For a finite binary string @math its logical depth @math for significance @math is the shortest running time of a program for @math of length @math . There is another definition of logical depth. We give a new proof that the two versions are close. There is an infinite sequence of strings of consecutive lengths such that for every string there is a @math such that incrementing @math by 1 makes the associated depths go from incomputable to computable. The maximal gap between depths resulting from incrementing appropriate @math 's by 1 is incomputable. The size of this gap is upper bounded by the Busy Beaver function. Both the upper and the lower bound hold for the depth with significance 0. As a consequence, the minimal computation time of the associated shortest programs rises faster than any computable function but not so fast as the Busy Beaver function.
The minimum time to compute a string by a @math -incompressible program was first considered in @cite_9 . This minimum time is called the logical depth at significance @math of the string concerned. Definitions, variations, discussion and early results can be found in the given reference. A more formal treatment as well as an intuitive approach was given in the textbook @cite_6 , Section 7.7. In @cite_8 the notion of computational depth is defined as @math (see definitions below). This would equal the negative logarithm of the expression @math in Definition if the following were proved. Since @cite_7 proved in the so-called Coding Theorem that @math up to a constant additive term it remains to prove @math up to a small additive term. The last equality is a major open problem in Kolmogorov complexity theory, see @cite_6 Exercises 7.6.3 and 7.6.4.
@cite_9 @cite_7 @cite_6 @cite_8
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We introduce Computational Depth, a measure for the amount of "nonrandom" or "useful" information in a string by considering the difference of various Kolmogorov complexity measures. We investigate three instantiations of Computational Depth: • Basic Computational Depth, a clean notion capturing the spirit of Bennett's Logical Depth. We show that a Turing machine M runs in time polynomial on average over the time-bounded universal distribution if and only if for all inputs x, M uses time exponential in the basic computational depth of x. • Sublinear-time Computational Depth and the resulting concept of Shallow Sets, a generalization of sparse and random sets based on low depth properties of their characteristic sequences. We show that every computable set that is reducible to a shallow set has polynomial-size circuits. • Distinguishing Computational Depth, measuring when strings are easier to recognize than to produce. We show that if a Boolean formula has a nonnegligible fraction of its satisfying assignments with low depth, then we can find a satisfying assignment efficiently. Abstract Given two infinite binary sequences A , B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A , modulo a constant. This relation, introduced in Nies (2005) [14] and denoted by A ≤ L K B , is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by ≤ L K are called L K degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K ’ sets. A well-known result from Nies (2005) [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤ L K , given any non-trivial Δ 2 0 sets X , Y there is a computably enumerable set A which is not K-trivial and it is below X , Y . This shows that the local structures of Σ 1 0 and Δ 2 0 Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from 0 ″ which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the Δ 2 0 or Σ 1 0 structures of LK degrees. The proofs introduce a new technique of permitting below a Δ 2 0 set that is not K -trivial, which is likely to have wider applications. In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number @math of repetitions, in the sense that no two strings in the code can give rise to the same string after @math repetitions. We propose explicit number theoretic constructions of such subsets. In the case of @math repetition, the proposed construction is asymptotically optimal. For @math , the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to @math repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number @math of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any @math repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible. This paper addresses the smallest grammar problem: What is the smallest context-free grammar that generates exactly one given string spl sigma ? This is a natural question about a fundamental object connected to many fields such as data compression, Kolmogorov complexity, pattern identification, and addition chains. Due to the problem's inherent complexity, our objective is to find an approximation algorithm which finds a small grammar for the input string. We focus attention on the approximation ratio of the algorithm (and implicitly, the worst case behavior) to establish provable performance guarantees and to address shortcomings in the classical measure of redundancy in the literature. Our first results are concern the hardness of approximating the smallest grammar problem. Most notably, we show that every efficient algorithm for the smallest grammar problem has approximation ratio at least 8569 8568 unless P=NP. We then bound approximation ratios for several of the best known grammar-based compression algorithms, including LZ78, B ISECTION, SEQUENTIAL, LONGEST MATCH, GREEDY, and RE-PAIR. Among these, the best upper bound we show is O(n sup 1 2 ). We finish by presenting two novel algorithms with exponentially better ratios of O(log sup 3 n) and O(log(n m sup * )), where m sup * is the size of the smallest grammar for that input. The latter algorithm highlights a connection between grammar-based compression and LZ77.