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

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 Request Respond Attack Peer Server Sensor Inform Action Ubiquitous Environment Human Reaction

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) alphabet 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 = e 0 e 1 … e t …  Interactive output streams  c = c 0 c 1 … c t … (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 E t-1 (c t-1 ) ∋ e t  C’s output at time t is depend on e 0 e 1 …e t-1 and c 0 c 1 …c t-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-silent silent t t+1 silent t+2 silent Non-silent 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 = e 0 e 1 … e t … and c = c 0 c 1 … c t … 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 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 E t-1 (c t-1 ) ∋ e t 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 17 Interactive Transduction EC ec 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 T C = {(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 T C (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 = T C

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 over {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 pref t (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 (M g ) which represents g with finite input stream  x = pref t (u)  M g simulates C  Output of c is a signal 0 or 1 M g writes corresponding symbol  Output of c is a silent symbol M g writes nothing  Output of c is #, M g 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 Sensor Inform Action Ubiquitous Environment Human Reaction

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

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 interactive systems and programs?