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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.

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A Theory of Interactive Computation

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  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

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