1 / 34

A Theory of Interactive Computation

This presentation by Choi and Chang-Beom from KAIST introduces a theory of interactive computation, exploring the model, computable relations, recognitions, generations, translations, and future works. It discusses the limitations of traditional models and the need to capture interactive properties.

halvorson
Télécharger la présentation

A Theory of Interactive Computation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 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. Ubiquitous Environment Peer Server Inform Reaction Action Sensor Human Examples of Inactive Systems Server Request Respond Attack Hacker 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) alphabet Environment (E) 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. t+1 t+2 t’ = t+3 silent silent silent silent silent Non-silent 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 t Non-silent silent 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 = 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

  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 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 E C e 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 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}ωand 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 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

  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 Ubiquitous Environment Peer Server Inform Reaction Action Sensor Human A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST

  32. Interactive Translations • Interactive components perform the online translationof 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? A Theory of Interactive Computation, presented by Choi, Chang-Beom, KAIST

More Related