CELL/SYMBOL COMPLEXITY OF TISSUE P SYSTEMS WITH SYMPORT/ANTIPORT RULES

We consider tissue P systems with symport/antiport rules and investigate their computational power when using only a (very) small number of symbols and cells. Even when using only one symbol, we need at most six (seven when allowing only one channel between a cell and the environment) cells to generate any recursively enumerable set of natural numbers. On the other hand, with only one cell we can only generate regular sets when using one channel with the environment, whereas one cell with two channels between the cell and the environment obtains computational completeness with five symbols. Between these extreme cases of one symbol and one cell, respectively, there seems to be a trade-off between the number of cells and the number of symbols. For example, for the case of tissue P systems with two channels between a cell and the environment we show that computational completeness can be obtained with two cells and three symbols as well as with three cells and two symbols, respectively. Moreover, we also show that some variants of tissue P systems characterize the families of finite or regular sets of natural numbers.

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When catalytic P systems with one catalyst can be computationally complete

Journal of Membrane Computing ◽

10.1007/s41965-021-00079-x ◽

2021 ◽

Author(s):

Artiom Alhazov ◽

Rudolf Freund ◽

Sergiu Ivanov

Keyword(s):

Computational Complexity ◽

P Systems ◽

Membrane Systems ◽

Computational Completeness ◽

Recursively Enumerable ◽

Recursively Enumerable Sets

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EXTENDED SPIKING NEURAL P SYSTEMS WITH DECAYING SPIKES AND/OR TOTAL SPIKING

International Journal of Foundations of Computer Science ◽

10.1142/s0129054108006248 ◽

2008 ◽

Vol 19 (05) ◽

pp. 1223-1234 ◽

Cited By ~ 11

Author(s):

RUDOLF FREUND ◽

MIHAI IONESCU ◽

MARION OSWALD

Keyword(s):

P Systems ◽

Extended Model ◽

Natural Numbers ◽

Spiking Neural P Systems ◽

Regular Sets

We consider extended variants of spiking neural P systems with decaying spikes (i.e., the spikes have a limited lifetime) and/or total spiking (i.e., the whole contents of a neuron is erased when it spikes). Although we use the extended model of spiking neural P systems, these restrictions of decaying spikes and/or total spiking do not allow for the generation or the acceptance of more than regular sets of natural numbers.

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OPTIMAL RESULTS FOR THE COMPUTATIONAL COMPLETENESS OF GEMMATING (TISSUE) P SYSTEMS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054105003388 ◽

2005 ◽

Vol 16 (05) ◽

pp. 929-942 ◽

Cited By ~ 1

Author(s):

RUDOLF FREUND ◽

MARION OSWALD ◽

ANDREI PĂUN

Keyword(s):

Theoretical Model ◽

General Model ◽

P Systems ◽

Optimal Result ◽

Recursively Enumerable Language ◽

Computational Completeness ◽

Minimal Weight ◽

Model Based ◽

Recursively Enumerable ◽

Enumerable Language

Gemmating P systems were introduced as a theoretical model based on the biological idea of the gemmation of mobile membranes. In the general model of extended gemmating P systems, strings are modified either by evolution rules in the membranes or while sending them to another membrane. We here consider the restricted variant of extended gemmating P systems with pre-dynamic rules where strings are only modified at the ends while sending them from one membrane to another one. In a series of papers the number of membranes being sufficient for obtaining computational completeness has steadily been decreased. In this paper we now prove the optimal result, i.e., gemmating P systems only using pre-dynamic rules are already computationally complete with three membranes, even in the non-extended case and with the minimal weight of rules possible. Moreover, we also show that for gemmating tissue P systems two cells suffice, and if we allow the environment to be fully involved in the communication of strings, even one cell together with the environment can manage the task to generate any recursively enumerable language.

