Irrationality is needed to compute with signal machines with only three speeds
Irrationality is needed to compute with
signal machines with only three speeds
To be presented at CiE '13
Jérôme Durand-Lose
Laboratoire d'Informatique Fondamentale d'Orléans,
Université d'Orléans, Orléans, FRANCE
Frac 2013 28 février & 1e mars 2013 LIRMM Montpellier
1 / 30
Irrationality is needed to compute with signal machines with only three speeds
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
2 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
3 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Time (N)
Time (R + )
Signals in cellular automata
Space (Z)
Space (R)
Signal (meta-signal)
Collision (rule)
4 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Vocabulary and example: nd the middle
Meta-signals (speed)
M (0)
Collision rules
M
M
5 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Vocabulary and example: nd the middle
Meta-signals (speed)
M (0)
div (3)
div
Collision rules
M
M
5 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Vocabulary and 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 }
5 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Vocabulary and example: nd the middle
back
M
hi
M
lo
M
div
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 }
5 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Vocabulary and example: nd the middle
M
M
back
hi
M
lo
M
div
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 }
5 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Complex behavior
6 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Complex behavior
6 / 30
Irrationality is needed to compute with signal machines with only three speeds
Signal machines
Complex behavior
6 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
7 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Accumulating
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
8 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Accumulating
Accumulations are quite common
9 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Computing
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
10 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Computing
Simulating a Turing machine (on a nite tape)
c
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c
b
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b
b
a
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b c a # ...
b c # ...
b b ...
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c # ...
b ...
b ...
b ...
b ...
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c ...
c ...
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#
q0 b
b
#
q0
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q0
q0
a
q0
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b
11 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Few speeds
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
12 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Few speeds
Minimality
Minimal number of
Accumulate 4
signals
to...
le
le
zig
zag
zig
ri
ri
13 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Few speeds
Minimality
Minimal number of
to...
Accumulate 4
Compute 13 meta-signals
(21 collision rules)
Cyclic tag system
signals
[Durand-Lose, 2011]
13 / 30
Irrationality is needed to compute with signal machines with only three speeds
Problematics
Few speeds
Minimality
Minimal number of
to...
Accumulate 4
Compute 13 meta-signals
(21 collision rules)
Cyclic tag system
signals
[Durand-Lose, 2011]
Minimal number of
Accumulate
speeds
to...
this talk
Compute
[Becker et al., 2013,
Durand-Lose, 2013]
this talk
[Durand-Lose, 2013]
13 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
14 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
2 speeds or less
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
15 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
2 speeds or less
1 or 2 speeds
1 speed
2 speeds
No collision at all
Bounded number of collisions
No accumulation
Not Turing-universal
16 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
4 speeds or more
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
17 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
4 speeds or more
Accumulation
le
le
zig
zag
zig
ri
ri
Computation
Accumulation
Turing-universal
18 / 30
Irrationality is needed to compute with signal machines with only three speeds
Simple cases
4 speeds or more
Computation simulation TM
Accumulation
c
q1
q1
c
b
le
le
zig
zag
zig
#̇
#
q1
q1
q1 b
ri
c
ri
c
#
q0
q0 #
c
Computation
c
q0
c
#
q0
#̇ #
q0
q0
q0
q0 b
b
#
q0
c
b
q0
q0
a
Accumulation
Turing-universal
q0
c
b
18 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
19 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Q
Rational (numbers) case ( )
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
20 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Q
Rational (numbers) case ( )
Rational case (Q)
Rational speed
Rational initial positions
Collisions at rational positions
as the solution of systems of two rational linear equations
Implemented in Java
Exact precision (on Q)
Tons of space-time diagrams
21 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Q
Rational (numbers) case ( )
Rational space-time diagrams
22 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Q
Rational (numbers) case ( )
Embedded in a mesh
Some gcd computations [Becker et al., 2013]
Embedded in a mesh [Becker et al., 2013, Durand-Lose, 2013]
Mesh
0
1
2
Results
No accumulation [Becker et al., 2013]
No computation [Durand-Lose, 2013]
3
23 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Irrational accumulating case
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
24 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Irrational accumulating case
Simple fractal construction
Fractal construction
cell
cell
ϕ-1
t
righ
t
left
cell
righ
1
left
border
Fractal
[Becker et al., 2013, Durand-Lose, 2013]
t
righ
left
ϕ
must satisfy ϕ1 = ϕ −1 1
ϕ
is the
t
righ
ϕ
seed
1
Golden ratio
25 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Irrational computing case
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
26 / 30
Irrationality is needed to compute with signal machines with only three speeds
3 speeds
Irrational computing case
How to enlarge the tape?
q0
q0
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c
c
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c
c
b
b
c
c
b
b
q0
q0
q0
b c a # ...
b c # ...
b b ...
c b ...
c # ...
b ...
b ...
b ...
b ...
c ...
c ...
c ...
enl
q1
b
q1
q0
t
righ c
left
c
q0
c
arg
eq1
q1
q1
b
c q1
t
righ
c
q1
q1
c
q0
t
righ
enl
arg
e
q0
c
q0
q0
q0
q0 b
q0
b
q0
c
b
q0
q0
a
border
c
c
c
c
c
c
c
b
b
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b
a
but do not generate it!
border
Use the fractal...
q0
c
b
27 / 30
Irrationality is needed to compute with signal machines with only three speeds
Results and future work
1
2
3
4
5
Signal machines
Problematics
Accumulating
Computing
Few speeds
Simple cases
2 speeds or less
4 speeds or more
3 speeds
Rational (numbers) case (Q)
Irrational accumulating case
Irrational computing case
Results and future work
28 / 30
Irrationality is needed to compute with signal machines with only three speeds
Results and future work
Results
Rational signal machines
Up to
( rational ratios)
At least 4 speeds to accumulate or compute
With an irrational ratio between initial distances
3 (rational) speeds are enough to accumulate and compute
Turing-universal 25-meta-signal 3-speed signal machine with
the Golden ratio
With an irrational ratio between speeds
Can be used to get an irrational ratio between distances
3 (rational) speeds are enough both to accumulate and
compute
normalization
29 / 30
Irrationality is needed to compute with signal machines with only three speeds
Results and future work
Future work
Use irrational values as oracle
Black hole (hyper-)computation
Analog computation?
30 / 30
Irrationality is needed to compute with signal machines with only three speeds
Results and future work
Becker, F., Chapelle, M., Durand-Lose, J., Levorato, V., and Senot,
M. (2013).
Abstract geometrical computation 8: Small machines, accumulations
& rationality.
Draft.
Durand-Lose, J. (2011).
Abstract geometrical computation 4: small Turing universal signal
machines.
Theoret. Comp. Sci.
, 412:5767.
Durand-Lose, J. (2013).
Irrationality is needed to compute with signal machines with only
three speeds.
In
CiE '13
, LNCS. Springer.
30 / 30