Towards a Characterization of P Systems with Minimal Symport / Antiport and Two Membrane s

1. 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 artiome.alhazov@estudiants.urv.cat

2. Key points of the systems definition • Maximal parallelism • Result at halting, in the elementary membrane

3. 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

4. e e s0 s1 sn......... s2 or or n times, n0 nothing here The Garbage is Unavoidable:If OP2(sym1,anti1) then0N()  N()NFIN • Suppose N()is infinite • Suppose N()contains 0: it halts • s0 is in region 1=> so is s1,..., sn • contradiction

5. c b a or 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). • (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 objectsinside 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.

6. UNIVERSALITY

7. 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)

8. Construction Notations: C={ck}, k{1,…,d}, Q’={qi’}, qi Q.

9. 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.

10. Preparation ck q0 Ic M S

11. Main stage: representation supply of states supply of instructions qi :current state supply of counters ck:counters

12. aj ql q’l a’j J0 J1 qi ck J2 dj bj q’l aj ql a’j J0 J1 qi J2 dj ck bj q’l aj ql a’j J0 J1 qi J2 no ck dj bj Main stage: +,-,=0

13. Termination J2 M F2 F3 F4 F5 F1 qf bj or dj T2 T1 S

15. 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}