Algorithmic complexity

What are the irreducible components of information that underlie all domains of knowledge and reality? This paper proposes a cross-disciplinary “factor analysis” of information to identify its prime factors—the minimal, universal dimensions from which all informational structures are built. Drawing on insights from physics, information theory, computer science, biology, cognitive science, and philosophy, we argue that five factors are foundational: (1) Distinction and Uncertainty, the binary differences that generate bits and reduce entropy; (2) Pattern and Complexity, the structural organization that distinguishes random data from ordered form; (3) Meaning and Intent, the semantic and pragmatic dimensions that make information significant and actionable; (4) Physical Embodiment and Correlation, the instantiation of information in material systems and its existence as relations among them; and (5) Processing and Dynamics, the computational and communicative transformations that allow information to flow and evolve. Each factor is irreducible, yet interdependent, collectively forming a universal framework for understanding information across natural, social, and artificial systems. Identifying these prime factors clarifies the ontology of information, unifies disparate disciplinary perspectives, and provides a foundation for future work in the philosophy of information, complexity science, and information technology.

The Kosmos Theory (KTS) proposes a unified view of reality, structured in island-universes, autonomous fragments of information that remain interconnected by holographic coherence. These universes constitute nuclei of a greater whole, in which physical, topological, and phenomenological dimensions are interwoven. Topology is formalized by the Klein bottle, which acts as an operator of fragmentation and stitching, dissolving the distinction between interior and exterior and allowing for holographic continuity between autonomous domains. On a physical level, the island formula preserves information in black holes, while topological black holes (TBHs) sustain the stability of information subsystems, and baryogenesis explains the matterantimatter asymmetry, reinterpreted as the Universe-Anti-universe duality. From a phenomenological standpoint, Cosmological Consciousness (CC) emerges as a unifying principle, generating the Holographic Mental Field (HMF), whose local condensation is the Unitary Phenomenal Self (UPS). Within this framework, both the Vital Universe and the central Island-Universe correspond to the UPS, functioning as a holographic nucleus where matter, energy, information, and consciousness are integrated. AdS/CFT holography formalizes the interface between physical reality and conscious experience, ensuring the complementarity between objectivity and subjectivity. Thus, the cosmos is neither merely physical nor merely mental, but a quantum-topological and phenomenological whole, in which each holographic fragment expresses and contains the whole, consolidating the UPS as the unifying instance of reality.

This paper constructs a unified theoretical framework for AI semantic dynamics, modeling the behavior of Large Language Models (LLMs) as dynamic evolutionary processes in high-dimensional semantic space. Based on the Unified Dynamic Approximation Equation (UDAE), we propose the fitting-reasoning continuous spectrum theory, explaining how AI systems dynamically adjust response strategies between the known and unknown. The research identifies three structural problems in modern LLMs: limitations of static approximation assumptions, repetitive defects in high-dimensional semantic matrices, and semantic convergence with crossdomain contamination in long-term dialogues. To address these issues, we design a four-module optimization architecture comprising Global Semantic Monitoring, Semantic Rebalancing, Hierarchical Memory Control, and Semantic Immune System, along with an enhanced Spectral Governor. Through theoretical analysis of mainstream models including GPT series, Tongyi Qianwen, Wenxin Yiyan, and Zhipu GLM, we validate the framework's explanatory power and predictive capability. This research provides theoretical foundations and engineering guidance for next-generation AI system design, promoting the paradigm shift from static fitting to dynamic intelligence in AI.

We present Computational Emergence Theory (CET)—a mathematically rigorous framework wherein physical reality emerges from self-organizing computational processes on infinite graph spaces, with fundamental laws arising as ergodic attractors in rule space. Unlike previous approaches, CET resolves four persistent problems: (1) the measure problem via novel σ-finite constructions on infinite configuration spaces, (2) emergence paradoxes through bootstrap consistency conditions, (3) conceptual
circularity in timeless formulations, and (4) computational irreducibility via attractor dynamics.

This paper introduces the Dynamic Rate Theory as a novel framework for analyzing the P vs. NP problem, integrating cognitive deconstruction methods with mathematical structural modeling. It proposes a shift from binary logical classification to a continuous dynamic rate spectrum that redefines problem solvability through rate-structure interactions. By framing computational complexity in terms of velocity of problem transformation and mapping, this framework bridges cognitive theory and formal computation, enabling a multi-dimensional continuum perspective on NP-completeness. The model demonstrates potential for reclassifying complexity problems through structural rate convergence, offering new possibilities for interdisciplinary analysis.

We consider several variations of the problems of covering a set of barriers (modeled as line segments) using sensors that can detect any intruder crossing any of the barriers. Sensors are initially located in the plane and they can relocate to the barriers. We assume that each sensor can detect any intruder in a circular area of fixed range centered at the sensor. Given a set of barriers and a set of sensors located in the plane, we study three problems: (i) the feasibility of barrier coverage, (ii) the problem of minimizing the largest relocation distance of a sensor (MinMax), and (iii) the problem of minimizing the sum of relocation distances of sensors (MinSum). When sensors are permitted to move to arbitrary positions on the barrier, the MinMax problem is shown to be strongly NP-complete for sensors with arbitrary ranges. We also study the case when sensors are restricted to use perpendicular movement to one of the barriers. We show that when the barriers are parallel, both the

We have proven that any implementation of the concept of `copy number' underlying Assembly Theory (AT) and its assembly index (Ai) is equivalent to Shannon entropy, and not fundamentally or methodologically different from algorithms like ZIP compression. Here we show that the weak empirical correlation between Ai and LZW, which the authors have offered as a defence is based on an incomplete and misleading experiment. When the experiment is completed, the fast asymptotic convergence to LZ compression and Shannon entropy is undeniable, just as their mathematical proof of equivalence remains undisputed. This contribution completes the theoretical and empirical demonstration that any variation of the copy-number concept underlying AT, which entails counting the number of object repetitions `to arrive at a measure for life,' is equivalent to statistical compression and Shannon entropy. We demonstrate that the authors' `we-are-better-because-we-are-worse' argument against compression does not withstand basic scrutiny, and that their empirical results separating organic from inorganic compounds have not only been previously reported, sans claims to unify physics and biology, but are driven solely by molecular length--which they did not control for. We show that Ai is a particular case of our BDM index, introduced almost a decade earlier, and that arguments attributing special stochastic properties to Ai are misleading, since the properties of Ai are not unique, but rather exactly the same as those that Shannon entropy possesses and for which it was designed in the first place--for the quantification of uncertainty--and which we have proven to be equivalent to Ai. Shannon entropy is already not only equipped with stochasticity but was designed for. This makes AT redundant especially when applied to their own experimental data.

We present an agnostic signal reconstruction method for zero-knowledge one-way communication channels in which a receiver aims to interpret a message sent by an unknown source about which no prior knowledge is available and to which no return message can be sent. