Tissue P Systems with Small Numbers of Symbols and Cells

1. Tissue P Systems with Small Numbers of Symbols and Cells Artiom ALHAZOV Rudolf FREUND Marion OSWALD Faculty of Informatics Vienna University of Technology Wien, Austria

2. Overview P Systems with Symport/Antiport Rules ● membrane systems / P systems - definition ● complexity issues ●selected results for P systems Tissue P Systems with Symport/Antiport Rules ● definition ● complexity issues ●results for tissue P systems Summary and Open Problems

3. Membrane Systems (P Systems) (invented by Gheorghe PǍUN , 1998) membrane structure multisets of objects • evolution / communication rules applied in • the maximally parallel mode • the sequential mode many variants computationally complete Gheorghe Păun: Membrane Computing - An Introduction. Springer-Verlag, Berlin, 2002. The P Systems Web Page: http://psystems.disco.unimib.it

4. Membrane structure [1[2[4 ]4 [5 ]5 ]2[3 ]3]1 elementary membrane region 4 5 2 3 1 skin membrane

5. P system with symport/antiport rules ( O,  ,w1, ... , wn, E, R1, ... , Rn, i0 ) O alphabet of objects (symbols);  membrane structure with n membranes; wi , 1  i  n, multiset over V in region i; E  V set of objects in the environment; Ri , 1  i  n, finite set of symport rules (x,in) or (x,out) and antiport rules (x,out;y,in) over V, x,y  O+assigned to membrane i; i0 output membrane.

6. derivation modes maximally parallel derivation mode choose a multiset of rules in such a way that after assigning objects from the environment and the regions to these rules not enough objects are left to add another rule which could be applied together with the chosen rules. sequential derivation mode only one rule is applied in each derivation step.

7. P system – derivation A derivation in the P system  works as follows: We start with the initial multisets wiin the regions inside the membranes; in the environment, all elements from E are available in an unbounded number. At any stage of the derivation, the rules assigned to the membranes are used according to the derivation mode.

8. P system – language Every number of objects from Oever appearing at the end of a halting computation in the ouput membrane i0 contributes to N(), the language generated by . • Attention: • in several variants to be found in the literature, • the ouput membrane i0 must be an elementary • membrane • by distinguishing between terminal and • nonterminal objects and taking only the • terminal objects in the ouput membrane i0 • garbage can be “eliminated”

9. P system – complexity issues • number of membranes • number of objects • weight of the rules • weight of (x,out;y,in)is (|x|,|y|) • number of rules • ... mostly studied in the literature so far: • number of membranes • weight of the rules

10. P systems – classic results Theorem. Any recursively enumerable language L can be generated by a P system with symport/antiport rules of weight (1,2) and (2,1) as well as (1,0) with only one membrane in the maximally parallel derivation mode. Theorem. P systems with symport/antiport rules (in an arbitrary membrane structure) in the sequential derivation mode exactly generate the matrix languages (sets of numbers generated by matrix grammars without appearance checking).

11. P systems – number of objects the first result concerning the number of objects (as well as the number of membranes): Gh.Păun, J.Pazos, M.J.PérezJiménez, A. Rodríguez-Patón: Symport/ antiport P systems with three objects areuniversal; downloadable from the P Systems Web Page: http://psystems.disco.unimib.it three objectsand four membranes were needed!

12. P systems – number of objects, ctd. Investigations were continued in the Third Brainstorming Week on Membrane Computing, Sevilla(Spain), January 31st - February 4th, 2005: • Alhazov, R. Freund: P systems with one • membrane andsymport / antiport rules of five • symbols are computationally complete. • M.A.Gutierrez-Naranjo, A. Riscos-Nunez, • F.J. Romero-Campero, D. Sburlan (Eds.): • Proceedings of the Third Brainstorming Week on • Membrane Computing, Sevilla(Spain), • January 31st - February 4th, 2005, 19 – 28.

13. Register machine M = (n,R,l0,lh) • n number of registers, • R finite set of instructions, injectively • labelled with elements from lab(M), • l0 initial/start label, and • lh final label.

14. Register machine – instructions The instructions are of the following forms: • l1:(ADD(r), l2, l3)Add 1 to the contents of • register r and proceed to instruction l2 or l3. • l1:(SUB(r), l2, l3) If register r is not empty, • then subtract 1 from its contents and go to • instruction l2, otherwise proceed to • instruction l3 . • lh:halt Stop the machine.

15. P systems – number of objects, newest results • Alhazov, R. Freund, M. Oswald: • Symbol/MembraneComplexity of P Systemswith • Symport / Antiport Rules. Pre-Proceedings • WMC6, Vienna, July 18 – 21, 2005. Main results: P systems with symport/antiport rules and s  2 objects as well as m  1 membranes can simulate register machines with max{ m(s-2), (m-1)(s-1) } registers (equality in case of s = m+1).

