Infinite hierarchies of conformonP systems

1
Infinite hierarchies of conformon-P systems
Pierluigi Frisco School of Mathematical and
Computer Sciences Heriot-Watt University Edinburgh
UK
WMC2006, Leiden 17-21/7/2006
2
Conformon-P systems
X
Z
1
2
Y
Z
3
W
4
3
Conformon-P systems
G, 5 R, 9 G, 2
G, 5 G, 2 ?r R, 9 R, 12

4
Conformon-P systems example
1
2
? 1
G, 4
4
3
G ? R
4
3
5
Conformon-P systems example
1
2
? 1
G, 1
4
3
G ? R
4
3
6
Conformon-P systems example
1
2
? 1
R, 0
R, 0
R, 3
G, 1
4
3
G ? R
4
3
7
Conformon-P systems example
1
2
? 1
R, 0
R, 0
R, 0
G, 4
4
3
G ? R
4
3
8
Conformon-P systems example
1
2
? 1
R, 0
R, 0
R, 0
4
G, 4
3
G ? R
4
3
9
Conformon-P systems example
1
2
? 1
R, 0
R, 3
R, 0
4
G, 1
3
G ? R
4
3
10
Conformon-P systems example
1
2
? 1
R, 0
R, 0
R, 3
4
G, 1
3
G ? R
4
3
11
Conformon-P systems example
1
2
? 1
R, 0
R, 0
G, 4
R, 3
4
3
G ? R
4
3
12
Conformon-P systems example
1
2
? 1
R, 0
R, 0
R, 3
4
3
G ? R
4
3
G, 4
13
Conformon-P systems module
R, 2
14
Conformon-P systems modules
only conformon A, ?, ? ? N can pass from
membrane 1 to membrane 2.
1
2
A, ?
?
a conformon with name A can interact with B
passing ? only if the value of A and B before the
interaction is ? and ? respectively, ?, ?, ? ? N.
A(?) ? B(?)
?
A ? B(?)
?
A(?) ? B
15
Counter machines
...
15
0
3
counters c1 c2 cn
states A, B, C, ..., Z
instructions
(A, c2, B) (A, c1-, V, W) (Z, halt)
16
Restricted counter machines
...
15
0
3
counters c1 c2 cn
states A, B, C, ..., Z
instructions
(A, c2, B) (A, c1-, V, W) (Z, halt)
(A, c2, c5-, B) (A, c1-, c2, V, W)
O. H. Ibarra. On membrane hierarchy in P systems.
Theoretical Computer Science, 334115-129, 2005.
17
Conformon-restricted basic conformon-P systems
Basic conformon-P systems (any conformon has a
finite number of occurrences) having a conformon
with a distinguished name, let us say c, and such
that one some (input) membranes contain only c
conformons in the initial configuration.
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Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 30
sB, 0
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28
5
2
c5, 7 c ? sB(28)
? 30
19
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 9
sB, 21
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28
5
2
c5, 7 c ? sB(28)
? 30
20
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
sB, 21
9
? 0
4
19
sA, 9
? 28
5
2
c5, 7 c ? sB(28)
? 30
21
Simulation (A, c2, c5-, B)
2
3
1
x, 3
c2, 5
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
sB, 19
9
? 0
4
19
sA, 9
? 28
5
2
c5, 7 c ? sB(28)
? 30
22
Simulation (A, c2, c5-, B)
2
3
1
c2, 5
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
x, 3
9
? 0
4
19
sA, 9
? 28
sB, 19
5
2
c5, 7 c ? sB(28)
? 30
23
Simulation (A, c2, c5-, B)
2
3
1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
x, 1
9
? 0
4
19
sA, 0
? 28
sB, 28
5
2
c5, 7 c ? sB(28)
? 30
24
Simulation (A, c2, c5-, B)
2
3
1
sA, 0
x, 1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28
5
2
c5, 7 c ? sB(28)
? 30
sB, 28
25
Simulation (A, c2, c5-, B)
2
3
1
sA, 0
x, 1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28
5
2
c5, 5 c ? sB(28)
? 30
sB, 30
26
Simulation (A, c2, c5-, B)
2
3
1
sA, 0
sB, 30
x, 1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28
5
2
c5, 5 c ? sB(28)
? 30
27
Deterministic simulation
(A, c2, c5-, B)
28
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 30
sB, 0
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28

• deterministic simulation

5

• fluctuations of the sum of the

values of the c conformons
2
c5, 7 c ? sB(28)
? 30
29
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 30
sB, 0
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28

• deterministic simulation
• fluctuations of the sum of the values of the c
  conformons

5
2
c5, 7 c ? sB(28)
? 30
30
Simulation (A, c2, c5-, B)
2
3
1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
x, 1
9
? 0
4
19
sA, 0
? 28

• deterministic simulation
• fluctuations of the sum of the values of the c
  conformons

sB, 28
5
2
c5, 7 c ? sB(28)
? 30
31
Simulation (A, c2, c5-, B)
2
3
1
sA, 0
x, 1
c2, 7
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28

• deterministic simulation
• fluctuations of the sum of the values of the c
  conformons

5
2
c5, 5 c ? sB(28)
? 30
sB, 30
32
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 30
sB, 0
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28

• deterministic simulation
• fluctuations of the sum of the values of the c
  conformons

5
2
c5, 7 c ? sB(28)
? 30

• presence of connected loops.

33
Simulation (A, c2, c5-, B)
2
3
1
x, 1
c2, 5
sA, 30
sB, 0
3
21
2
2
sB(21) ? x(1)
x ? c2
x, 1
9
? 0
4
19
? 28

• deterministic simulation
• fluctuations of the sum of the values of the c
  conformons

5
2
c5, 7 c ? sB(28)
? 30

• presence of connected loops.

34
Conformon-restricted basic conformon-P systems
Basic conformon-P systems (any conformon has a
finite number of occurrences) having a conformon
with a distinguished name, let us say c, and such
that one some (input) membranes contain only c
conformons in the initial configuration.
... to put the c conformons in different input
membranes seems to be a must ...
Theorem 1 conformon-restricted basic
conformon-P systems induce an infinite
hierarchy on the number of membranes.
35
Theorem 1 proof
36
Membrane-restricted basic conformon-P systems
Basic conformon-P systems (any conformon has a
finite number of occurrences) in which the number
of input membrane is restricted to one and the
set of names of input conformons is bounded.
... to have as many different names of input
conformons as many counters in the simulated
restricted counter machine seems to be a must ...
Theorem 2 membrane-restricted basic conformon-P
systems induce an infinite hierarchy on the
number of input conformons.
37
Final remarks

• maximal parallelism is not necessary to obtain
  infinite hierarchies

• is it possible to have deterministic conformon-P
  systems?

38
Final remarks

• maximal parallelism is not necessary to obtain
  infinite hierarchies

• is it possible to have deterministic conformon-P
  systems?

• loops as measure of complexity

39
Final remarks

• maximal parallelism is not necessary to obtain
  infinite hierarchies

• is it possible to have deterministic conformon-P
  systems?

• loops as measure of complexity

• prove the must.

... to put the c conformons in different input
membranes seems to be a must ...
... to have as many different names of input
conformons as many counters in the simulated
restricted counter machine seems to be a must ...
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