Hyperbolic geometry is an essential part of theoretical astrophysics and cosmology. Besides specialists of these domains, many specialists of new domains start to show a growing interest
both to hyperbolic geometry and to cellular automata. This is especially the case in biology and computer science.

This book gives the reader a deep and efficient introduction to an algorithmic approach to hyperbolic geometry. It focuses the attention on the possibilities to obtain in this frame the power of computing everything a computer can compute, that is to say: universality.

The minimal ways to get universality are investigated in a large family of tilings of the hyperbolic plane. In several cases the best results are obtained.In all cases, the results are close to the theoretical best values. This gives rise to fantastic illustrations: the results are jewels in all meanings of the word.

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Maurice MARGENSTERN is professor emeritus at the University of Lorraine, he is a member of LITA, the research unit of computer science in the campus of Metz of this university. Professor Margenstern is amongst top world experts in theory of computation, mathematical machines and geometry. He is a pioneer
in cellular automata in hyperbolic spaces.

Hyperbolic geometry is an essential part of theoretical astrophysics and cosmology. Besides specialists of these domains, many specialists of new domains start to show a growing interest both to hyperbolic geometry and to cellular automata. This is especially the case in biology and computer science. This book gives the reader a deep and efficient introduction to an algorithmic approach to hyperbolic geometry. It focuses the attention on the possibilities to obtain in this frame the power of computing everything a computer can compute, that is to say: universality. The minimal ways to get universality are investigated in a large family of tilings of the hyperbolic plane. In several cases the best results are obtained.In all cases, the results are close to the theoretical best values. This gives rise to fantastic illustrations: the results are jewels in all meanings of the word. ------------------------ Maurice MARGENSTERN is professor emeritus at the University of Lorraine, he is a member of LITA, the research unit of computer science in the campus of Metz of this university. Professor Margenstern is amongst top world experts in theory of computation, mathematical machines and geometry. He is a pioneer in cellular automata in hyperbolic spaces.

This book constitutes the refereed proceedings of the 5th International Conference on Machines, Computations, and Universality, MCU 2007, held in Orleans, France, September 10-13, 2007.

The 18 revised full papers presented together with 9 invited papers were carefully reviewed and selected. The topics include Turing machines, register machines, word processing, cellular automata, tiling of the plane, neural networks, molecular computations, BSS machines, infinite cellular automata, real machines, and quantum computing.

In this volume, the reader will ?rst?nd the invited talks givenatthe conference. Then, in a second part, he/she will ?nd the contributions which were presented attheconferenceafterselection.Inbothcases, papersaregiveninthealphabetic order of the authors. MCU 2004 was the fourth edition of the conference in theoretical computer science, Machines, Computations and Universality, formerly, Machines etcalculs universels. The ?rst and the second editions, MCU 1995 and MCU 1998, were organized by Maurice Margenstern, respectively in Paris and in Metz (France). The third edition, MCU 2001, was the ?rst one to be organized outside France and it was held in Chi, sin? au (Moldova). Its co-organizers were Maurice Marg- sternandYuriiRogozhin.TheproceedingsofMCU2001werethe?rstto appear in Lecture Notes in Computer Science, see LNCS 2055. From its very beginning, the MCU conference has been an international s- enti?c event. For the fourth edition, Saint Petersburg was chosen to hold the meeting. The success of the meeting con?rmed that the choice was appropriate. MCU 2004 also aimed at high scienti?c standards. We hope that this v- ume will convince the reader that this tradition of the previous conferences was also upheld by this one. Cellular automata and molecular computing are well represented in this volume. And this is the case for quantum computing, formal languages and the theory of automata too. MCU 2004 also did not fail its tra- tion to provide our community with important results on Turing machines. Also a new feature of the Saint Petersburg edition was the contributions on analog models and the presence of unconventional model

In the ?rst part of the present volume of LNCS, the reader will ?nd the invited talks given at the MCU 2001 conference. In the second part, he/she will ?nd the contributions that were presented at the conference after selection. In both cases, papers are arranged in the alphabetical order of the authors. MCU 2001 is the third conference in theoretical computer science, Machines, computations and universality, formerly, Machines et calculs universels. Both previous conferences, MCU 95 and MCU 98, were organized by Maurice M- genstern in Paris and in Metz (France), respectively. From the very beginning, MCU conferences have been an international sci- ti?c event. For the third conference, in order to stress that aspect, it was decided to hold it outside France. Moldova was chosen thanks to the close cooperation between the present chairmen of MCU 2001. MCU 2001 also aims at high scienti?c standards. We hope that the present volume will convince the reader that the tradition of previous conferences have been upheld by this one. Cellular automata and molecular computing are well represented in this volume. And this is also the case for quantum computing, f- mal languages, and the theory of automata. MCU 2001 does not fail its tradition of providing our community with important results on Turing machines."