Desarrollo de competencias orientadas al cliente

We can now apply our main theorem, Theorem 15, to carry over results that rely on the notion of reduction graph for legal strings. To illustrate this, we characterize successfulness for realistic overlap graphs in any given S ⊆ {Gnr, Gpr, Gdr}. To accomplish this, we also use results from [13] (or Chapter 13 in [12]). The notions of successful reduction, string negative rule and graph negative rule used in this section are defined in [12].

First we restate a theorem of [6]. Theorem 16

Letube a legal string, andN be the number of components inRu. Then every successful reduction ofuhas exactlyN−1string negative rules.

Due to the “weak equivalence” of the string pointer reduction system and the graph pointer reduction system, proved in Chapter 11 of [12], we can, using Theorem 15, restate Theorem 16 in terms of graph reduction rules.

Theorem 17

Let γ be a realistic overlap graph, andN be the number of components in Rγ. Then every successful reduction ofγ has exactlyN −1 graph negative rules. As an immediate consequence we get the following corollary. It provides an answer to an open problem formulated in Chapter 13 in [12]: to provide a graph theoretic characterization of successfulness in{Gpr, Gdr}. However, note that our answer is only for the case whereγ is arealistic overlap graph.

Corollary 18

Let γ be a realistic overlap graph. Then γ is successful in {Gpr, Gdr} iff Rγ is connected.

Example 11

Every successful reduction of the overlap graph of Example 9 has exactly one graph negative rule. For examplegnr₂gpr₄ gpr₅gpr₇ gpr₆gpr₃ is a successful reduction of this overlap graph.

With the help of [13] (or Chapter 13 in [12]) and Corollary 18, we are ready to complete the characterization of successfulness for realistic overlap graphs in any givenS ⊆ {Gnr, Gpr, Gdr}.

Theorem 19

Letγbe a realistic overlap graph. Thenγ is successful in:

• {Gnr}iffγ is a discrete graph with only negative vertices.

• {Gnr, Gpr} iff each component ofγ that consists of more than one vertex contains a positive vertex.

120 Discussion

• {Gnr, Gpr, Gdr}.

• {Gdr} iff all vertices ofγ are negative andRγ is connected.

• {Gpr} iff each component of γ contains a positive vertex and Rγ is con- nected.

• {Gpr, Gdr} iffRγ is connected. Proof

The cases where Gnr ∈ S (cf. the first four cases in the theorem) are known from [13], and case{Gpr, Gdr} holds by Corollary 18. Case{Gdr}({Gpr}, resp.) is obtained by combining the results of cases {Gnr, Gdr} ({Gnr, Gpr}, resp.) and {Gpr, Gdr}. Note that if γ has an isolated negative vertex, then Rγ is not connected.

5.9

Discussion

We have shown a way to directly construct the reduction graph of a realistic string (up to isomorphism) from its overlap graphγ. This allows one to (directly) determine the number n of graph negative rules that are necessary to reduceγ

successfully. Surprisingly, although a lot a structural information is lost in the overlap graph (compared to a string representation), this information can be re- trieved from the overlap graph via its reduction graph. The main result allows for a complete characterization of successfulness ofγin any givenS⊆ {Gnr, Gpr, Gdr}

by extending [13] for the cases where Gnr 6∈ S. It remains an open problem to find a (direct) method to determine this numbernfor overlap graphsγin general (not just for realistic overlap graphs).

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Membrane Systems with

Proteins Embedded in

Membranes

Abstract

Membrane computing is a biologically inspired computational paradigm. Moti- vated by brane calculi we investigate membrane systems which differ from con- ventional membrane systems by the following features: (1) biomolecules (proteins) can move through the regions of the systems, and can attach onto (and de-attach from) membranes, and (2) membranes can evolve depending on the attached mole- cules. The evolution of membranes is performed by using rules that are motivated by the operation of pinocytosis (the pino rule) and the operation of cellular drip- ping (the drip rule) that take place in living cells.

We show that such membrane systems are computationally universal. We also show that if only the second feature is used then one can generate at least the family of Parikh images of the languages generated by programmed grammars without appearance checking (which contains non-semilinear sets of vectors).

If, moreover, the use of pino/drip rules is non-cooperative (i.e., not dependent on the proteins attached to membranes), then one generates a family of sets of vectors that is strictly included in the family of semilinear sets of vectors.

We also consider a number of decision problems concerning reachability of configurations and boundness.

6.1

Introduction

Membrane computing is a biologically inspired computational paradigm intro- duced by Gh. Păun in 1998, [20]. The model is based on a hierarchical structure of nested membranes, inspired by the structure of living cells. In each region

128 Introduction

(enclosed by a membrane) some objects are present, modeling the presence of molecules inside the compartments of living cells. Moreover, each region has an associated set of multiset rewriting rules. These rules are motivated by chemical reactions that occur inside the regions of living cells. Membranes play a crucial role in living cells: the cell membrane separates, and hence protects the cell from its environment and the inner membranes delimit the structure of various organelles of the cell, e.g., the nuclear membrane separates the nucleus from the rest of the cell.

Membranes are not only “containers” but they also regulate the flow of mole- cules into and out of the cell. This is facilitated by proteins that are embedded in membranes and which provide channels for the transport of molecules through membranes.

In brane calculi, presented in [5], several operations (pino, exo, phago, mate, drip, bud) involving membranes with embedded proteins are considered and for- malized in the framework of process calculi. The important difference with mem- brane computing is that the evolution of the system happenson the membranes andnot inside the compartments (regions) delimited by them. The computational power of several brane calculi operations has been investigated in [4] where univer- sality has been obtained for systems usingphago andexo. In [6] these operations from brane calculi have been represented in the membrane computing framework and then studied by using tools from formal language theory.

In this chapter we investigate operations involving membranes with embedded proteins, but we also add the ability of proteins to attach/de-attach to/from the membranes, and also to move through the membranes. Hence, in our case, the evolution of the system takes place both on the membranes and inside the regions, which is natural from a biological point of view.

More specifically, we considerprotein-membrane rules – rules that modify the structure of (the membranes of) the system where the modifications are based on the multisets of proteins embedded in the membranes (we say that such multisets mark the membranes). In particular, we consider thepino anddriprules inspired by the operation ofpinocytosisand the operation of cellulardripping, respectively. Both pinocytosis and dripping split off a membrane from another membrane, however, in pinocytosis, this new (empty) membrane is found inside the original membrane, while in dripping, this new membrane is found outside the original membrane. We also useprotein movement rules, that model the attachment, de- attachment and movement of the proteins. Also these rules are applied according to the proteins marking the involved membranes. The protein movement rules do not change the membrane structure of the system, but they can change the multisets of embedded proteins marking the membranes of the system.

The chapter is structured in the following way. In Section 6.2 we provide preliminaries concerning formal languages, recalling in particular the definition of programmed grammars often used in the proofs. In Section 6.3 we recall the formal definition of pino and drip rules, and introduce the protein movement rules, and in Section 6.4 we introduce membrane systems based on these rules – the model

is calledmembrane system with marked membranes, protein-membrane rules, and protein movement rules, abbreviated asPpp system.

In Section 6.5, we investigate the computational power of Ppp systems which use only protein movement rules, and in Section 6.6 of Ppp systems using only pino (or drip) rules. In Section 6.7, we discuss Ppp systems using both types of rules. In Section 6.8 we prove several decidability results concerning reachability of configurations and boundness ofPpp systems with pino, drip rules, and protein movement rules. In the last section we discuss the results obtained in this chapter and formulate a number of research directions.