16. P Systems with Antiport Rules and a Small Number of Objects and Membranes objects NRE 5 NRE (new) 4 at least undecid. 3 at least NREG 2 1 NFIN 1 2 3 4 5 6 membranes

17. Tissue P System (m, O, w1,... ,wn, E, ch, (R(i,j))(i,j) ch, i0 ) m number of cells; O alphabet of objects (symbols); wi , 1  i  n, multiset over V in cell i; E  V set of objects in the environment; ch set of channels between cells (environment) R(i,j), (i,j) ch, finite set of symport/antiport rules over O, x,y  O+assigned to channel (i,j); we simply write x/y for the rules; i0 output cell.

18. Tissue P System – conventions cells with an arbitrary graph structure for the connections betweencells (not necessarily a tree as in P systems) and cells with the environment Conventions: E = V all objects are available in an unlimited number in the environment; (i,j) ch implies i  j i0 = 1 the first cell is always the output cell. (m, O, w1,... ,wn, ch, (R(i,j))(i,j) ch)

19. Tissue P System – derivations A derivation in the P system  works as follows: We start with the initial multisets wiin the cells. In each derivation step, all channels in parallel executeonesymport/antiport rule (if possible).

20. Tissue P System – languages Every number of objects from Oever appearing at the end of a halting computation in the ouput cell 1 contributes to N(), the language generated by . NOnt’Pmand NOntPm family of languages (sets of natural numbers) generated by tissue P systems with n objects and m cells/ with only one channel out of { (i,j), (j,i) } for each pair (i,j) with i  j.

21. Tissue P System – example 1 Let L be an arbitrary finite set of natural numbers.  = ( 1, {a}, w1, { (1,0) }, R(1,0) ) w1 = amm := max { i | ai  L } +1 R(1,0) = {am /ai | ai  L } 1 N() = L

22. Tissue P System – example 2 G = ({ Xi | 1  i  n }, { a }, P, X1 ) regular grammar over one-letter alphabet { a }; productions in P of the form Xi aXj and Xn .  = ( 2, {a}, , aa, { (0,2), (2,0), (2,1) }, R(0,2), R(2,0), R(2,1) ) R(2,0) = { a2i /a2j+1 | Xi aXj P } R(0,2) = { a2n+2 /a }, R(2,1) = { a/ }, N() = Ps(L(G)) 1 2

23. Tissue P system – complexity issues • number of cells • number of objects • weight of the rules • weight of x/yis (|x|,|y|) • number of rules • ... mostly studied in the literature so far: • number of cells • weight of the rules

24. Tissue P systems – number of objects the first result concerning the number of objects (as well as the number of cells): R.Freund, M.Oswald: Tissue P systems with symport/ antiportrules of one symbol are computationally complete; downloadable from the P Systems Web Page: http://psystems.disco.unimib.it Theorem. NRE = NO1t‘P6 = NO1tP7

25. Tissue P systems – number of objectsnewest results Theorem. NRE = NO2t‘P3 Theorem. NRE = NO2tP3 !! Theorem. NRE = NO3t‘P2 Theorem. NRE = NO4tP2

26. Tissue P systems – number of objectsproof ideas - simulate register machine • encode the labels di+1 of the register machine • as powers of a symbol p in such a way that •  the sum of two codes is larger than the • largest code c(lh), lh = d(t-1)+1; •  the distance g between two codes allows • for using one symbol p for appearance • checking and numbers of p > 1 to detect an • incorrect application of rules (g=3 sufficient); • linear encoding c(x) = gx + gdt

27. Tissue P systems – number of objectsresults for one cell Theorem. NRE = NO5t‘P1 Theorem. NREG = NOntP1 for n  2 Theorem. NFIN = NO1tP1

28. tP Systems with Antiport Rules and a Small Number of Objects and Cells objects NRE 5 NREG 4 at least NREG 3 > NFIN 2 1 NFIN 1 2 3 4 5 6 7 cells

29. t‘P Systems with Antiport Rules and a Small Number of Objects and Cells objects NRE 5 4 at least NREG 3 2 1 NFIN 1 2 3 4 5 6 7 cells

30. SUMMARY ►we have shown that only a few cells and objects are needed to obtain NRE ►with only one channel between a cell and the environment and with only one cell we only obtain NREG ►with only one channel between a cell and the environment we obtain less power than with two channels between a cell and the environment

31. OPEN PROBLEMS ►with only one object, how many cells are needed to obtain NRE ►with two channels between a cell and the environment and with only one cell, how many objects are needed to obtain NRE ►with only two cells, how many objects are needed to obtain NRE

32. Workshop on Membrane Computing WMC6 will take place in VIENNA July 18 to 21, 2005

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