Resumen de Kolmogorov randomnes and its applications to structural complexity theory
THE THEORY OF KOLMOGOROV COMPLEXITY HAS PROVED TO BE VERY USEFUL IN THEORETICAL COMPUTER SCIENCE AND SPECIALLY IN COMPLEXITY THEORY, ONE OF THE MAIN REASONS IS THAT IT PROVIDES A PRECISE DEFINITION OF "RAMDOM STRING" AND ENSURES THE EXISTENCE AND ABUNDANCE OF THESE STRINGS.
THIS WORK DESCRIBES SEVERAL APPLICATIONS OF KOLMOGOROV RANDOMNESS TO STRUCTURAL COMPLEXITY THEORY. WE SHOW HOW RANDOM STRINGS CAN BE USED TO POINT OUT LIMITATIONS IN THE COMPUTATIONAL POWER OF A MACHINE TYPE. IN SOME CASES, OUR RESULTS CAN BE OBTAINED BY OTHER METHODS, BUT ALMOST INVARIABLY OUR PROOF ARE SHORTER AND CLEANER. IN OTHER CASES, ONLY THE SIMPLIFICATIONS INTRODUCED BY THE USE OF KOLMOGOROV RANDOMNESS ALLOWS US TO SOLVE OPEN PROBLEMS THAT COULD NOT BE SOLVED BY MORE CONVENTIONAL TECHNIQUES.
IN CHAPTER 3 WE PRESENT TECHNIQUES TO SHOW SEPARATIONS BETWEEN RELATIVIZED COMPLEXITY CLASSES.
SOMOE OF THE IDEAS WE EXPLOIT ARE: ADAPTATIVE VR. NON ADAPTATIVE COMPUTATION; NONDETERMISM VS.
CO-NONDETERMINISM; THE USE OF "PASWORDS" TO PROTECT INFORMATION HIDDEN IN THE ORACLE; AND THE LIMITATION OF RESOURCE-BOUNDED MACHINES TO EXTRACT INFORMATION FROM RANDOM ORACLES. WE USE AS EXAMPLES THE SPACE-BOUNDED CLASSES PSPACE AND NPSPACE AND THE TIME-BOUNDED CLASS EXPTIME, AND SHOW THAT EVERY NONCONTRADICTORY INCLUSION RELATION BETWEEN THESE TREE CLASSES IS TRUE RELATIVE TO SOME ORACLE.
CHAPTER 4 PRESENTS VARIOUS APPLICATIONS TO COMPUTATIONAL LEARNING, MORE PRECISELY, TO THE MODEL OF EXACT LEARNING VIA QUERIES. BOTH DETERMINISTIC AND RANDOMIZED LEARNERS ARE DISCUSSED. USING KOLMOGOROV -RANDOMNESS ARGUMENTS, WE THEN SHOW SOME LOWER BOUNDS ON THE COMPLEXITY OF THIS COMPUTATIONAL PROBLEM WHEN VARIOUS TYPES OF QUERIES ARE ALLOWED. THIS YIELSD ALTERNATIVE PROOFS OF SOME NON-LEARNABILITY RESULTS.
IN CHAPTER 5 WE GENERALIZA THE PROBLEM OF LEARNING TO THAT THE FINDING SHORT DESCRIPTIONS FOR ARBITRARY SETS IN P/POLY. WE FIND VERY TIGH LOWER AND UPPER BOUNDS FOR THE COMPL
