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

## 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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### 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])....

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### 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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...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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##### References

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

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