### Abstract

First, we consider P systems with active membranes, hence with the possibility that the membranes can be divided, with non-cooperating evolution rules (the objects always evolve separately). These systems are known to be able to solve NP-complete problems in linear time. Here we give a normal form theorem for such systems: their computational universality is preserved even if only the elementary membranes are divided. The possibility of solving SAT in linear time is preserved only when non-elementary membranes may also be divided under the influence of objects in their region. Second, we consider a slight generalization, namely, we allow that a membrane can produce by division both a copy of itself and a copy of a membrane with a different label; again, only elementary membranes may be divided. In this case, we prove that the hierarchy on the maximal number of membranes present in the system collapses: three membranes at a time are sufficient in order to characterize the recursively enumerable sets of vectors of natural numbers. This result is optimal, two membranes are shown not to be sufficient. Third, we consider P systems with cooperating rules (several objects may evolve together). Making use of this powerful feature, we show that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges). The degree of cooperation is minimal: two objects at a time.

Original language | English |
---|---|

Pages (from-to) | 61-85 |

Number of pages | 25 |

Journal | Theoretical Computer Science |

Volume | 324 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Sep 16 |

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### Keywords

- Membrane computing
- Recursively enumerable language
- SAT problem
- Universality

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*324*(1), 61-85. https://doi.org/10.1016/j.tcs.2004.03.053

**On the power of membrane division in P systems.** / Pǎun, Gheorghe; Suzuki, Yasuhiro; Tanaka, Hiroshi; Yokomori, Takashi.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 324, no. 1, pp. 61-85. https://doi.org/10.1016/j.tcs.2004.03.053

}

TY - JOUR

T1 - On the power of membrane division in P systems

AU - Pǎun, Gheorghe

AU - Suzuki, Yasuhiro

AU - Tanaka, Hiroshi

AU - Yokomori, Takashi

PY - 2004/9/16

Y1 - 2004/9/16

N2 - First, we consider P systems with active membranes, hence with the possibility that the membranes can be divided, with non-cooperating evolution rules (the objects always evolve separately). These systems are known to be able to solve NP-complete problems in linear time. Here we give a normal form theorem for such systems: their computational universality is preserved even if only the elementary membranes are divided. The possibility of solving SAT in linear time is preserved only when non-elementary membranes may also be divided under the influence of objects in their region. Second, we consider a slight generalization, namely, we allow that a membrane can produce by division both a copy of itself and a copy of a membrane with a different label; again, only elementary membranes may be divided. In this case, we prove that the hierarchy on the maximal number of membranes present in the system collapses: three membranes at a time are sufficient in order to characterize the recursively enumerable sets of vectors of natural numbers. This result is optimal, two membranes are shown not to be sufficient. Third, we consider P systems with cooperating rules (several objects may evolve together). Making use of this powerful feature, we show that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges). The degree of cooperation is minimal: two objects at a time.

AB - First, we consider P systems with active membranes, hence with the possibility that the membranes can be divided, with non-cooperating evolution rules (the objects always evolve separately). These systems are known to be able to solve NP-complete problems in linear time. Here we give a normal form theorem for such systems: their computational universality is preserved even if only the elementary membranes are divided. The possibility of solving SAT in linear time is preserved only when non-elementary membranes may also be divided under the influence of objects in their region. Second, we consider a slight generalization, namely, we allow that a membrane can produce by division both a copy of itself and a copy of a membrane with a different label; again, only elementary membranes may be divided. In this case, we prove that the hierarchy on the maximal number of membranes present in the system collapses: three membranes at a time are sufficient in order to characterize the recursively enumerable sets of vectors of natural numbers. This result is optimal, two membranes are shown not to be sufficient. Third, we consider P systems with cooperating rules (several objects may evolve together). Making use of this powerful feature, we show that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges). The degree of cooperation is minimal: two objects at a time.

KW - Membrane computing

KW - Recursively enumerable language

KW - SAT problem

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=4344596902&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344596902&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2004.03.053

DO - 10.1016/j.tcs.2004.03.053

M3 - Article

VL - 324

SP - 61

EP - 85

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1

ER -