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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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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?
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 7
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
on
men
t
Human
Reactio
n
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 9
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 10
A Model of Interactive Computation
Component (C)
Environment (E)
alp
hab
et
Alphabet Σ = {0, 1, τ, #}
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 11
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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)
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 13
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 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’ > t and vice versa
Non-sile
nt
silent
t
silent
silent
t+1
silent
silent
t+2
silent
Non-sile
nt
t’ = t+3
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 15
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 16
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 17
Interactive Transduction
E Ce c
ω-transducer on infinite sequence
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 18
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 19
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}
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 20
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 21
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 22
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 23
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 24
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 25
Theorem 2
Proof => : Thm 1 <= Design a component C
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 26
Theorem 3 Interactiveness is recursively undecidable
Proof Cantor’s Diagonal argument
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 27
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 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.
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 29
Definitions
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 30
Lemma
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 31
Interactive Generations
Proves that interactive generation and interactive recognition is dual
Peer Server
SensorIn
form
Action
Ub
iqu
itou
s
En
vir
on
men
t
Human
Reactio
n
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 32
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
A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST 33
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 34
Conclusion
It requires knowledge of Basic Automata Theory Omega Language Theory
Future works How about nonuniformly evolving of interactiv
e systems and programs?