On the Executability of Interactive Computation. June 23, 2016 Where innovation starts
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1 On the Executability of Interactive Computation Bas Luttik Fei Yang June 23, 2016 Where innovation starts
2 Outline 2/37 From Computation to Interactive Computation Executability - an Integration of Computability and Concurrency Comparison of RTMs and ITMs RTM with advice
3 Outline 3/37 From Computation to Interactive Computation Executability - an Integration of Computability and Concurrency Comparison of RTMs and ITMs RTM with advice
4 Turing machine and Computers 4/37
5 Turing machine and Computers 4/37 Church Turing Thesis: every computable function can be computed with a Turing Machine
6 Classical Theory of Computation 5/37 Functions on natural numbers;
7 Classical Theory of Computation 5/37 Functions on natural numbers; No interaction;
8 Classical Theory of Computation 5/37 Functions on natural numbers; No interaction; Termination.
9 A Model for Interactive Computation A theory of interactive computation 6/37 Jan van Leeuwen and Jiří Wiedermann
10 A Model for Interactive Computation A theory of interactive computation 6/37 Jan van Leeuwen and Jiří Wiedermann
11 A Model for Interactive Computation A theory of interactive computation 6/37 Jan van Leeuwen and Jiří Wiedermann The input and output forms two infinite streams: i 0 i 1... and o 0 o 1...
12 Stream Translation (ω-translation) 7/37 Definition Let be an alphabet of data symbols (usually we take 0, 1), and let ω be the set of all infinite streams over. A stream translation (or an ω-translation) is a function φ : ω ω.
13 Interactive Computation 8/37 Classical theory Functions on natural numbers; No interaction; Termination.
14 Interactive Computation 8/37 Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation);
15 Interactive Computation 8/37 Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment;
16 Interactive Computation 8/37 Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment; No termination.
17 Interactive Turing machine 9/37 Definition (Interactive Turing machine) An interactive Turing machine (ITM) with a single work tape is a triple I = (Q, I, q in ), where 1. Q is its set of states; 2. I : Q D τ Q D {L, R} τ is a (partial) transition function; and 3. q in Q is its initial state.
18 Interactive Turing machine 9/37 Definition (Interactive Turing machine) An interactive Turing machine (ITM) with a single work tape is a triple I = (Q, I, q in ), where 1. Q is its set of states; 2. I : Q D τ Q D {L, R} τ is a (partial) transition function; and 3. q in Q is its initial state. Initially, the tape is empty. A computation of an ITM is an infinite sequence of transitions (q in, ˇ ) = (q 0, δ 0 ) i 0/o 0 I (q 1, δ 1 ) i 1/o 1 I.
19 Interactively Computable ω-translation 10/37 The set of infinite runs of an ITM result in an ω-translation: ω ω We call such translation interactively computable ω-translation.
20 Outline 11/37 From Computation to Interactive Computation Executability - an Integration of Computability and Concurrency Comparison of RTMs and ITMs RTM with advice
21 A Mathematical Characterisation of Behaviour 12/37
22 Labelled Transition Systems 13/37 A: set of actions; τ: a special action (/ A), for unobservable actions. A τ = A {τ}. Definition An A τ -labelled transition system is a triple (S,, ) where 1. S is a set of states; 2. S A τ S is a transition relation; and 3. S is an initial state.
23 Interaction in Concurrency Theory 14/37 Parallel composition of components C1 C 2 Communication happens between predefined channels: c?d If C 1 C 1 and C c!d 2 C 2, then C τ 1 C 2 C 1 C 2
24 Behavioural Equivalence 15/37
25 Behavioural Equivalence 15/37 In classical theory computation: equivalence between functions
26 Behavioural Equivalence 15/37 In classical theory computation: equivalence between functions In concurrency: behavioural equivalence
27 Computation + Concurrency Theory Computability 16/37
28 Computation + Concurrency Theory Computability +Concurrency 16/37
29 Computation + Concurrency Theory Computability +Concurrency = Executability 16/37
30 Computation + Concurrency Theory 16/37 Computability +Concurrency = Executability Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment; No termination.
31 Computation + Concurrency Theory 16/37 Computability +Concurrency = Executability Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment; No termination. Executability Labelled transition systems;
32 Computation + Concurrency Theory 16/37 Computability +Concurrency = Executability Classical theory Functions on natural numbers; No interaction; Termination. Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment; No termination. Executability Labelled transition systems; Interaction: between parallel components
33 Computation + Concurrency Theory Computability +Concurrency = Executability Classical theory Functions on natural numbers; No interaction; Termination. 16/37 Interactive computation Functions on infinite streams (ω-translation); Interaction: computation components and environment; No termination. Executability Labelled transition systems; Interaction: between parallel components / department No of mathematics termination. and computer science
34 Reactive Turing Machines 17/37 A: set of actions; τ: a special action (/ A), for unobservable actions. A τ = A {τ}. A reactive Turing machine (RTM) is a classical Turing machine with an action from some set A τ associated with every transition. So RTMs have two types of transitions: 1. s a[d/e]m t means externally observable, as execution of a 2. s τ[d/e]m t means internal, unobservable transition M is ether moving left or moving right
35 Labelled Transition System of an RTM 18/37 We associate with every configuration (control state, tape instance) a state, and associate with every execution step a labelled transition.
36 Executability and Behavioural Equivalence 19/37 A transition system is called executable if it is behaviourally equivalent to the transition system of an RTM.
37 Executability and Behavioural Equivalence 19/37 A transition system is called executable if it is behaviourally equivalent to the transition system of an RTM. The notion executability varies for different types of behavioural equivalences.
38 Outline 20/37 From Computation to Interactive Computation Executability - an Integration of Computability and Concurrency Comparison of RTMs and ITMs RTM with advice
39 Difference in Semantics 21/37
40 Difference in Semantics 21/37 1. Does ITMs give rise to an executable transition system? 2. Does RTMs give rise to an interactively computable ω-translation?
41 Associate ITMs with LTSs 22/37
42 Associate ITMs with LTSs 23/37
43 Associate ITMs with LTSs 24/37
44 Associate ITMs with LTSs 25/37
45 Executability Results 26/37 Theorem For every ITM I there exists an RTM M, such that their transition systems are behaviourally equivalent.
46 Restrict RTMs to ω-translations 27/37
47 Restrict RTMs to ω-translations 28/37
48 Restrict RTMs to ω-translations 29/37
49 Executable ω-translation 30/37 Theorem An ω-translation is executable iff it is interactively computable.
50 Conclusion 31/37
51 Outline 32/37 From Computation to Interactive Computation Executability - an Integration of Computability and Concurrency Comparison of RTMs and ITMs RTM with advice
52 ITM with Advice (ITM/A) 33/37 Advice function f : N N
53 Advice as a process 34/37
54 Expressiveness of RTM/A 35/37 Theorem If T is a boundedly branching labelled transition system, then there exists an RTM/A [M A f ] C such that T ([M A f ] C ) b T. Theorem If T is a countable labelled transition system, then there exists an RTM/A [M A f ] C such that T ([M A f ] C ) b T.
55 Future work 36/37 1. Uncomputable ω-translations unexecutable transition systems
56 Future work 36/37 1. Uncomputable ω-translations unexecutable transition systems 2. Complexity of ω-translations complexity of transition systems
57 Future work 36/37 1. Uncomputable ω-translations unexecutable transition systems 2. Complexity of ω-translations complexity of transition systems 3. Hierarchy of ω-translations Hierarchy of transition systems
58 37/37 Thank you!
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