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Signal machines : localization of isolated accumulation Signal machines : localization of isolated accumulation Jrme Durand-Lose Laboratoire dInformatique Fondamentale dOrlans, Universit dOrlans, Orlans, FRANCE 6 mars 2011

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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

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Signal machines : localization of isolated accumulation

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

“Nice regular drawings”

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

“Nice regular drawings”

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

“Nice regular drawings”

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

“Nice regular drawings”

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

“Nice regular drawings”

Lines: traces of signals Space-time diagrams of signal machines

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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

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Example: find the middle

M M

Meta-signals (speed) M (0) Collision rules

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Example: find the middle

div M M

Meta-signals (speed) M (0) div (3) Collision rules

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Example: find the middle

div M M hi lo M

Meta-signals (speed) M (0) div (3) hi (1) lo (3) Collision rules { div, M } → { M, hi, lo }

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Example: find the middle

div M l

  • M

M hi b a c k M

Meta-signals (speed) M (0) div (3) hi (1) lo (3) back (-3) Collision rules { div, M } → { M, hi, lo } { lo, M } → { back, M }

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Example: find the middle

div M l

  • M

h i back M M M

Meta-signals (speed) M (0) div (3) hi (1) lo (3) back (-3) Collision rules { div, M } → { M, hi, lo } { lo, M } → { back, M } { hi, back } → { M }

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Known results

Turing computations [Durand-Lose, 2011] TM run

1 0 1 # qi 0 0 1 # qi 0 1 1 # qi 0 1 1 # qi 1 → 0 1 1 # q3 →1 0 1 1 # qi 0 → 0 1 1 # q2 →0 0 1 1 # qi 0 →

Simulation

1 ← − qi − → qi − → qi 1 − → q

i

# 1 get 1 get get get 1 1 (qi, #) # 15 / 39

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Known results

Turing computations [Durand-Lose, 2011] Analog computations Computable analysis [Weihrauch, 2000] [Durand-Lose, 2010a] Blum, Shub and Smale model [Blum et al., 1989] [Durand-Lose, 2008] TM run

1 0 1 # qi 0 0 1 # qi 0 1 1 # qi 0 1 1 # qi 1 → 0 1 1 # q3 →1 0 1 1 # qi 0 → 0 1 1 # q2 →0 0 1 1 # qi 0 →

Simulation

1 ← − qi − → qi − → qi 1 − → q

i

# 1 get 1 get get get 1 1 (qi, #) # 16 / 39

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Known results

Turing computations [Durand-Lose, 2011] Analog computations Computable analysis [Weihrauch, 2000] [Durand-Lose, 2010a] Blum, Shub and Smale model [Blum et al., 1989] [Durand-Lose, 2008] “Black hole” implementation [Durand-Lose, 2009] TM run

1 0 1 # qi 0 0 1 # qi 0 1 1 # qi 0 1 1 # qi 1 → 0 1 1 # q3 →1 0 1 1 # qi 0 → 0 1 1 # q2 →0 0 1 1 # qi 0 →

Simulation

1 ← − qi − → qi − → qi 1 − → q

i

# 1 get 1 get get get 1 1 (qi, #) # 17 / 39

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Geometric primitives: accelerating and bounding time

Normal Shrunk

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Geometric primitives: accelerating and bounding time

Normal Shrunk Iterated

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Rational signal machines and isolated accumulations

Q signal machine all speed are in Q all initial positions are in Q ⇒ all location remains in Q Space and time location Easy to compute Simplest accumulation

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Rational signal machines and isolated accumulations

Q signal machine 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 Accumulation?

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Signal machines : localization of isolated accumulation Signal machines and isolated accumulations

Rational signal machines and isolated accumulations

Q signal machine 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 (Σ0

2 in the arithmetic hierarchy)

[Durand-Lose, 2006] Accumulation?

