Generalized Gandy-Pun-Rozenberg machines for tile systems and - - PowerPoint PPT Presentation

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generalized gandy p un rozenberg machines for tile

Generalized Gandy-Pun-Rozenberg machines for tile systems and - - PowerPoint PPT Presentation

Mar 18, 2023 159 likes •267 views

Generalized Gandy-Pun-Rozenberg machines for tile systems and cellular automata Adam Obtuowicz Institute of Mathematics, Polish Academy of Sciences Warsaw, Poland Adam Obtuowicz, IMPAN, Warsaw, Poland Generalized Gandy-Pun-Rozenberg

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Generalized Gandy-Păun-Rozenberg machines for tile systems and cellular automata

Adam Obtułowicz Institute of Mathematics, Polish Academy of Sciences Warsaw, Poland

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Membrane computing

The paper proposes an extension of membrane computing towards modelling those systems whose underlying topology evolves in a more complicated way than by membrane division and membrane creation.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Generalized G–P–R machines

We introduce a concept of a generalized Gandy–Păun–Rozenberg machine, briefly called a generalized G–P–R machine, which is aimed to be applied for modelling various systems of multidimensional tile-like compartments (cells) with common or overlapping parts (tile faces) of compartment boundaries by graph rewriting.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Generalized G–P–R machines

The above systems of multidimensional tile-like compartments comprise the underlying tile systems of cellular automata (see anywhere for cellular automata on multidimensional grids), of the DNA based self-assembly systems, of the general self-assembly systems for certain purposes, the tile systems appearing in tile logic, and in geometrical or topological programming.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Explanatory example

Configuration C of hexagonal tiles is transformed by simultaneous application of two graph rewriting rules R1, R2.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Explanatory example

C: x b a x z R1: x

ld tile to be

relabelled x/a b d new tile to be introduced R2: z z/c

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Explanatory example

The result of transformation a b a a c b b d common face to keep the resulting configuration in a hexagonal grid

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Explanatory example

The graph rewriting rules R1, R2 should be completed by an auxiliary gluing rule: x z x/a z/c b d common face of new tiles to respect common face of new tiles.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Definition

A generalized G–P–R machine is a system of transformation (processing) of finite labelled graphs by simultaneous application of graph rewriting rules with respect to auxiliary gluing rules, and its mathematical definition is given in categorical terms of diagrams and their colimits in categories

f labelled graphs.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines

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Looking forward

Looking forward, since self-assembly is an important attribute

f life, the generalized G–P–R machines, being systems

equivalent to Turing machines (via their representation by Gandy machines) and aimed to model self-assembly systems like, may serve for modelling computable approximations of life, whenever it is not computable.

Adam Obtułowicz, IMPAN, Warsaw, Poland Generalized Gandy-Păun-Rozenberg machines