Plain Kolmogorov Complexity Research Articles

Relating and Contrasting Plain and Prefix Kolmogorov Complexity

In (Bauwens and Shen, J. Symb. Log. 79(2), 620---632, 2013) a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. We argue that the proof technique is useful to simplify existing proofs and to solve open questions. We present a short proof of a result due to Robert Solovay that relates plain and prefix complexity: K(x)=C(x)+CC(x)+O(CCC(x))C(x)=K(x)?KK(x)+O(KKK(x)),$\begin {array}{ccc} K(x) &=& C(x) + CC(x) + O(CCC(x)) C(x) &=& K(x) - KK(x) + O(KKK(x)), \end {array}$ (here CC(x) denotes C(C(x)), etc.). We show that there exist ? such that liminfC(?1??n)?C(n)$\liminf C(\omega _{1}{\dots } \omega _{n}) - C(n)$ is infinite and liminfK(?1??n)?K(n)$\liminf K(\omega _{1}{\dots } \omega _{n}) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals, answering Question 1 in Barmpalias (Bull. Symb. Log. 19(3), 2013). We answer a question from Bienvenu (Laurent Bienvenu, personal communication 2011): some 2-random sequence has a family of initial segments with bounded plain deficiency (i.e. |x|?C(x) is bounded) and unbounded prefix deficiency (i.e. |x|+K(|x|)?K(x) is unbounded). Finally, we show that there exists no monotone relation between probability and expectation bounded randomness deficiency, answering Question 1 in Bienvenu et al. (Proceedings of the Steklov Institute of Mathematics, 274(1), 34---89, 2011).

Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs

Miller (J Symb Log 69(3):907---913, 2004, http://projecteuclid.org/euclid.jsl/1096901774) and independently Nies et al. (J Symb Log 70(2):515---535, 2005) gave a complexity characterization of 2-random sequences in terms of plain Kolmogorov complexity $${C(\cdot)}$$C(·) : they are sequences that have infinitely many initial segments with O(1)-maximal plain complexity (among the strings of the same length). Later Miller (Notre Dame J Form Log 50(4):381---391, 2009) showed that prefix complexity $${K(\cdot)}$$K(·) can also be used in a similar way: a sequence is 2-random if and only if it has infinitely many initial segments with O(1)-maximal prefix complexity (which is n + K(n) for strings of length n). The known proofs of these results are quite involved; in this paper we provide elementary proofs. Miller (J Symb Log 69(3):907---913, 2004, http://projecteuclid.org/euclid.jsl/1096901774) also gave a quantitative version of the first result: the $${\mathbf{0}'}$$0?-randomness deficiency of a sequence $${\omega}$$? equals lim inf$${_n [n - C(\omega_1\dots\omega_n)] + O(1)}$$n[n-C(?1??n)]+O(1). Our simplified proof can also be used to prove this. We show (and this seems to be a new result) that a similar quantitative result is also true for prefix complexity: $${\mathbf{0}'}$$0?-randomness deficiency equals lim inf$${_n [n + KP(n) - K(\omega_1\dots\omega_n)]+ O(1)}$$n[n+KP(n)-K(?1??n)]+O(1).

Universality of quantum Turing machines with deterministic control

A simple notion of quantum Turing machine with deterministic, classical control is proposed and shown to be powerful enough to compute any unitary transformation that is computable by a finitely generated quantum circuit. An efficient universal machine with the s-m-n property is presented. The BQP class is recovered. A robust notion of plain Kolmogorov complexity of quantum states is proposed and compared with those previously reported in the literature.

Initial segment complexities of randomness notions

Schnorr famously proved that Martin-Löf-randomness of a sequence A can be characterised via the complexity of Aʼs initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that a set is 2-random (that is, Martin-Löf random relative to the halting problem K) iff there is no function f such that for all m and all n>f(m) it holds that C(A(0)A(1)…A(n))⩽n−m; before the proof of this equivalence the notion defined via the latter condition was known as Kolmogorov random.In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strong randomness (also known as weak 2-randomness), Kurtz randomness relative to K, Martin-Löf randomness of PA-incomplete sets, and strong Kurtz randomness; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is Martin-Löf random and PA-incomplete iff there is no A-recursive function f such that for all m and all n>f(m) it holds that C(A(0)A(1)…A(n))⩽n−m. The characterisation for strong randomness relates to functions which are the concatenation of an A-recursive function executed after a K-recursive function; this solves an open problem of Nies.In addition to this, characterisations of a similar style are also given for Demuth randomness, weak Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines.

