Finite Versus Infinite Contributions to an Eternal Dilemma

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.

Preface.- Tribute to Professor Solomon Marcus.- Smoothing Data: When Finite and Infinite Dimensional Help Each Other.- Rearrangements of Real Functions Derived from the Combinatorics of Young Tableaux.- A Possibilistic Distance for Sequences of Equal and Unequal Length.- Solving Problems with Finite Test Sets.- State Complexity of Regular Languages: Finite Versus Infinite.- A Century of Controversy over the Foundations of Mathematics.- Finite Versus Infinite in Singularity Dynamics.- An Interesting Serendipitous Real Number.- Algebraic Representations of Regular Array Languages.- Rough Set Processing of Vague Information Using Fuzzy Similarity Relations.- Relativized Code Concepts and Multi-Tube DNA Dictionaries.- Uniformly Scattered Factors.- Splicing Normalization and Regularity.- Infinitely Many Infinities.- Finite and Infinite in Geography.- Ultrafilters, Dictators, and Gods.- On the Behaviour of Semiautomata.- Infinity - An Outline of Conceptions in Mythology, Cosmology and Natural Philosophy.- Finite Versus Infinite Neural Computation.- On Information-Theoretical Aspects of Relational Databases.- A Version of ? for Which ZFC Cannot Predict a Single Bit.- On the Power of Reading the Whole Infinite Input Tape.- Finite and Infinite in Logic.