# A Characterisation of NL Using Membrane Systems without Charges and Dissolution

TL;DR: It turns out that the computational power of some systems is lowered from P to NL, so it seems that the tighter uniformities are more reasonable for these systems.

Abstract: We apply techniques from complexity theory to a model of biological cellular membranes known as membrane systems or P-systems. Like circuits, membrane systems are defined as uniform families. To date, polynomial time uniformity has been the accepted uniformity notion for membrane systems. Here, we introduce the idea of using AC 0and L -uniformities and investigate the computational power of membrane systems under these tighter conditions. It turns out that the computational power of some systems is lowered from P to NL , so it seems that our tighter uniformities are more reasonable for these systems. Interestingly, other systems that are known to be lower bounded by P are shown to retain their computational power under the new uniformity conditions. Similarly, a number of membrane systems that are lower bounded by PSPACE retain their power under the new uniformity conditions.

## Summary (2 min read)

### 1 Introduction

- Membrane systems [12] are a model of computation inspired by living cells.
- In the active membrane model it is also possible for a membrane to completely dissolve, and for a membrane to divide into two child membranes.
- The authors also show that the PSPACE lower and upper bounds mentioned above still hold under these restricted uniformity conditions.
- So far the authors have shown that four models, that characterise P when polynomial time uniformity is used, actually only characterise NL when restricted to be AC0 uniform.
- Interestingly, the authors also show that two other polynomial time uniform membrane system that are known [9] to be lower bounded by P actually retain this P lower bound when restricted to be AC0 uniform.

### 2 Membrane Systems

- In this section the authors define membrane systems and complexity classes.
- The authors also introduce the notion of AC0 uniformity for membrane systems.

### 2.1 Recogniser membrane systems

- Active membranes systems are membrane systems with membrane division rules.
- Division rules can either only act on elementary membranes, or else on both elementary and non-elementary membranes.
- These rules are applied according to the following principles: – All the rules are applied in maximally parallel manner.
- The result of the computation (a solution to the instance) is “yes” if a distinguished object yes is expelled during the computation, otherwise the result is “no”.
- Therefore, the following interpretation holds: given a fixed initial configuration, a confluent membrane system nondeterministically chooses one from a number of valid configuration sequences, but all of them must lead to the same result.

### 2.2 Complexity classes

- Here the authors introduce the notion of AC0 uniformity to membrane systems.
- Previous work on the computational complexity of membrane systems used (Turing machine) polynomial time uniformity [14].
- – Each ΠX(n) is confluent: all computations of ΠX(n) with the same input x of size n give the same result; either always “yes” or else always “no”.
- The authors denote by AM0+ne the classes of membrane systems with active membranes, and both non-elementary and elementary membrane division and no charges.
- The authors now show that the use of AC0 uniformity does not change this lower bound.

### 3 NL Upper bound on active membranes without dissolution rules

- Previously the upper bound on all active membrane systems without dissolution was P [5].
- The authors give an overview rather than the full details.
- The authors make the observation that the graph GΠ can be constructed in deterministic logspace, and even in AC0.
- Since the authors have shown that the problem of simulating a membrane system without charges and without dissolution can be encoded as an NL-complete problem they have proved Theorem 1.

### 4 NL lower bound for semi-uniform active membranes without dissolution

- The algorithm works by having each edge in the problem instance graph represented as a membrane.
- The initial multisets are all empty except Mcount = {c2n+1}.
- The authors also have a counter that counts down in parallel with the above steps.
- Note that the authors encode the edges of the graph as membranes, rather than objects.
- In the membrane computing framework, for uniform membrane systems, inputs must be specified as objects.

### 4.1 PARITY lower bound for uniform active membranes without dissolution

- The previous proof gave a lower bound for a semi-uniform membrane system.
- The authors show that PARITY ∈ PMCAM0−d,+u by providing an AC0 uniform membrane system that can solve instances of the problem.
- PARITYis the problem of telling whether the number of 1 symbols in the input word is odd.
- A type (a) rule is created mapping every even object with i “1” symbols to the odd object with i− 1 “1” symbols in it.
- The AC0 uniformity machine (a CRAM) rearranges the input word w by moving all 1 symbols to the left and all 0 symbols to the right, to give w′.

### 5 P lower bound on uniform families of active membrane systems with dissolving rules

- In this section the authors show that does not happen for all models with at least P power.
- Naturally this result also holds for the semi-uniform case.
- The resulting membrane system directly solves the instance of CVP in polynomial time.
- The authors simulate multiple fanouts by outputting multiple copies of the resulting truth value of each gate.
- The output of a gate moves up through the layers of the membrane system until it reaches the correct gate according to its tag.

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##### Citations

13 citations

### Cites background from "A Characterisation of NL Using Memb..."

...AC or L), then we conjecture that a P lowerbound can be found by improving a result in [6]....

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...What is the lowerbound on the power of the systems that we consider? If P uniformity is used, then we get a trivial P lowerbound [6]....

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...Given a (properly encoded) set of rules for a membrane system Π, the dependency graph GΠ is created in logspace [6]....

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9 citations

### Cites background from "A Characterisation of NL Using Memb..."

...In this paper and others [33, 34, 35, 36, 37], we have put forward the idea of exploring the power of membrane systems under tight uniformity conditions....

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9 citations

### Cites background from "A Characterisation of NL Using Memb..."

...It is also widely investigated how certain restrictions on P systems with active membrane affect the computation power of these systems (see for example [6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 25])....

[...]

6 citations

6 citations

### Cites background or methods from "A Characterisation of NL Using Memb..."

...Membrane systems have been introduced as a computational model inspired by cellular biology [9], and have been later applied to the description of biological systems [6]....

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...Various possibilities how to “go beyond Turing” to be already found in the literature are discussed in [9]; most of the definitions and results for red-green Turing machines are taken from this paper....

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...To get the reader familiar with the basic idea of red-green automata, we give a short sketch of the proofs for some well-known results (see [9]):...

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...Various classes of membrane systems (also called P systems) have been defined in [9], while several applications of these systems are described in [3]....

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...We assume that the reader is familiar with the basics of formal language theory and membrane computing; for more information, we refer to the monograph [10], and the handbooks [9] and [8]....

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##### References

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### "A Characterisation of NL Using Memb..." refers methods in this paper

...STCON is also known as PATH [17] and REACHABILITY [12]....

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...STCON is also known as PATH [17] and REACHABILITY [12]....

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