Limit complexities revisited

2013

The famous Gödel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T . Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us to prove new interesting theorems. This result (cf. [She06]) answers a question recently asked by Lipton [LR11]. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE). This result partially answers the question asked in [She06]. We then study the axiomatic power of the statements of type "the Kolmogorov complexity of x exceeds n" (where x is some string, and n is some integer) in general. They are Π1 (universally quantified) statements of Peano arithmetic. We show (Theorem 5) that by adding all true statements of this type, we obtain a theory that proves all true Π1-statements, and also provide a more detailed classification. In particular, as Theorem 7 shows, to derive all true Π1-statements it is enough to add one statement of this type for each n (or even for infinitely many n) if strings are chosen in a special way. On the other hand, one may add statements of this type for most x of length n (for every n) and still obtain a weak theory (Theorem 10). We also study other logical questions related to "random axioms" (hierarchy with respect to n, Theorem 8 in Section 3.3, independence in Section 3.6, etc.). Finally, we consider a theory that claims Martin-Löf randomness of a given infinite binary sequence. This claim can be formalized in different ways. We show that different formalizations are closely related but not equivalent, and study their properties.

Bulletin of Symbolic Logic, 2013

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

Information Processing Letters, 2003

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativization of the Cantor topology, the U δ -topology introduced by Staiger [RAIRO Inform. Théor. 21 (1987) 147-173]. This topology is also metric, but the distance between two sequences does not depend on their longest common prefix (Cantor metric), but on the number of their common prefixes in a given language U . The resulting space is complete, but not always compact. We will show how to derive a computable set U from a universal Martin-Löf test such that the set of non-random sequences is nowhere dense in the U δ -topology. As a byproduct we obtain a topological characterization of the set of random sequences. We also show that the Law of Large Numbers, which fails with respect to the usual topology, is true for the U δ -topology.

Journal of Symbolic Logic, 2005

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅ of the probability that the output be in some set O ⊆ 2 ≤ω under complexity assumptions about O. §1. Randomness in the spirit of Rice's theorem for computability. Let 2 * be the set of all finite strings in the binary alphabet 2 = {0, 1}. Let 2 ω be the set of all infinite binary sequences. For X ⊆ 2 ω the Lebesgue set-theoretic measure of X is denoted by µ(X ). For a particular string s ∈ 2 * , µ(s2 ω ) = 2 −|s| . If X ⊆ 2 * is a prefix-free set then µ(X2 ω ) = s∈X 2 −|s| ≤ 1. As usual (cf.

Journal of Logic and Computation, 2007

We investigate notions of randomness in the space C½2 N of non-empty closed subsets of f0, 1g N . A probability measure is given and a version of the Martin-Lo¨f test for randomness is defined. Å 0 2 random closed sets exist but there are no random Å 0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log 2 ð4=3Þ. A random closed set has no n-c.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If T n ¼ T \ f0, 1g n , then for any random closed set ½T where T has no dead ends, KðT n Þ ! n À Oð1Þ but for any k, KðT n Þ 2 nÀk þ Oð1Þ, where K() is the prefix-free complexity of 2 f0, 1g à .

Journal of Complexity, 1996

This paper introduces a new complexity measure for binary sequences, the tree complexity. The tree complexity of a sequence grows asymptotically like O(2 2 h) (h the height of the tree) for random sequences. Functions in F 2 [[x]] can be identified with their coefficient sequence. Under this aspect we will show that the tree complexity is O(1) for all algebraic sequences in F ȍ 2. This doubly exponential gap may serve as an indicator of ''simply'' structured sequences and furthermore it defines certain classes within the vast set of transcendental sequences.

2000

This document contains lecture notes of an introductory course on Kolmogorov complexity. They cover basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), notion of randomness (Martin-Löf randomness, Mises–Church randomness), Solomonoff universal a priori probability and their properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity) and applications (incompressibility method in computational complexity theory, incompleteness theorems).

