Tissue P Systems with Small Numbers of Symbols and Cells

Faculty of Informatics Vienna University of Technology

Wien, Austria

Rudolf FREUND

Marion OSWALD

Artiom ALHAZOV

P Systems with Symport/Antiport Rules

Overview

● membrane systems / P systems - definition● complexity issues ● selected results for P systems

Tissue P Systems with Symport/Antiport Rules

Summary and Open Problems

● results for tissue P systems

● definition● complexity issues

(invented by Gheorghe PǍUN , 1998)

membrane structuremultisets of objectsevolution / communication rules applied in • the maximally parallel mode• the sequential mode

many variants computationally complete

Membrane Systems (P Systems)

Gheorghe Păun: Membrane Computing - An Introduction. Springer-Verlag, Berlin, 2002.

The P Systems Web Page: http://psystems.disco.unimib.it

1

2 3

4 5

skin membrane

elementarymembrane

region

Membrane structure [1 [2 [4 ]4 [5 ]5 ]2 [3 ]3 ]1

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.

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.

derivation modes

maximally parallel derivation mode

sequential derivation modeonly one rule is applied in each derivation step.

P system – derivationA derivation in the P system works as follows:

We start with the initial multisets wi in 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.

P system – languageEvery number of objects from O ever 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”

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 - ...

- number of membranes- weight of the rules

mostly studied in the literature so far:

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 modeexactly generate the matrix languages (sets of numbers generated by matrix grammars without appearance checking).

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. Pérez Jiménez, A. Rodríguez-Patón: Symport/ antiport P systems with three objects are universal;downloadable from the P Systems Web Page: http://psystems.disco.unimib.it

three objects and four membranes were needed!

P systems – number of objects, ctd.

Investigations were continued in the Third Brainstorming Week on Membrane Computing, Sevilla (Spain), January 31st - February 4th, 2005:

A. Alhazov, R. Freund: P systems with one membrane and symport / 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.

Register machine

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

M = (n,R,l0,lh)

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.

P systems – number of objects, newest resultsA. Alhazov, R. Freund, M. Oswald: Symbol/Membrane Complexity of P Systems with Symport / Antiport Rules. Pre-ProceedingsWMC6, Vienna, July 18 – 21, 2005.

Main results:P systems with symport/antiport rules ands 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).

P Systems with Antiport Rules and a Small Number of Objects and Membranes

NFIN

NRE

at leastNREG

NRE(new)

objects

membranes1 2 3 4 5

1

2

3

4

5

at leastundecid.

6

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.

Tissue P System – conventions cells with an arbitrary graph structure for the connections between cells (not necessarily a tree as in P systems) and cells with the environment

(m, O, w1,... ,wn, ch, (R(i,j))(i,j) ch)

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.

Conventions:

Tissue P System – derivations

In each derivation step,

all channels in parallel

execute one symport/antiport rule (if possible).

A derivation in the P system works as follows:

We start with the initial multisets wi in the cells.

Tissue P System – languages Every number of objects from O ever appearing at the end of a halting computation in the ouput cell 1 contributes to N(), the language generated by .

NOnt’Pm and 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.

Tissue P System – example 1

= ( 1, {a}, w1, { (1,0) }, R(1,0) )

w1 = am m := max { i | ai L } +1

Let L be an arbitrary finite set of natural numbers.

R(1,0) = {am /ai | ai L }

N() = L 1

Tissue P System – example 2

= ( 2, {a}, , aa, { (0,2), (2,0), (2,1) }, R(0,2), R(2,0), R(2,1) )

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 .

R(2,0) = { a2i /a2j+1 | Xi aXj P }

N() = Ps(L(G))

R(0,2) = { a2n+2 /a }, R(2,1) = { a/ },

1 2

Tissue P system – complexity issues- number of cells- number of objects- weight of the rules weight of x/y is (|x|,|y|)- number of rules - ...

- number of cells- weight of the rules

mostly studied in the literature so far:

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 / antiport rules of one symbol arecomputationally complete;downloadable from the P Systems Web Page: http://psystems.disco.unimib.it

Theorem. NRE = NO1t‘P6 = NO1tP7

Tissue P systems – number of objectsnewest results

Theorem. NRE = NO2t‘P3

Theorem. NRE = NO2tP3 !!

Theorem. NRE = NO3t‘P2

Theorem. NRE = NO4tP2

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

Tissue P systems – number of objectsresults for one cell

Theorem. NRE = NO5t‘P1

Theorem. NREG = NOntP1 for n 2

Theorem. NFIN = NO1tP1

tP Systems with Antiport Rules and a Small Number of Objects and Cells

NFIN

NRE

>NFIN

NREG

objects

1 2 3 4 5

1

2

3

4

5

at leastNREG

6 cells7

t‘P Systems with Antiport Rules and a Small Number of Objects and Cells

NFIN

NRE

objects

1 2 3 4 5

1

2

3

4

5

at leastNREG

6 cells7

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

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

WMC6

will take place in VIENNA

July 18 to 21, 2005

Workshop on Membrane Computing

THANK YOU

FOR YOUR ATTENTION