Topological arguments for Kolmogorov complexity

Theoretical Computer Science, 2002

We consider for a real number the Kolmogorov complexities of its expansions with respect to di erent bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the preÿxes of their expansions with respect to di erent bases r and b are related in a way that depends only on the relative information of one base with respect to the other.

2020

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997), as well as for sizes of subgroups and projections of sets (Chan, Yeung, Romashchenko, Shen, Vereshchagin, 1998--2002). This parallelism started with the Kolmogorov-Levin formula (1968) for the complexity of pairs of strings with logarithmic precision. Longpre (1986) proved a version of this formula for space-bounded complexities. In this paper we prove an improved version of Longpre's result with a tighter space bound, using Sipser's trick (1980). Then, using this space bound, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead.

Theoretical Computer Science, 2012

We present a measure of string complexity, called I-complexity, computable in linear time and space. It counts the number of different substrings in a given string. The least complex strings are the runs of a single symbol, the most complex are the de Bruijn strings. Although the I-complexity of a string is not the length of any minimal description of the string, it satisfies many basic properties of classical description complexity. In particular, the number of strings with I-complexity up to a given value is bounded, and most strings of each length have high I-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.

2010

This is a short introduction to Kolmogorov complexity and information theory. The interested reader is referred to the literature, especially the textbooks [CT91] and [LV97] which cover the fields of information theory and Kolmogorov complexity in depth and with all the necessary rigor. They are well to read and require only a minimum of prior knowledge.

1995

We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resource-bounded measure introduced by [17]. From this we conclude that R t is not Turing-complete for EXP . This contrasts the resource unbounded setting. There R is Turing-complete for co-RE . We show that the class of sets to which R t bounded truthtable reduces, has p 2 -measure 0 (therefore, measure 0 in EXP ). This answers an open question of Lutz, giving a natural example of a language that is not weaklycomplete for EXP and that reduces to a measure 0 class in EXP . It follows that the sets that are p btt -hard for EXP have p 2 -measure 0. 1 Introduction One of the main questions in complexity theory is the relation between complexity classes, such as for example P ; NP , and EXP . It is well known that ...

Annals of Pure and Applied Logic, 2004

We investigate the initial segment complexity of random reals. Let K() denote preÿx-free Kolmogorov complexity. A natural measure of the relative randomness of two reals and ÿ is to compare complexity K(n) and K(ÿ n). It is well-known that a real is 1-random i there is a constant c such that for all n, K(n) ¿ n − c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the ÿne behaviour of K(n) for random. Following work of Downey, Hirschfeldt and LaForte, we say that 6 K ÿ i there is a constant O(1) such that for all n, K(n) 6 K(ÿ n) + O(1). We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real so that lim sup n K(n) − K(n) = ∞ where is Chaitin's random real. One is based upon (unpublished) work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that ({ÿ : ÿ 6 K }) = 0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n = m then the initial segment complexities of the natural n-and m-random sets (namely ∅(n−1) and ∅(m−1)) are di erent. The techniques introduced in this paper have already found a number of other applications.

The Computer Journal, 1999

We briefly discuss the origins, main ideas and principal applications of the theory of Kolmogorov complexity.

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.

2010

We study the relationship between complexity cores of a language and the descriptional complexity of the characteristic sequence of the language based on Kolmogorov complexity. Intuitively, a complexity core is a set of hard instances of a language, i.e. instances which cannot be decided in polynomial time. Kolmogorov complexity measures the information content of a string by the length of the shortest program which prints the string. Time-bounded Kolmogorov complexity looks at the length of a shortest program which prints the string within a specified time bound. We prove that a recursive set A has a complexity core if for all constants c, the computational depth (the difference between time-bounded and unbounded Kolmogorov complexities) of the characteristic sequence of A up to length n is larger than c infinitely often. We also show that if a language has a complexity core of exponential density, then it cannot be accepted in average polynomial time, when the strings are distributed according to a time bounded version of the universal distribution.