Transactions of the American Mathematical Society, 2008

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤ K Y to mean that (∀n) K(X n) ≤ K(Y n) + O(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ 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 vL-degrees. Unlike the K-degrees, many basic properties of the vL-degrees are easy to prove. We show that X ≤ K Y implies X ≤ vL Y , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤ C , the analogue of ≤ K for plain Kolmogorov complexity.

Theory of Computing Systems, 2012

We prove the formula C (a, b) = K (a| C (a, b))+C (b|a, C (a, b))+O(1) that expresses the plain complexity of a pair in terms of prefix-free and plain conditional complexities of its components. The well known formula from Shannon information theory states that H(ξ , η) = H(ξ) + H(η|ξ). Here ξ , η are random variables and H stands for the Shannon entropy. A similar formula for algorithmic information theory was proven by Kolmogorov and Levin [5] and says that C (a, b) = C (a) + C (b|a) + O(logn), where a and b are binary strings of length at most n and C stands for Kolmogorov complexity (as defined initially by Kolmogorov [4]; now this version is usually called plain Kolmogorov complexity). Informally, C (u) is the minimal length of a program that produces u, and C (u|v) is the minimal length of a program that transforms v to u; the complexity C (u, v) of a pair (u, v) is defined as the complexity of some standard encoding of this pair. This formula implies that I(a : b) = I(b : a) + O(logn) where I(u : v) is the amount of information in u about v defined as C (v) − C (v|u); this property is often called "symmetry of information". The term O(log n), as was noted in [5], cannot be replaced by O(1). Later Levin found an O(1)-exact version of this formula that uses the so-called prefix-free version of complexity: K (a, b) = K (a) + K (b|a, K (a)) + O(1); this version, reported in [2], was also discovered by Chaitin [1]. In the definition of prefix-free complexity we restrict ourselves to self-delimiting programs: reading a program from left to right, the interpreter determines where it ends. See, e.g., [7] for the definitions and proofs of these results. In this note we provide a somewhat similar formula for plain complexity (also with O(1)-precision): Theorem 1. C (a, b) = K (a| C (a, b)) + C (b|a, C (a, b)) + O(1). Proof. The proof is not difficult after the formula is presented. The ≤-inequality is a generalization of the inequality C (x, y) ≤ K (x) + C (y) and can be proven in the same way. Assume that p is a self-delimiting program that maps C (a, b) to a, and q is a (not necessarily self-delimiting) program that maps a and C (a, b) * B. Bauwens is supported by Fundaçao para a Ciência e a Tecnologia by grant (SFRH/BPD/75129/2010), and is also partially supported by project CSI 2 (PTDC/EIAC/099951/2008). A. Shen is supported in part by projects NAFIT ANR-08-EMER-008-01 grant and RFBR 09-01-00709a. Author are grateful to their colleagues for interesting discussions and to the anonymous referees for useful comments.

arXiv (Cornell University), 2012

Péter Gács showed (Gács 1974) that for every n there exists a bit string x of length n whose plain complexity C (x) has almost maximal conditional complexity relative to x, i.e., C (C (x)|x) ≥ log n − log (2) n − O(1). (Here log (2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n − O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| − C (x) ≤ c and |x| + K (|x|) − K (x) ≥ log (2) |x| − c log (3) |x|. In fact, the results of Solovay and Gács are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n − C (x) ≤ c and n + K (n) − K (x) ≥ K (K (n)|n) − 3 K (K (K (n)|n)|n) − c. We also prove a close upper bound K (K (n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) − K(x) ≥ log (2) |x| − O(log (3) |x|). Introduction. Plain Kolmogorov complexity C (x) of a binary string x was defined in [6] as the minimal length of a program that computes x. (See, e.g., [4, 8, 12, 15] for more details.) It was clear from the beginning that this complexity function is not computable: no algorithm can compute C (x) given x. In [3] (see also [4, 8]) a stronger non-uniform version of this result was proven: for every n there exists a string x of length n such that conditional complexity C (C (x)|x), Bruno Bauwens is supported by the Portuguese science foundation FCT (SFRH/BPD/75129/2010); and partially supported by CSI 2