Enumerations of the Kolmogorov function

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is an k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f . If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity: Email: [email protected]. Deptartment of Computer and Information Sciences, Temple University, 1805 North Broad Street, Philadelphia PA 19122, USA. Research performed in part at NEC and the Institute for Advanced Study. Supported in part by a State of New Jersey grant and by the National Science Foundation under grants CCR-0049019 and CCR-9877150. Email: [email protected]. CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. Partially supported by the EU through the 5th framework program FET. Email: [email protected]. Department of Computer Science, University of Massachusetts Boston, Boston, MA 02125, USA. Email: [email protected]. Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA. Research performed in part at NEC Research Institute. Email: [email protected]. Institut für Informatik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany. Email: [email protected]. Computer Science Department, UTEP, El Paso, TX 79968, USA. Email: [email protected]. Institute of New Techologies, Nizhnyaya Radishevskaya, 10, Moscow, 109004, Russia. The work was partially supported by Russian Foundation for Basic Research (grants N 04-01-00427, N 02-01-22001) and Council on Grants for Scientific Schools. Email: [email protected]. National ICT Australia LTD, Sydney Research Laboratory at Kensington, The University of New South Wales, Sydney NSW 2052, Australia. Research supported by the Deutsche Forschungsgemeinschaft (DFG), Heisenberg grant Ste 967/1-1 while previously working at the Universität Heidelberg. Email: [email protected]. Institute for Language Logic and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.