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International Conference on Machines, Computations, and Universality

MCU 2018: Machines, Computations, and Universality pp 1–33Cite as

Control Mechanisms for Array Grammars on Cayley Grids

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10881))

Abstract

In this paper, the computational power of several control mechanisms for specific variants of (sequential, isometric) array grammars generating arrays on Cayley grids of finitely presented groups is investigated. Using \(\#\)-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules or activation and blocking of rules the same computational power is obtained as when using arbitrary array productions.

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Acknowledgements

The current paper has been inspired by the discussions and the exchange of ideas with many of my friends and co-authors, yet I can only mention some of them being most influential for my work: I already became interested in the concepts of regulated rewriting during my studies even a decade before the book of Jürgen Dassow and Gheorghe Păun Regulated Rewriting in Formal Language Theory appeared in 1989, see [8], and collected the main results on control mechanisms for the string case in a comprehensive way.

Yet since the beginning, I have always tried to also apply these control mechanisms to grammars dealing with other objects as graphs, for instance, see [19], or d-dimensional arrays, for example, see [24], the first paper with Gheorghe, as well as [16]. Already more than twenty-five years ago, together with Jürgen I started the investigation of a general framework, yet it took a long time before the initial ideas condensed in A General Framework for Regulated Rewriting Based on the Applicability of Rules, see [21].

Array grammars have also been an interesting topic in picture generation and acceptance, and the first paper together with Jürgen and Gheorghe on Cooperating Array Grammar Systems was exploring some possibilities of how to use controlled array grammars for picture generation. Cooperating systems (of grammars) are another interesting way of controlling the application of rules by allowing different sets of rules to be applied according to a given strategy as, for example, to work as long as possible or a certain number of steps on the underlying object (an array in this case), also see [17].

Variants of array grammars were investigated to be applied for character recognition, for example, see [9], which paper started a very fruitful cooperation with Henning Fernau. The topic of character recognition was continued together with Markus Holzer, see [11,12,13]. Recently the cooperation with Henning (and his team in Trier) has been resumed with several papers dealing with control mechanisms for several variants of array grammars, for example, see [2, 14, 15].

With the concept of membrane systems introduced by Gheorghe Păun, see [27, 28], the Handbook of Membrane Computing, a new model emerged, controlling the applicability of rules by using the hierarchical membrane structure where together with the application of a rule the current array can be sent to the outer or an inner membrane, which makes other sets of rules applicable, for example, see [18], the first paper on that topic for the sequential derivation mode with arrays as underlying objects, as well as [14, 20], the contributions to MCU 2013.

From the beginning of this century, Marion Oswald, my colleague and friend working with me at the TU Wien for a very long period, had become my favorite co-author, especially in the area of P systems, but she also co-authored several papers on regulated rewriting, especially [21] on A General Framework for Regulated Rewriting Based on the Applicability of Rules, and on variants of array grammars, especially the main papers introducing Array Grammars and Automata on Cayley Grids, see [22, 23].

More recently, my friends from Moldova Artiom Alhazov and Sergiu Ivanov have become the motivating force to continue the research on new control mechanisms, not only in the area of P systems, but also with respect to new control mechanisms, especially also for array grammars, for instance, see [2, 14, 20]. The new concept of activation and blocking of rules in its final form has been developed during the Brainstorming Week on Membrane Computing this year in Sevilla mainly together with Sergiu, and different facets of this new control mechanism are presented in several papers, see [3,4,5], including this paper.

Both the list of references and the list of colleagues who contributed to my research on control mechanisms and/or on array grammars are far from being complete. I want to express my deep respect and my gratitude to all my colleagues and friends who helped me to develop new ideas and concepts, some of them presented in this paper.

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  1. Faculty of Informatics, TU Wien, Favoritenstraße 9–11, 1040, Vienna, Austria

    Rudolf Freund

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  1. Rudolf Freund
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  1. Université d'Orléans, Orleans, France

    Jérôme Durand-Lose

  2. Université Paris Est, Creteil, France

    Sergey Verlan

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Freund, R. (2018). Control Mechanisms for Array Grammars on Cayley Grids. In: Durand-Lose, J., Verlan, S. (eds) Machines, Computations, and Universality. MCU 2018. Lecture Notes in Computer Science(), vol 10881. Springer, Cham. https://doi.org/10.1007/978-3-319-92402-1_1

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  • DOI: https://doi.org/10.1007/978-3-319-92402-1_1

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