Towards a Characterization of P Systems with Minimal Symport/Antiport and Two Membranes Artiom Alhazov 1,2, Yurii Rogozhin 2 1 Institute of Mathematics.

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}