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Signal machines : localization of isolated accumulation Necessary conditions on the coordinates of isolated accumulations

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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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

  • n each collision, print the date

increasing computable sequence of rational numbers (converges iff there is an accumulation)

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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 right left zag zig right left z i g right left z a g zig right left

z i g right left zag zig right left

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

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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.,

  • n∈N

|xn+1 − xn| converges

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Signal machines : localization of isolated accumulation Manipulating c.e. and d-c.e. real numbers

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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Signal machines : localization of isolated accumulation Manipulating c.e. and d-c.e. real numbers

Encoding

For d-c.e. real numbers x =

  • i∈N

zi 2i , zi ∈ Z the sequence i → zi is computable and

  • i∈N
  • zi

2i

  • converges

For c.e. real numbers identical but zi ∈ N zi in signed unary representation

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Signal machines : localization of isolated accumulation Manipulating c.e. and d-c.e. real numbers

TM outputting the infinite sequence

Run wait between each zi

# # qi 1 # q1 1 0 # q2 1 0 1 qi 1 1 1 q1 →1 1 1 1 q3 →1 1 1 1 q4 wait

Simulation

− → q1 # − → q2 # ← − qi 1 ← − q

i

− → q1 1 1 1 # 1 ← − q3 1 1 1 # 1 1 1 wait #

Shrunk to output in bounded time Simulation and shrinking structure stop after each value

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Signal machines : localization of isolated accumulation Accumulating at c.e. and d-c.e. real numbers

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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Signal machines : localization of isolated accumulation Accumulating at c.e. and d-c.e. real numbers

Two-level scheme

Control/inner structure Provide the data for accumulating

Control/inner structure TM generating the sequence zi Outer structure shrinks & moves resume

Outer structure shrink and move the whole structure accumulation

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Signal machines : localization of isolated accumulation Accumulating at c.e. and d-c.e. real numbers

Temporal coordinate

Wait the corresponding time Constant (up to scale) delay before outer structure action total delay is rational and should be previously subtracted

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Signal machines : localization of isolated accumulation Accumulating at c.e. and d-c.e. real numbers

Spatial coordinate

Move left or right, more or less

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Signal machines : localization of isolated accumulation Accumulating at c.e. and d-c.e. real numbers

Examples

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Signal machines : localization of isolated accumulation Conclusion

1

Signal machines and isolated accumulations

2

Necessary conditions on the coordinates of isolated accumulations

3

Manipulating c.e. and d-c.e. real numbers

4

Accumulating at c.e. and d-c.e. real numbers

5

Conclusion

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Signal machines : localization of isolated accumulation Conclusion

Results Isolated accumulations happen at d-c.e. spacial and c.e. temporal coordinates Accumulation at any c.e. temporal coordinate is possible Accumulation at any d-c.e. spacial coordinate is possible Perspectives Uncorrelate space and time coordinate it is possible for computable coordinates [Durand-Lose, 2010b] Higher order isolated accumulations Non isolated accumulations

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Signal machines : localization of isolated accumulation Conclusion

Ambos-Spies, K., Weihrauch, K., and Zheng, X. (2000). Weakly computable real numbers.

  • J. Complexity, 16(4):676–690.

Blum, L., Shub, M., and Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines.

  • Bull. Amer. Math. Soc., 21(1):1–46.

Durand-Lose, J. (2006). Forcasting black holes in abstract geometrical computation is highly unpredictable. In Cai, J.-Y., Cooper, S. B., and Li, A., editors, Theory and Applications of Models of Computations (TAMC ’06), number 3959 in LNCS, pages 644–653. Springer. Durand-Lose, J. (2008). Abstract geometrical computation with accumulations: Beyond the Blum, Shub and Smale model.

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Signal machines : localization of isolated accumulation Conclusion

In Beckmann, A., Dimitracopoulos, C., and Löwe, B., editors, Logic and Theory of Algorithms, 4th Conf. Computability in Europe (CiE ’08) (abstracts and extended abstracts of unpublished papers), pages 107–116. University of Athens. Durand-Lose, J. (2009). Abstract geometrical computation 3: Black holes for classical and analog computing.

  • Nat. Comput., 8(3):455–472.

Durand-Lose, J. (2010a). Abstract geometrical computation 5: embedding computable analysis.

  • Nat. Comput.

Special issue on Unconv. Comp. ’09. Durand-Lose, J. (2010b). The coordinates of isolated accumulations [includes] computable real numbers.

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Signal machines : localization of isolated accumulation Conclusion

In Ferreira, F., Guerra, H., Mayordomo, E., and Rasga, J., editors, Programs, Proofs, Processes, 6th Int. Conf. Computability in Europe (CiE ’10) (abstracts and extended abstracts of unpublished papers), pages 158–167. CMATI, U. Azores. Durand-Lose, J. (2011). Abstract geometrical computation 4: small Turing universal signal machines.

  • Theoret. Comp. Sci., 412:57–67.

Weihrauch, K. (2000). Introduction to computable analysis. Texts in Theoretical computer science. Springer, Berlin.

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