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Variants of derivation modes for which catalytic P systems with one catalyst are computationally complete

Journal of Membrane Computing ◽

10.1007/s41965-021-00085-z ◽

2021 ◽

Author(s):

Artiom Alhazov ◽

Rudolf Freund ◽

Sergiu Ivanov ◽

Sergey Verlan

Keyword(s):

Computational Complexity ◽

P Systems ◽

P System ◽

Membrane Systems ◽

Energy Control ◽

Computational Completeness ◽

Current Configuration ◽

Recursively Enumerable ◽

Completeness Result ◽

Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining even computational completeness with only one catalyst. Last year we could show that the derivation mode $$max_{objects}$$ m a x objects , where we only take those multisets of rules which affect the maximal number of objects in the underlying configuration one catalyst is sufficient for obtaining computational completeness without any other ingredients. In this paper we follow this way of research and show that one catalyst is also sufficient for obtaining computational completeness when using specific variants of derivation modes based on non-extendable multisets of rules: we only take those non-extendable multisets whose application yields the maximal number of generated objects or else those non-extendable multisets whose application yields the maximal difference in the number of objects between the newly generated configuration and the current configuration. A similar computational completeness result can even be obtained when omitting the condition of non-extendability of the applied multisets when taking the maximal difference of objects or the maximal number of generated objects. Moreover, we reconsider simple P system with energy control—both symbol and rule energy-controlled P systems equipped with these new variants of derivation modes yield computational completeness.

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Some representations of Diophantine sets

Journal of Symbolic Logic ◽

10.2307/2272746 ◽

1972 ◽

Vol 37 (3) ◽

pp. 572-578 ◽

Cited By ~ 5

Author(s):

Raphael M. Robinson

Keyword(s):

Universal Quantifier ◽

Natural Numbers ◽

Recursively Enumerable Set ◽

Minor Error ◽

Integer Coefficients ◽

Recursively Enumerable ◽

Recursively Enumerable Sets ◽

Image Position ◽

A Minor

A set D of natural numbers is called Diophantine if it can be defined in the formwhere P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the formwhere P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.

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Arithmetical representation of recursively enumerable sets

Journal of Symbolic Logic ◽

10.2307/2268755 ◽

1956 ◽

Vol 21 (2) ◽

pp. 162-186 ◽

Cited By ~ 18

Author(s):

Raphael M. Robinson

Keyword(s):

Recursive Function ◽

Natural Numbers ◽

General Recursive Function ◽

Recursively Enumerable Set ◽

Integer Coefficients ◽

Primitive Recursive Function ◽

Recursively Enumerable ◽

Image Position ◽

The Right ◽

Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

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On the Power of Small Size Insertion P Systems

International Journal of Computers Communications & Control ◽

10.15837/ijccc.2011.2.2175 ◽

2011 ◽

Vol 6 (2) ◽

pp. 266 ◽

Cited By ~ 2

Author(s):

Alexander Krassovitskiy

Keyword(s):

P Systems ◽

Computational Power ◽

Recursively Enumerable Language ◽

Recursively Enumerable ◽

Enumerable Language ◽

The Family ◽

Context Free

In this article we investigate insertion systems of small size in the framework of P systems. We consider P systems with insertion rules having one symbol context and we show that they have the computational power of context-free matrix grammars. If contexts of length two are permitted, then any recursively enumerable language can be generated. In both cases a squeezing mechanism, an inverse morphism, and a weak coding are applied to the output of the corresponding P systems. We also show that if no membranes are used then corresponding family is equal to the family of context-free languages.

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CD GRAMMAR SYSTEMS WITH REGULAR START CONDITIONS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054108005942 ◽

2008 ◽

Vol 19 (04) ◽

pp. 767-779

Author(s):

RUDOLF FREUND ◽

MARION OSWALD

Keyword(s):

Finite Index ◽

Hybrid Mode ◽

Computational Completeness ◽

Recursively Enumerable ◽

The Family ◽

Grammar Systems ◽

Regular Sets ◽

Classical Derivation ◽

Recursively Enumerable Languages

We consider cooperating distributed grammar systems with the components working in different derivation modes as well as with regular sets as additional start conditions for the components. With the classical derivation modes ≤ k and = k as well as with the internally hybrid mode (≥ ℓ∧ ≤ k) we obtain a characterization of the family of recursively enumerable languages even with only one component, with the derivation modes *, t, and ≥ k as well as with the internally hybrid mode (t∧ ≥ k) two components working in the same mode and only one common regular set for both components yield computational completeness. For the internally hybrid modes (t∧ ≤ k) and (t∧ = k) we only obtain languages of finite index, but combining one component working in one of these modes (t∧ ≤ k) and (t∧ = k) with a component working in one of the modes * and ≥ k we again obtain a characterization of the family of recursively enumerable languages.

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