Finite Versus Infinite

1. Front Matter

   Pages i-x

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2. Smoothing Data: When Finite and Infinite Dimensional Help Each Other

   • Umberto Amato, Dan Tudor Vuza

   Pages 1-16

3. Rearrangements of Real Functions Derived from the Combinatorics of Young Tableaux

   • Hari Bercovici

   Pages 17-26

4. A Possibilistic Distance for Sequences of Equal and Unequal Length

   • Massimo Borelli, Andrea Sgarro

   Pages 27-38

5. Solving Problems with Finite Test Sets

   • Cristian S. Calude, Helmut Jürgensen, Shane Legg

   Pages 39-52

6. State Complexity of Regular Languages: Finite versus Infinite

   • Cezar Câmpeanu, Kai Salomaa, Sheng Yu

   Pages 53-73

7. A Century of Controversy over the Foundations of Mathematics

   • Gregory J. Chaitin

   Pages 75-100

8. Finite Versus Infinite in Singularity Dynamics

   • Florin Diacu

   Pages 101-117

9. An Interesting Serendipitous Real Number

   • John Ewing, Ciprian Foias

   Pages 119-126

10. Algebraic Representations of Regular Array Languages

    • Rudolf Freund, Alexandru Mateescu, Arto Salomaa

    Pages 127-148

11. Rough Set Processing of Vague Information Using Fuzzy Similarity Relations

    • Salvatore Greco, Benedetto Matarazzo, Roman Slowinski

    Pages 149-173

12. Relativized Code Concepts and Multi-Tube DNA Dictionaries

    • Tom Head

    Pages 175-186

13. Uniformly Scattered Factors

    • Lucian Ilie, Ion Petre, Grzegorz Rozenberg

    Pages 187-198

14. Splicing Normalization and Regularity

    • Vincenzo Manca

    Pages 199-215

15. Infinitely Many Infinities

    • Carlos Martín-Vide

    Pages 217-230

16. Finite and Infinite in Geography

    • Georges Nicolas

    Pages 231-237

17. Ultrafilters, Dictators, and Gods

    • Piergiorgio Odifreddi

    Pages 239-245

18. On the Behaviour of Semiautomata

    • Sergiu Rudeanu

    Pages 247-256

19. Infinity — An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy

    • Jouko Seppänen

    Pages 257-283

20. Finite Versus Infinite Neural Computation

    • Hava T. Siegelmann

    Pages 285-299

The finite - infinite interplay is central in human thinking, from ancient philosophers and mathematicians (Zeno, Pythagoras), to modern mathe­ matics (Cantor, Hilbert) and computer science (Turing, Godel). Recent developments in mathematics and computer science suggest a) radically new answers to classical questions (e. g. , does infinity exist?, where does infinity come from?, how to reconcile the finiteness of the human brain with the infinity of ideas it produces?), b) new questions of debate (e. g. , what is the role played by randomness?, are computers capable of handling the infinity through unconventional media of computation?, how can one approximate efficiently the finite by the infinite and, conversely, the infinite by finite?). Distinguished authors from around the world, many of them architects of the mathematics and computer science for the new century, contribute to the volume. Papers are as varied as Professor Marcus' activity, to whom this volume is dedicated. They range from real analysis to DNA com­ puting, from linguistics to logic, from combinatorics on words to symbolic dynamics, from automata theory to geography, and so on, plus an incursion into the old history of conceptions about infinity and a list of philosophical "open problems". They are mainly mathematical and theoretical computer science texts, but not all of them are purely mathematical.

• Algebra
• Automata Theory
• Cantor
• Finite
• Mathematical Logic
• Neural Computing
• Problem-solving
• combinatorics
• complexity
• databases
• function
• logic
• mathematics

• Department of Computer Science, University of Auckland, Auckland, New Zealand

  Cristian Calude

• Institute of Mathematics, Romanian Academy of Sciences, Bucharest, Romania

  Gheorghe Păun

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