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
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
2 / 39
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
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
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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
Dened by
bottom: initial conguration
lines: signals
meta-signals
end-points: collisions
rules
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Example: nd the middle
Meta-signals (speed)
M (0)
Collision rules
M
M
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Example: nd the middle
Meta-signals (speed)
M (0)
div (3)
Collision rules
div
M
M
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Example: nd the middle
Meta-signals (speed)
M (0)
div (3)
hi (1)
lo (3)
M
div
hi
lo
M
M
Collision rules
{ div, M }
→
{ M, hi, lo }
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Example: nd the middle
Meta-signals (speed)
M (0)
div (3)
hi (1)
back
M
hi
lo (3)
M
lo
M
div
M
back (-3)
Collision rules
{ div, M }
{ lo, M }
→ { M, hi, lo }
→ { back, M }
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Example: nd the middle
Meta-signals (speed)
M (0)
div (3)
M
M
back
hi
M
M
lo (3)
back (-3)
lo
M
div
hi (1)
Collision rules
{ div, M }
{ lo, M }
{ hi, back
→ { M, hi, lo }
→ { back, M }
} → { M }
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Known results
0→
0 1 1
0 1 1
0→
0 1 1
0 1 1
1→
Simulation
qi
#
q2 →0
#
qi
#
q3 →1
#
qi
#
0 1 1
qi
0 1 1 #
qi
0 0 1 #
qi
1 0 1 #
ge
t
0
1
1
ge
t
(qi , #)
[Durand-Lose, 2011]
TM run
ge
t
ge
t
1
1
0
#
−
qi→
Turing computations
−q i
→
−q i
→
1
−
qi←
1
#
0
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Known results
Analog computations
0→
0 1 1
0 1 1
0→
Computable analysis
0 1 1
[Weihrauch, 2000]
0 1 1
[Durand-Lose, 2010a]
Blum, Shub and Smale
model [Blum et al., 1989]
[Durand-Lose, 2008]
1→
Simulation
qi
#
q2 →0
#
qi
#
q3 →1
#
qi
#
0 1 1
qi
0 1 1 #
qi
0 0 1 #
qi
1 0 1 #
ge
t
0
1
1
ge
t
(qi , #)
[Durand-Lose, 2011]
TM run
ge
t
ge
t
1
1
0
#
−
qi→
Turing computations
−q i
→
−q i
→
1
−
qi←
1
#
0
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Signal machines : localization of isolated accumulation
Signal machines and isolated accumulations
Known results
Analog computations
0→
0 1 1
0 1 1
0→
Computable analysis
0 1 1
[Weihrauch, 2000]
0 1 1
[Durand-Lose, 2010a]
Blum, Shub and Smale
model [Blum et al., 1989]
[Durand-Lose, 2008]
1→
Simulation
qi
#
q2 →0
#
qi
#
q3 →1
#
qi
#
0 1 1
qi
0 1 1 #
qi
0 0 1 #
qi
1 0 1 #
ge
t
0
1
1
ge
t
(qi , #)
[Durand-Lose, 2011]
TM run
ge
t
ge
t
1
1
0
#
−
qi→
Turing computations
−q i
→
−q i
→
1
−
qi←
1
#
0
Black hole implementation
[Durand-Lose, 2009]
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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
Iterated
Shrunk
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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
Simplest accumulation
Q
all initial positions are in
⇒
all location remains in
Q
Q
Space and time location
Easy to compute
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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
Accumulation?
Q
all initial positions are in
⇒
all location remains in
Q
Q
Space and time location
Easy to compute
Not so easy to guess
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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
Accumulation?
Q
all initial positions are in
⇒
all location remains in
Q
Q
Space and time location
Easy to compute
Not so easy to guess
Forecasting any accumulation
Highly undecidable
0
(Σ2 in the arithmetic hierarchy)
[Durand-Lose, 2006]
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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
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
23 / 39
Signal machines : localization of isolated accumulation
Necessary conditions on the coordinates of isolated accumulations
Temporal coordinate
Q-signal
Q
machine
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 i 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
left
zig
zig
zag
left zig
zag
right
right
leftzig
left
zag
left
zig
right
right
leftzig
right
right
Drifts by 1, 2 and 4
With all speeds positive
the left most coordinate is increasing (and computable)
converges i 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
dierence of two
c.e.
real number
form a eld
[Ambos-Spies et al., 2000]
these are exactly the limits of a computable sequence of
rational numbers that converges
X
n ∈N
weakly eectively, i.e.,
|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
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
27 / 39
Signal machines : localization of isolated accumulation
Manipulating c.e. and d-c.e. real numbers
Encoding
For
d-c.e.
real numbers
x=
the sequence
i → zi
c.e.
X zi i
2
converges
real numbers
identical but
zi
i ∈N
, zi ∈ Z
i
2
is computable and
i ∈N
For
X zi
zi ∈ N
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 innite sequence
Run
Simulation
wait between
wait
#
1
1
1
1
−q 3
←
1
1
1
#
1
1 →
−q 1
#
1
−q i
←
−q i
←
0
1
−q 2
0→
−
q1→
zi
each
q4 wait
1 1 1
q3 →1
1 1 1
q1 →1
1 1 1
qi
1 0 1
q2
1 0 #
q1
1 #
qi
# #
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
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
30 / 39
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
zi
Control/inner structure
Outer structure
TM generating the sequence
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
34 / 39
Signal machines : localization of isolated accumulation
Conclusion
1
Signal machines and isolated accumulations
2
Necessary conditions on the coordinates of isolated
accumulations
c.e.
3
Manipulating
4
Accumulating at
5
Conclusion
and
c.e.
d-c.e.
and
real numbers
d-c.e.
real numbers
35 / 39
Signal machines : localization of isolated accumulation
Conclusion
Results
Isolated accumulations happen at
d-c.e.
spacial and
c.e.
temporal coordinates
c.e. temporal coordinate is possible
Accumulation at any d-c.e. spacial coordinate is possible
Accumulation at any
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):676690.
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):146.
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 644653. 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 107116. University of Athens.
Durand-Lose, J. (2009).
Abstract geometrical computation 3: Black holes for classical and
analog computing.
Nat. Comput., 8(3):455472.
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 158167. CMATI, U. Azores.
Durand-Lose, J. (2011).
Abstract geometrical computation 4: small Turing universal signal
machines.
Theoret. Comp. Sci.
, 412:5767.
Weihrauch, K. (2000).
Introduction to computable analysis
.
Texts in Theoretical computer science. Springer, Berlin.
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