Complexity Of Strings Research Articles

Efficiency of non-identical double helix patterns in minimizing ropelength of torus knots

The ropelength of a knotted string with volume is defined as the ratio of the length of its central curve to the radius of its sectional disc. In a physical context, achieving minimal ropelength corresponds to a state of minimal potential energy, and geometrically, it signifies a tightly-packed conformation. The quest to establish a connection between the topological complexity of knotted strings and their minimal ropelength has persisted into recent years. In this paper, we introduce a new upper bound on the minimal ropelength of (2, n)-torus knots and links: Rop(T(2, n)) ≤ 7.3163 Cr(T(2, n)) + 17.1657. This upper bound is derived from a torus knot conformation constructed based on a tightened pattern of double helix with non-identical radii of winding. A comparative analysis with conformations generated from a superhelix and a circular helix underscores the efficiency of the non-identical double helix pattern, particularly when it appears as a long repeated motif in knotted strings.

Randomness and initial segment complexity for measures

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures.We begin with examples, and provide a robustness property related to Solovay tests. The initial segment complexity of a measure μ at a length n is defined as the μ-average over the descriptive complexity of strings of length n, in the sense of either C or K. We relate this weak randomness notion for a measure to the growth of its initial segment complexity. We show that a maximal growth implies the weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss C-triviality and K-triviality for measures and relate these two notions with each other. Here, triviality means that the initial segment complexity grows as slowly as possible.We show that every measure that is Martin-Löf random in the sense of Hoyrup and Rojas is Martin-Löf absolutely continuous; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions.

The Maximal Complexity of Quasiperiodic Infinite Words

A quasiperiod of a finite or infinite string is a word whose occurrences cover every part of the string. An infinite string is referred to as quasiperiodic if it has a quasiperiod. We present a characterisation of the set of infinite strings having a certain word q as quasiperiod via a finite language Pq consisting of prefixes of the quasiperiod q. It turns out its star root Pq* is a suffix code having a bounded delay of decipherability. This allows us to calculate the maximal subword (or factor) complexity of quasiperiodic infinite strings having quasiperiod q and further to derive that maximally complex quasiperiodic infinite strings have quasiperiods aba or aabaa. It is shown that, for every length l≥3, a word of the form anban (or anbban if l is even) generates the most complex infinite string having this word as quasiperiod. We give the exact ordering of the lengths l with respect to the achievable complexity among all words of length l.

Estimating Algorithmic Information Using Quantum Computing for Genomics Applications

Inferring algorithmic structure in data is essential for discovering causal generative models. In this research, we present a quantum computing framework using the circuit model, for estimating algorithmic information metrics. The canonical computation model of the Turing machine is restricted in time and space resources, to make the target metrics computable under realistic assumptions. The universal prior distribution for the automata is obtained as a quantum superposition, which is further conditioned to estimate the metrics. Specific cases are explored where the quantum implementation offers polynomial advantage, in contrast to the exhaustive enumeration needed in the corresponding classical case. The unstructured output data and the computational irreducibility of Turing machines make this algorithm impossible to approximate using heuristics. Thus, exploring the space of program-output relations is one of the most promising problems for demonstrating quantum supremacy using Grover search that cannot be dequantized. Experimental use cases for quantum acceleration are developed for self-replicating programs and algorithmic complexity of short strings. With quantum computing hardware rapidly attaining technological maturity, we discuss how this framework will have significant advantage for various genomics applications in meta-biology, phylogenetic tree analysis, protein-protein interaction mapping and synthetic biology. This is the first time experimental algorithmic information theory is implemented using quantum computation. Our implementation on the Qiskit quantum programming platform is copy-left and is publicly available on GitHub.

Algorithmic Complexity of Multiplex Networks

Multilayer networks preserve full information about the different interactions among the constituents of a complex system, and have recently proven quite useful in modelling transportation networks, social circles, and the human brain. A fundamental and still open problem is to assess if and when the multilayer representation of a system provides a qualitatively better model than the classical single-layer aggregated network. Here we tackle this problem from an algorithmic information theory perspective. We propose an intuitive way to encode a multilayer network into a bit string, and we define the complexity of a multilayer network as the ratio of the Kolmogorov complexity of the bit strings associated to the multilayer and to the corresponding aggregated graph. We find that there exists a maximum amount of additional information that a multilayer model can encode with respect to the equivalent single-layer graph. We show how our complexity measure can be used to obtain low-dimensional representations of multidimensional systems, to cluster multilayer networks into a small set of meaningful super-families, and to detect tipping points in the evolution of different time-varying multilayer graphs. Interestingly, the low-dimensional multiplex networks obtained with the proposed method also retain most of the dynamical properties of the original systems, as demonstrated for instance by the preservation of the epidemic threshold in the multiplex SIS model. These results suggest that information-theoretic approaches can be effectively employed for a more systematic analysis of static and time-varying multidimensional complex systems.

Asymptotic Analysis of the kth Subword Complexity.

