Abstract geometrical computation for Black hole computation

Transcription

1 Abstract geometrical computation for Black hole computation Laboratoire d Informatique Fondamentale d Orléans, Université d Orléans, Orléans, FRANCE Computations on the continuum Lisboa, June 17th, 25

2 Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

3 Black hole computation Cellular automata Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

4 Black hole model Black hole computation Cellular automata Observer [Hogarth 94 & ] [Etesi & Nemeti 2] 1. Observer at the edge

5 Black hole model Black hole computation Cellular automata Observer [Hogarth 94 & ] [Etesi & Nemeti 2] 1. Observer at the edge 2. Machine sent into the black hole infinitely accelerated

6 Black hole model Black hole computation Cellular automata Observer [Hogarth 94 & ] [Etesi & Nemeti 2] 1. Observer at the edge 2. Machine sent into the black hole infinitely accelerated 3. Message sent by the machine received by the observer within a bounded delay

7 Malament-Hogarth space-time Observer s life-line Black hole computation Cellular automata Possible messages Maximum delay Black hole Machine infinite time ahead of it Message indicates the result of the computation After the delay, the observer knows whether the computation stops Any recursively enumerable problem can be decided!

8 Related models Black hole computation Cellular automata Main idea: infinitely many iterations on a sub-time-scale Can be achieved with a transfinite ordinal scale as in: Infinite time Turing machines [Hamkins 2] Or with a Zeno sub-scale as in: Piecewise constant derivative systems [Asarin & Maler 95, Bournez 99] We use the last approach

9 Black hole computation Cellular automata Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

10 Black hole computation Cellular automata Starting from discrete model... Cellular automata Q Z bi-infinite line of states Q < continuous dynamical system over Q Z local parallel synchronous uniform updating of Q Z

11 Black hole computation Cellular automata... with discrete space-time diagrams [HSC1, Fig. 7] [BNR91, Fig. 7] [Siw1, Fig. 5]

12 Black hole computation Cellular automata... with discrete space-time diagrams [LN9, Fig. 4] [LN9, Fig. 3]

13 Black hole computation Cellular automata... with discrete space-time diagrams [Got66, Fig. 3] [Got66, Fig. 6]

14 Black hole computation Cellular automata... with discrete space-time diagrams [VMP7, Fig. 1] [VMP7, Fig. 2] [VMP7, Fig. 3]

15 % + + / /,-/,- -!!!!! " " Black hole computation Cellular automata... with discrete space-time diagrams [Fis65, Fig. 2] * < * * [Maz96, Fig. 8] Bits 1 of the multiplier Bits of the multiplier Bit "end " of the multiplier Bits 1 of the multiplicand Bits of the multiplicand Bit "end" of the multiplicand Limits of a computation Bits in transit throught the network Bits 1 in transit throught the network Time f(n) Time f(5) E + + **+ )** ) Time f(4) ) ) Time f(3) (() && '' (( %&& '' Time f(2) $$% Time f (1) $$ Time / / / / /.... D1 / E-1 ## ## (2n, 2t(n)) Cell ;: < = A B B 5 D EFD G B? 9 H 5 B 9 H;I J K L I J M M N E1 T [MT99, Fig. 18] D 2 Signal T Signal D k / 2 Signal D k Signal E Signal E - 1 Signal E + 1

16 Black hole computation Cellular automata (Discrete) Space-time diagrams Observation Implementation Discrete lines Interpretation Discretization Lines on the plane Dynamics Conception

17 Black hole computation Cellular automata (Discrete) Space-time diagrams Observation Implementation Discrete lines Interpretation Discretization Dynamics Lines on the plane Model on its own Conception

18 Definition Restriction Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

19 Continuous space-time, Definition Restriction Z N R R + (or Q Q + ) Time Space

20 Continuous space-time, signals Definition Restriction Z N R R + (or Q Q + ) Signal (Meta-signal, position) Position (x, t) Time Meta-signal µ = (ι, ν) Space

21 Definition Restriction Continuous space-time, signals and collisions Z N R R + (or Q Q + ) Signal (Meta-signal, position) Position (x, t) Time Collision (Rule, position) Meta-signal µ = (ι, ν) Rule ρ = {µ i } i {µ + j } j Space