Time-Bounded Kolmogorov Complexity and Solovay Functions

A Solovay function is an upper bound g for prefix-free Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n. We obtain natural examples of computable Solovay functions by showing that for some constant c 0 and all computable functions t such that c 0 n?t(n), the time-bounded version K t of K is a Solovay function. By unifying results of Bienvenu and Downey and of Miller, we show that a right-computable upper bound g of K is a Solovay function if and only if ? g =?2?g(n) is Martin-Lof random. We obtain as a corollary that the Martin-Lof randomness of the various variants of Chaitin's ? extends to the time-bounded case in so far as $\Omega _{ \textnormal{K}^{t}}$ is Martin-Lof random for any t as above. As a step in the direction of a characterization of K-triviality in terms of jump-traceability, we demonstrate that a set A is K-trivial if and only if A is O(g(n)?K(n))-jump traceable for all computable Solovay functions g. Furthermore, this equivalence remains true when the universal quantification over all computable Solovay functions in the second statement is restricted either to all functions of the form K t for some function t as above or to a single function K t of this form. Finally, we investigate into the plain Kolmogorov complexity C and its time-bounded variant C t of initial segments of computably enumerable sets. Our main theorem here asserts that every high c.e. Turing degree contains a c.e. set B such that for any computable function t there is a constant c t >0 such that for all m it holds that C t (B?m)?c t ?m, whereas for any nonhigh c.e. set A there is a computable time bound t and a constant c such that for infinitely many m it holds that C t (A?m)?logm+c. By similar methods it can be shown that any high degree contains a set B such that C t (B?m)?+ m/4. The constructed sets B have low unbounded but high time-bounded Kolmogorov complexity, and accordingly we obtain an alternative proof of the result due to Juedes et al. (Theor. Comput. Sci. 132(1---2):37---70, 1994) that every high degree contains a strongly deep set.

Axiomatizing Kolmogorov Complexity

We revisit the axiomatization of Kolmogorov complexity given by Alexander Shen, currently available only in Russian language. We derive an axiomatization for conditional plain Kolmogorov complexity. Next we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). In addition we prove that the analogue of Shen's axiomatic system fails to characterize prefix-free Kolmogorov complexity.

The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

We call A weakly low for K if there is a c such that K A ( σ ) ≥ K ( σ ) − c for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K if and only if Chaitin's Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.

Limit Complexities Revisited

The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from Vereshchagin (Theor. Comput. Sci. 271(1–2):59–67, 2002) saying that lim sup n C(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′(x), the plain Kolmogorov complexity with 0′-oracle. Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree and measure of effectively open sets, and also to improve results of Muchnik (Theory Probab. Appl. 32:513–514, 1987) about limit frequencies. As a by-product, we get a criterion of 0′ Martin-Löf randomness (called also 2-randomness) proved in Miller (J. Symb. Log. 69(2):555-584, 2004): a sequence ω is 2-random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y)≥|y|−c. (In the 1960ies this property was suggested in Kolmogorov, IEEE Trans. Inf. Theory IT-14(5):662–664, 1968, as one of possible randomness definitions; its equivalence to 2-randomness was shown in Miller, J. Symb. Log. 69(2):555-584, 2004.) Miller (J. Symb. Log. 69(2):555-584, 2004) and Nies et al. (J. Symb. Log. 70(2):515–535, 2005) proved another 2-randomness criterion: ω is 2-random if and only if C(x)≥|x|−c for some c and infinitely many prefixes x of ω. We show that the low-basis theorem can be used to get alternative proofs of our results on Kolmogorov complexity and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define $X\leq _K Y$ to mean that $(\forall n)\; K(X\upharpoonright n)\leq K(Y\upharpoonright n)+O(1)$. The equivalence classes under this relation are the $K$-degrees. We prove that if $X\oplus Y$ is $1$-random, then $X$ and $Y$ have no upper bound in the $K$-degrees (hence, no join). We also prove that $n$-randomness is closed upward in the $K$-degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the $\textit {vL}$-degrees. Unlike the $K$-degrees, many basic properties of the $\textit {vL}$-degrees are easy to prove. We show that $X\leq _K Y$ implies $X\leq _{\textit {vL}} Y$, so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for $\leq _C$, the analogue of $\leq _K$ for plain Kolmogorov complexity. Two other interesting results are included. First, we prove that for any $Z\in 2^\omega$, a $1$-random real computable from a $1$-$Z$-random real is automatically $1$-$Z$-random. Second, we give a plain Kolmogorov complexity characterization of $1$-randomness. This characterization is related to our proof that $X\leq _C Y$ implies $X\leq _{\textit {vL}} Y$.

Every 2-random real is Kolmogorov random

Abstract.We study reals with infinitely many incompressible prefixes. Call A ∈ 2ωKolmogorov random if . where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf. Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse—proved by Nies. Stephan and Terwijn [11]—this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.