Patterns within strings enable us to extract vital information regarding a string’s randomness. Understanding whether a string is random (Showing no to little repetition in patterns) or periodic (showing repetitions in patterns) are described by a value that is called the kth Subword Complexity of the character string. By definition, the kth Subword Complexity is the number of distinct substrings of length k that appear in a given string. In this paper, we evaluate the expected value and the second factorial moment (followed by a corollary on the second moment) of the kth Subword Complexity for the binary strings over memory-less sources. We first take a combinatorial approach to derive a probability generating function for the number of occurrences of patterns in strings of finite length. This enables us to have an exact expression for the two moments in terms of patterns’ auto-correlation and correlation polynomials. We then investigate the asymptotic behavior for values of . In the proof, we compare the distribution of the kth Subword Complexity of binary strings to the distribution of distinct prefixes of independent strings stored in a trie. The methodology that we use involves complex analysis, analytical poissonization and depoissonization, the Mellin transform, and saddle point analysis.

Exact Constructive and Computable Dimensions

In this paper we derive several results which generalise the constructive dimension of (sets of) infinite strings to the case of exact dimension. We start with proving a martingale characterisation of exact Hausdorff dimension. Then using semi-computable super-martingales we introduce the notion of exact constructive dimension of (sets of) infinite strings. This allows us to derive several bounds on the complexity functions of infinite strings, that is, functions assigning to every finite prefix its Kolmogorov complexity. In particular, it is shown that the exact Hausdorff dimension of a set of infinite strings lower bounds the maximum complexity function of strings in this set. Furthermore, we show a result bounding the exact Hausdorff dimension of a set of strings having a certain computable complexity function as upper bound. Obviously, the Hausdorff dimension of a set of strings alone without any computability constraints cannot yield upper bounds on the complexity of strings in the set. If we require, however, the set of strings to be Σ2-definable several results upper bounding the complexity by the exact Hausdorff dimension hold true. First we prove that for a Σ2-definable set with computable dimension function one can construct a computable – not only semi-computable – martingale succeeding on this set. Then, using this result, a tight upper bound on the prefix complexity function for all strings in the set is obtained.

ON REALS WITH -BOUNDED COMPLEXITY AND COMPRESSIVE POWER

AbstractThe (prefix-free) Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some ‘standard’ lowness notions for reals: A isK-trivial if its initial segments have the lowest possible complexity and A is low forKif using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold only up to all${\rm{\Delta }}_2^0$orders, and call the new notions${\rm{\Delta }}_2^0$-bounded K-trivialand${\rm{\Delta }}_2^0$-bounded low for K. Several of the ‘nice’ properties ofK-triviality are lost with this weakening. For instance, the new weaker definitions both give uncountable set of reals. In this paper we show that the weaker definitions are no longer equivalent, and that the${\rm{\Delta }}_2^0$-boundedK-trivials are cofinal in the Turing degrees. We then compare them to other previously studied weakenings, namelyinfinitely-often K-trivialityandweak lowness for K(in each, the defining inequality must hold up to a constant, but only for infinitely many inputs). We show that${\rm{\Delta }}_2^0$-boundedK-trivial implies infinitely-oftenK-trivial, but no implication holds between${\rm{\Delta }}_2^0$-bounded low forKand weakly low forK.

The Minimum Oracle Circuit Size Problem

We consider variants of the minimum circuit size problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MSCP QBF is known to be complete for PSPACE under ZPP reductions. We show that it is not complete under logspace reductions, and indeed it is not even hard for TC 0 under uniform AC 0 reductions. We obtain a variety of consequences that follow if oracle versions of MCSP are hard for various complexity classes under different types of reductions. We also prove analogous results for the problem of determining the resource-bounded Kolmogorov complexity of strings, for certain types of Kolmogorov complexity measures.

Finite state incompressible infinite sequences

In this paper we define and study finite state complexity of finite strings and infinite sequences as well as connections between these complexity notions to randomness and normality. We show that the finite state complexity does not only depend on the codes for finite transducers, but also on how the codes are mapped to transducers. As a consequence we relate the finite state complexity to the plain (Kolmogorov) complexity, to the process complexity and to prefix-free complexity. Working with prefix-free sets of codes we characterise Martin-Löf random sequences in terms of finite state complexity: the weak power of finite transducers is compensated by the high complexity of enumeration of finite transducers. We also prove that every finite state incompressible sequence is normal, but the converse implication is not true. These results also show that our definition of finite state incompressibility is stronger than all other known forms of finite automata based incompressibility, in particular the notion related to finite automaton based betting systems introduced by Schnorr and Stimm. The paper concludes with a discussion of open questions.

Algorithmic complexity for psychology: a user-friendly implementation of the coding theorem method.