22 Continuous space-time diagrams Definition Restriction

23 Continuous space-time diagrams Definition Restriction

24 Definition Restriction Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

25 Definition Restriction Zeno artifact for black hole implementation Finite duration infinitely many collisions

26 Unwanted space-time diagrams Definition Restriction Unwanted because The number of signals is bursting to infinity (free creation of mater/energy) Difficulty (if not impossibility) to define continuation there

27 Restriction Definition Restriction Idea: Associate to each meta-signal a minimal amount of energy Ensure that no energy is created Collision must not create energy

28 Restriction Definition Restriction Energy: µ E(µ) N ρ = {µ i } i {µ + j } j, E(µ i ) E(µ + i ) E( configuration ) = E( existing signals ) Total energy quantified and bounded The total number of signals is bounded Special case The number of signals is preserved by any collision

29 Back to the space-time diagrams Definition Restriction

30 Back to the space-time diagrams Definition Restriction OK Energy is lost in the accumulation

31 Back to the space-time diagrams Definition Restriction OK Energy is lost in the accumulation

32 Back to the space-time diagrams Definition Restriction OK Energy is lost in the accumulation Can a Turing-computation be carried out with such a restriction?

33 2-counter automata simulation Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

34 2-counter automata simulation 2-counter automata (or 2-register machine) beg: B++ A-- IF A!= beg1 IF B!= imp beg1: A-- IF A!= beg pair: B-- A++ IF B!= pair IF A!= beg imp: B-- A++ A++ IF B!= imp1 IF A!= beg imp1: B-- A++ A++ A++ IF B!= imp1 IF A!= beg Turing-universal A, B counters (values in N) Operations A++ B++ A-- B-- A!= m B!= m a configuration (n, a, b)

35 2-counter automata simulation Encoding (n, a, b) into a space-time diagram Position encoding of a and b zero a ba b a b a b one a b Encoding of configurations a = < a b= zero n one a b zero n a one b <b zero n b one a zero n a b one zero n b a one A set of signals for each line of instruction

36 Implementing n A-- 2-counter automata simulation time zero n n a = n+1 one a zero 1 n 1 n Other instructions are implemented similarly 3 n n 3 a 1 a < a n+1 1 a one

37 Examples 2-counter automata simulation a=1 b= a=3 b= a=5 b=

38 Restriction and halting 2-counter automata simulation Restriction is always satisfied but... what about halting? The instruction turns into a yes/no signal leaving on the left

39 2-counter automata simulation Theorem (Energy preserving) signal machines can simulate any 2-counter automaton Theorem (Energy preserving) signal machines can carry out any Turing computation Turing-universal model of computation

40 2-counter automata simulation Theorem (Energy preserving) signal machines can simulate any 2-counter automaton Theorem (Energy preserving) signal machines can carry out any Turing computation Turing-universal model of computation All is done with rational positions manipulable by classical Turing machines

41 2-counter automata simulation How to embed this into a Zeno artifact?

42 Straining Iterated shrinking Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

43 Providing a strain Straining Iterated shrinking

44 Providing a shrinking Straining Iterated shrinking Two consecutive strains with the same directions coefficient 1/2 on one direction then the other Iterating possible if spatially bounded

45 Straining Iterated shrinking Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

46 Iterating shrinking Straining Iterated shrinking (for a spatially bounded computation)

47 Bounding delay Straining Iterated shrinking Simulation & iterated shrinking construction satisfy the restriction Accepts Rejects Does not stop Y N accumulation accumulation accumulation Bounding signals indicate when it is too late to get any answer

48 Introduction Black hole computation Cellular automata Definition Restriction 2-counter automata simulation Straining Iterated shrinking Conclusion

49 Results New model of computation Turing computation power Computation on the continuum (space and time) Geometric model where geometric constructions allow Zeno effects Similarity with the Black hole model Rational numbers are enough to get all this (i.e. distinction lies in continuity and not in cardinality)

50 Work in progress Second (and higher) accumulation infinite amount of energy at start lifting the restriction (hierarchy climbing) Super Turing-computability through real positions Analog computation through real positions