Signal machines : localization of isolated accumulation Signal machines : localization of isolated accumulation Jérôme Durand-Lose Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, Orléans, FRANCE 6 mars 2011 — Journées Calculabilités — Paris 1 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations 1 Necessary conditions on the coordinates of isolated 2 accumulations Manipulating c.e. and d-c.e. real numbers 3 Accumulating at c.e. and d-c.e. real numbers 4 Conclusion 5 2 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Signal machines and isolated accumulations 1 Necessary conditions on the coordinates of isolated 2 accumulations Manipulating c.e. and d-c.e. real numbers 3 Accumulating at c.e. and d-c.e. real numbers 4 Conclusion 5 3 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” 4 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” 5 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” 6 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” 7 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” Lines: traces of signals Space-time diagrams of signal machines 8 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations “Nice regular drawings” Lines: traces of signals Space-time diagrams of signal machines Defined by bottom: initial configuration lines: signals � meta-signals end-points: collisions � rules 9 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Example: find the middle Meta-signals (speed) M (0) Collision rules M M 10 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Example: find the middle Meta-signals (speed) M (0) div (3) Collision rules div M M 11 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Example: find the middle Meta-signals (speed) M (0) div (3) hi (1) lo (3) hi lo M Collision rules M div M { div, M } → { M, hi, lo } 12 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Example: find the middle Meta-signals (speed) M (0) div (3) hi (1) lo (3) b a c k M back (-3) hi M o l M Collision rules div M { div, M } → { M, hi, lo } { lo, M } → { back, M } 13 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Example: find the middle Meta-signals (speed) M (0) div (3) hi (1) M M lo (3) back M back (-3) i h o l M Collision rules div M { div, M } → { M, hi, lo } { lo, M } → { back, M } { hi, back } → { M } 14 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Known results Turing computations TM run Simulation [Durand-Lose, 2011] 0 → q i 0 1 1 # 0 q 2 → 0 get 1 0 1 1 # ( q i , # ) 1 0 → q i get 0 1 1 # get q 3 → 1 get 0 1 1 # # 1 → q i → 1 − q i 1 0 0 1 1 # → − q i q i 0 1 1 # # → − q i q i 1 0 0 1 # 0 ← q i − q i 1 1 0 1 # 15 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Known results Turing computations TM run Simulation [Durand-Lose, 2011] 0 → q i 0 1 1 # 0 q 2 → 0 get 1 0 1 1 # ( q i , # ) Analog computations 1 0 → q i get 0 1 1 # Computable analysis get q 3 → 1 [Weihrauch, 2000] get 0 1 1 # # 1 → q i → 1 − [Durand-Lose, 2010a] q i 1 0 0 1 1 # → − q i q i Blum, Shub and Smale 0 1 1 # model [Blum et al., 1989] # → − q i q i 1 0 0 1 # 0 [Durand-Lose, 2008] ← q i − q i 1 1 0 1 # 16 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Known results Turing computations TM run Simulation [Durand-Lose, 2011] 0 → q i 0 1 1 # 0 q 2 → 0 get 1 0 1 1 # ( q i , # ) Analog computations 1 0 → q i get 0 1 1 # Computable analysis get q 3 → 1 [Weihrauch, 2000] get 0 1 1 # # 1 → q i → 1 − [Durand-Lose, 2010a] q i 1 0 0 1 1 # → − q i q i Blum, Shub and Smale 0 1 1 # model [Blum et al., 1989] # → − q i q i 1 0 0 1 # 0 [Durand-Lose, 2008] ← q i − q i 1 1 0 1 # “Black hole” implementation [Durand-Lose, 2009] 17 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Geometric primitives: accelerating and bounding time Normal Shrunk 18 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Geometric primitives: accelerating and bounding time Normal Iterated Shrunk 19 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Rational signal machines and isolated accumulations Q signal machine Simplest accumulation all speed are in Q all initial positions are in Q ⇒ all location remains in Q Space and time location Easy to compute 20 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Rational signal machines and isolated accumulations Q signal machine Accumulation? all speed are in Q all initial positions are in Q ⇒ all location remains in Q Space and time location Easy to compute Not so easy to guess 21 / 39
Signal machines : localization of isolated accumulation Signal machines and isolated accumulations Rational signal machines and isolated accumulations Q signal machine Accumulation? all speed are in Q all initial positions are in Q ⇒ all location remains in Q Space and time location Easy to compute Not so easy to guess Forecasting any accumulation Highly undecidable ( Σ 0 2 in the arithmetic hierarchy) [Durand-Lose, 2006] 22 / 39
Signal machines : localization of isolated accumulation Necessary conditions on the coordinates of isolated accumulations Signal machines and isolated accumulations 1 Necessary conditions on the coordinates of isolated 2 accumulations Manipulating c.e. and d-c.e. real numbers 3 Accumulating at c.e. and d-c.e. real numbers 4 Conclusion 5 23 / 39
Signal machines : localization of isolated accumulation Necessary conditions on the coordinates of isolated accumulations Temporal coordinate Q -signal machine Q on computers/Turing machine exact representation exact operations exact computations by TM (and implanted in Java) Simulation near an isolated accumulation on each collision, print the date � increasing computable sequence of rational numbers (converges iff there is an accumulation) 24 / 39
Signal machines : localization of isolated accumulation Necessary conditions on the coordinates of isolated accumulations Spacial coordinate Static deformation by adding a constant to each speed zig zig left left zag right z right a g left left zig i g z right right zig left right zag left z i g right Drifts by 1, 2 and 4 With all speeds positive the left most coordinate is increasing (and computable) converges iff there is an accumulation correction by subtracting the date times the drift 25 / 39
Signal machines : localization of isolated accumulation Necessary conditions on the coordinates of isolated accumulations c.e. real number limit of a convergent increasing computable sequence of rational numbers no bound on the convergence rate represents a c.e. set (of natural numbers) stable by positive integer multiplication but not by subtraction d-c.e. real number difference of two c.e. real number form a field [Ambos-Spies et al., 2000] these are exactly the limits of a computable sequence of rational numbers that converges weakly effectively , i.e. , � | x n + 1 − x n | converges n ∈ N 26 / 39
Signal machines : localization of isolated accumulation Manipulating c.e. and d-c.e. real numbers Signal machines and isolated accumulations 1 Necessary conditions on the coordinates of isolated 2 accumulations Manipulating c.e. and d-c.e. real numbers 3 Accumulating at c.e. and d-c.e. real numbers 4 Conclusion 5 27 / 39
Signal machines : localization of isolated accumulation Manipulating c.e. and d-c.e. real numbers Encoding For d-c.e. real numbers z i � x = 2 i , z i ∈ Z i ∈ N the sequence i → z i is computable and � z i � � � � converges � � 2 i i ∈ N For c.e. real numbers identical but z i ∈ N z i in signed unary representation 28 / 39
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