Kolmogorov-Chaitin complexity has long been believed to be impossible to approximate when it comes to short sequences (e.g. of length 5-50). However, with the newly developed coding theorem method the complexity of strings of length 2-11 can now be numerically estimated. We present the theoretical basis of algorithmic complexity for short strings (ACSS) and describe an R-package providing functions based on ACSS that will cover psychologists' needs and improve upon previous methods in three ways: (1) ACSS is now available not only for binary strings, but for strings based on up to 9 different symbols, (2) ACSS no longer requires time-consuming computing, and (3) a new approach based on ACSS gives access to an estimation of the complexity of strings of any length. Finally, three illustrative examples show how these tools can be applied to psychology.

The Sum 2 KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite

We consider two quantities that measure complexity of binary strings: KM(x) is defined as the negative logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ? such that the sum of 2KM(x)?K(x) over all prefixes x of ? is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.

Calculating Kolmogorov complexity from the output frequency distributions of small Turing machines.

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all binary strings of length and for most strings of length by running all Turing machines with 5 states and 2 symbols ( with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings.Additional material can be found at the Algorithmic Nature Group website at http://www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.

Jeffrey Shallit and Ming-Wei Wang. Automatic complexity of strings. Journal of Automata, Languages and Combinatorics, vol. 6 (2001), pp. 537–554. - Cristian S. Calude, Kai Salomaa and Tania K. Roblot. Finite-state complexity and randomness. Theoretical Computer Science, vol. 412 (2011), no. 41, pp. 5668–5677. - Cristian S. Calude, Kai Salomaa and Tania K. Roblot. State-size hierarchy for finite-state complexity. International

Jeffrey Shallit and Ming-Wei Wang. Automatic complexity of strings. Journal of Automata, Languages and Combinatorics, vol. 6 (2001), pp. 537–554. - Cristian S. Calude, Kai Salomaa and Tania K. Roblot. Finite-state complexity and randomness. Theoretical Computer Science, vol. 412 (2011), no. 41, pp. 5668–5677. - Cristian S. Calude, Kai Salomaa and Tania K. Roblot. State-size hierarchy for finite-state complexity. International Journal of Foundations of Computer Science, vol. 23 (2012), no. 1, pp. 37–50. - Volume 18 Issue 4

Finite state complexity

In this paper we develop a version of Algorithmic Information Theory (AIT) based on finite transducers instead of Turing machines; the complexity induced is called finite-state complexity. In spite of the fact that the Universality Theorem (true for Turing machines) is false for finite transducers, the Invariance Theorem holds true for finite-state complexity. We construct a class of finite-state complexities based on various enumerations of the set of finite transducers. In contrast with descriptional complexities (plain, prefix-free) from AIT, finite-state complexity is computable and there is no a priori upper bound for the number of states used for minimal descriptions of arbitrary strings. Upper and lower bounds for the finite-state complexity of arbitrary strings, and for strings of particular types, are given and incompressible strings are studied.

Effective Complexity of Stationary Process Realizations

The concept of effective complexity of an object as the minimal description length of its regularities has been initiated by Gell-Mann and Lloyd. The regularities are modeled by means of ensembles, which is the probability distributions on finite binary strings. In our previous paper [1] we propose a definition of effective complexity in precise terms of algorithmic information theory. Here we investigate the effective complexity of binary strings generated by stationary, in general not computable, processes. We show that under not too strong conditions long typical process realizations are effectively simple. Our results become most transparent in the context of coarse effective complexity which is a modification of the original notion of effective complexity that needs less parameters in its definition. A similar modification of the related concept of sophistication has been suggested by Antunes and Fortnow.

ON THE QUANTUM KOLMOGOROV COMPLEXITY OF CLASSICAL STRINGS

We show that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding string. It follows that quantum complexity is an extension of classical complexity to the domain of quantum states. This is true even if we allow a small probabilistic error in the quantum computer's output. We outline a mathematical proof of this statement, based on an inequality for outputs of quantum operations and a classical program for the simulation of a universal quantum computer.

Two-pattern strings II—frequency of occurrence and substring complexity

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses T λ of two-pattern strings of scope λ, λ = 1 , 2 , … . Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency f λ ( n ) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on { a , b } : we show that lim n → ∞ f λ ( n ) = 0 , but that for strings of lengths n ⩽ 2 λ , two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C 1 ( k ) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim k → ∞ ( C 1 ( k ) − k ) = ∞ .

The Kolmogorov complexity of infinite words

We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is placed on bounds on the complexity of strings in constructively given subsets of the Cantor space. Finally, we compare the Kolmogorov complexity to the subword complexity of infinite strings.

How many strings are easy to predict?

It is well known in the theory of Kolmogorov complexity that most strings cannot be compressed; more precisely, only exponentially few (Θ (2 n − m )) binary strings of length n can be compressed by m bits. This paper extends the ‘incompressibility’ property of Kolmogorov complexity to the ‘unpredictability’ property of predictive complexity. The ‘unpredictability’ property states that predictive complexity (defined as the loss suffered by a universal prediction algorithm working infinitely long) of most strings is close to a trivial upper bound (the loss suffered by a trivial minimax constant prediction strategy). We show that only exponentially few strings can be successfully predicted and find the base of the exponent.