Towards a Characterization of P Systems with Minimal Symport / Antiport and Two Membrane s
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Transcript of Towards a Characterization of P Systems with Minimal Symport / Antiport and Two Membrane s
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, [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
• OP1(sym1,anti2/1) is computationally complete
• OP1(sym3) is complete modulo 7 addditional objects
•OP2(sym1,anti1) and OP2(sym2) are complete modulo some additional objects
The Garbage is Unavoidable:If OP2(sym1,anti1) then0N() N()NFIN
• Suppose N()is infinite
• Suppose N()contains 0: it halts
• s0 is in region 1=> so is s1,..., sn
• contradictionnothing here
s0s1
n times, n0
s2sn.........or
e
or
e
Symport Garbage: If OP2(sym2) then 0N() 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 aEI0 : (ab,in)R1 (bO-E).
b aor
c•(b,out): cannot halt with empty region 2.•(bc,out), cO-E: c does not stay in the environment(otherwise it does not help increasing the number of objectsinside the system).•Still, c cannot end up in region 1 if b is there. Add c to I0 and repeat.•(bc,out), cE: 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.
UNIVERSALITY
RENantisymOPN
FINNFSEGN
FREN),anti(symNOP
nnkNkSEGN
NNFIN{NFINN
NNRENREN
NRELLkRENk
11121
010
1112
10
0
1
),(
.
where,
}.0|}|{{
},0|
}.0|{
},|{ :Notations
only Proving
3. Theorem
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
Ic
M S
q0
Main stage: representation
qi :current statesupply of counters
ck :counters
supply of statessupply of instructions
Mai
n st
age:
+,-
,=0
aj
qi
bj
a’j
J2
J1
J0
ck
dj
q’lql
aj
qi
bj
a’j
J2
J1
J0
ckdj
q’l ql
aj
qi
bj
a’j
J2
J1
J0
no ck dj
q’l ql
Termination
S
qf
T2 T1
F1F2 F3 F4 F5
J2 M
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}nN}