## Journal of Symbolic Logic

### Enumerations of the Kolmogorov function

#### Abstract

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 a 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:

• For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n.
• For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem.
• There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A.
The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity.

Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g:

• For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f.
• For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f.
Furthermore, we deal with the resource-bounded case and give characterizations for the class S₂p introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP.
• S₂p is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g.
• PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity.
• EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g.
Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P=NP.

#### Article information

Source
J. Symbolic Logic Volume 71, Issue 2 (2006), 501-528.

Dates
First available in Project Euclid: 2 May 2006

http://projecteuclid.org/euclid.jsl/1146620156

Digital Object Identifier
doi:10.2178/jsl/1146620156

Mathematical Reviews number (MathSciNet)
MR2225891

Zentralblatt MATH identifier
05043304

#### Citation

Beigel, Richard; Buhrman, Harry; Fejer, Peter; Fortnow, Lance; Grabowski, Piotr; Longpré, Luc; Muchnik, Andrej; Stephan, Frank; Torenvliet, Leen. Enumerations of the Kolmogorov function. J. Symbolic Logic 71 (2006), no. 2, 501--528. doi:10.2178/jsl/1146620156. http://projecteuclid.org/euclid.jsl/1146620156.

#### References

• Andris Ambainis, Harry Buhrman, William Gasarch, BelaKalyanasundaram, and Leen Torenvliet, The communication complexity of enumeration, elimination and selection, Proceedings 15th IEEE Conference on Computational Complexity,2000, pp. 44--53.
• Amihood Amir, Richard Beigel, and William Gasarch, Some connections between bounded query classes and nonuniform complexity, Proceedings of the 5th Annual Conference on Structure in Complexity Theory,1990, pp. 232--243.
• Janis M. Bārzdi\c nš, Complexity of programs to determine whether natural numbers not greater than $n$ belong to a recursively enumerable set, Soviet Mathematics Doklady, vol. 9 (1968), pp. 1251--1254.
• Richard Beigel, Query-limited reducibilities, Ph.D. thesis, Department of Computer Science, Stanford University,1987.
• Richard Beigel, Harry Buhrman, Peter A. Fejer, Lance Fortnow, Piotr Grabowski, Luc Longpré, Andrej A. Muchnik, Frank Stephan, and Leen Torenvliet, Enumerations of the Kolmogorov function, Electronic Colloquium on Computational Complexity (ECCC), (015),2004.
• Richard Beigel, William Gasarch, John Gill, and Jim Owings Jr., Terse, superterse and verbose sets, Information and Computation, vol. 103 (1993), pp. 68--85.
• Harry Buhrman and Leen Torenvliet, Randomness is hard, SIAM Journal on Computing, vol. 30 (2000), no. 5, pp. 1485--1501.
• Jin-Yi Cai and Lane Hemachandra, Enumerative counting is hard, Information and Computation, vol. 82 (1989), no. 1, pp. 34--44.
• --------, A note on enumerative counting, Information Processing Letters, vol. 38 (1991), no. 4, pp. 215--219.
• Ran Canetti, More on BPP and the polynomial-time hierarchy, Information Processing Letters, vol. 57 (1996), no. 5, pp. 237--241.
• Gregory Chaitin, On the length of programs for computing finite binary sequences, Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 547--569.
• --------, Information-theoretical characterizations of recursive infinite strings, Theoretical Computer Science, vol. 2 (1976), pp. 45--48.
• Rod Downey, Denis Hirschfeldt, André Nies, and Frank Stephan, Trivial reals, Proceedings of the 7th and 8th Asian Logic Conferences (R. Downey et al., editors), World Scientific, River Edge,2003, pp. 103--131.
• Richard Friedberg and Hartley Rogers, Reducibilities and completeness for sets of integers, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 117--125.
• William Gasarch and Frank Stephan, A techniques oriented survey of bounded queries, Models and computability: Invited papers from Logic Colloquium 1997 -- European Meeting of the Association for Symbolic Logic, Leeds, July 1997 (Cooper and Truss, editors), Cambridge University Press,1999, pp. 117--156.
• Juris Hartmanis, Generalized Kolmogorov complexity and the structure of feasible computations, Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science,1983, pp. 439--445.
• Carl G. Jockusch Jr. and Richard Shore, Pseudo-jump operators I: the r.e. case., Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599--609.
• Carl G. Jockusch Jr. and Robert I. Soare, $\Pi^0_1$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
• Andrei Kolmogorov, Three approaches for defining the concept of information quantity, Problems of Information Transmission, vol. 1 (1965), pp. 1--7.
• Antonín Kučera and Sebastiaan A. Terwijn, Lowness for the class of random sets, Journal of Symbolic Logic, (1999), no. 64, pp. 1396--1402.
• Martin Kummer, On the complexity of random strings, Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science (C. Puech and R. Reischuk, editors), Lecture Notes in Computer Science, vol. 1046, Springer,1996, pp. 25--36.
• Martin Kummer and Frank Stephan, Effective search problems, Mathematical Logic Quarterly, vol. 40 (1994), pp. 224--236.
• Ming Li and Paul M.B. Vitányi, An introduction to Kolmogorov complexity and its applications, second ed., Graduate Texts in Computer Science, Springer-Verlag,1997.
• Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan, Algebraic methods for interactive proof systems, Journal of the Association for Computing Machinery, vol. 39 (1992), no. 4, pp. 859--868.
• André Nies, Lowness properties of reals and randomness, Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274--305.