Towards a Characterization of P Systems
with Minimal Symport/Antiport and Two
Membranes
Artiom Alhazov 1,2, Yurii Rogozhin 2
1 Institute of Mathematics and Computer Science
Academy of Sciences of Moldova
{artiom,rogozhin}@math.md
2 Research Group on Mathematical Linguistics
Rovira i Virgili University, Tarragona, Spain
[email protected]
Key points of the systems
definition
• Maximal parallelism
• Result at halting, in the elementary
membrane
The power of some classes of P
systems with small symport/antiport
• OP3(sym1,anti1) and OP3(sym2) are
computationally complete
•OP2(sym1,anti1) and OP2(sym2) are complete
modulo some additional objects
• OP1(sym1,anti2/1) is computationally
complete
• OP1(sym3) is complete modulo 7
addditional objects
The Garbage is Unavoidable:
If OP2(sym1,anti1) then
0N()  N()NFIN
• Suppose N()
is infinite
e
sn......... s2
or
• Suppose N()
contains 0: it halts
• s0 is in region 1=>
so is s1,..., sn
• contradiction
s1
n times, n0
e
or
nothing here
s0
Symport Garbage: If OP2(sym2)
then 0N()  N()NFIN
• Call I0 the set of objects from O-E that we know must be in the environment at
halting with empty region 2; I0 :=Ø.
• Assume N() is infinite: it is necessary (though not sufficient)
to bring in some object aEI0 : (ab,in)R1 (bO-E).
c
or
•(b,out): cannot halt with empty region 2.
•(bc,out), cO-E: c does not stay in the environment
(otherwise it does not help increasing the number of objects
inside the system).
•Still, c cannot end up in region 1 if b is there. Add c to I0 and repeat.
•(bc,out), cE: repeat with c instead of a.
•If a system can increase the number of objects inside it, then it cannot halt
without any objects in region 2.
b a
UNIVERSALITY
Notations : N k RE  {k  L | L  NRE },
N1 RE  {N  NRE | 0  N }.
N 0 FIN  {N  NFIN | 0  N },
N 0 SEG1  {{k  N | k  n} | n  0}.
Theorem 3. NOP2(sym1,anti1 )  N1 RE  F , where
N 0 SEG1  F  N 0 FIN .
Proving only
N1OP2 ( sym1 , anti1 )  N1 RE
Outline of the proof
• We simulate a d -counter automaton
M=(d, Q, q0 , qf , P).
Q={qi |0 ≤ i ≤ f } states, q0 initial, qf final,
P finite set of instructions.
“increment” j: (qi  ql ,k+)
“decrement” j: (qi  ql ,k-)
“test for zero” j: (qi  ql ,k=0)
Construction
Notations: C={ck}, k{1,…,d}, Q’={qi’}, qi  Q.
The functioning of this system may
be split into three stages:
• preparation of the system for the
computation.
• simulation of instructions of the counter
automaton.
• terminating the computation.
Preparation
ck
q0
Ic
M
S
Main stage: representation
supply of states
supply of instructions
qi :current state
supply of counters
ck :counters
aj
a’j
J1
qi
Main stage: +,-,=0
ql
q’l
J0
ck
J2
dj
bj
aj
a’j
J0
q’l
ql
J1
qi
J2
dj
bj
aj
a’j
ck
J0
J1
qi
J2
bj
no ck dj
q’l
ql
Termination
J2
F1
M
F2
F3 F4
F5
qf
T2
T1
S
bj or dj
Conclusion
• P systems with minimal symport/antiport and two
membranes: the optimal result is obtained.
– One additional object in the output membrane is
necessary and sufficient for computational completeness.
• For P systems with two membranes and symport
of weight 2,
– one object is necessary; sufficiency is open.
• It is still open what finite sets containing zero can
be generated by these systems
– Conjecture: {{m|m≤n}nN}