior

A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
5
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
6
Examples of Inactive Systems
Sensor
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
Action
Human
Ubiquitous Environment
Inform
Hacker
Peer Server
Attack
Request
Respond
Server
7
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
8
Content







Introduction
A Model of Interactive Computation
Interactively Computable Relations
Interactive Recognitions
Interactive Generations
Interactive Translations
Conclusion and Future works
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
9
A Model of Interactive Computation
Component (C)
alphabet
Environment (E)
Alphabet Σ = {0, 1, τ, #}
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
10
Definitions



C : Component
E : Environment
Alphabet : Σ = {0, 1, τ, #}
 0, 1 : actual symbols
 τ : silent or empty symbol
 # : fault or error symbolτ

Interactive input streams
 e = e0e1 … et …

Interactive output streams
 c = c0c1 … ct …
(if C’s output is c then C is interactive component )
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
11
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)
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
12
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
τ
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
13
Interactiveness

For all times t, when E sends a nonsilent 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
t
t+1
t+2
t’ = t+3
Non-silent
silent
silent
silent
silent
silent
silent
Non-silent
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
14
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
15
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 E’s actual behavior, there is an
algorithmic way to verify afterwards that a
sequence could have been generated by E
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
16
Interactive Transduction
E
e
c
C
ω-transducer on
infinite sequence
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
17
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
18
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
over {0,1}
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
19
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
20
Content







Introduction
A Model of Interactive Computation
Interactively Computable Relations
Interactive Recognitions
Interactive Generations
Interactive Translations
Conclusion and Future works
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
21
Interactively Computable Relations

Interactive computations can be view
as classical, monotonic computations
taken to infinity
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
22
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

A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
23
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
24
Theorem 2

Proof
 => : Thm 1
 <= Design a component C
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
25
Theorem 3

Interactiveness is recursively undecidable

Proof
 Cantor’s Diagonal argument
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
26
Content







Introduction
A Model of Interactive Computation
Interactively Computable Relations
Interactive Recognitions
Interactive Generations
Interactive Translations
Conclusion and Future works
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
27
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.
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
28
Definitions
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
29
Lemma
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
30
Interactive Generations

Proves that interactive generation and
interactive recognition is dual
Inform
Sensor Action Human
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
Ubiquitous Environment
Peer Server
31
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


A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
32
Content







Introduction
A Model of Interactive Computation
Interactively Computable Relations
Interactive Recognitions
Interactive Generations
Interactive Translations
Conclusion and Future works
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
33
Conclusion

It requires knowledge of
 Basic Automata Theory
 Omega Language Theory

Future works
 How about nonuniformly evolving of
interactive systems and programs?
A Theory of Interactive Computation, presented by Choi, Chang-Beom,
KAIST
34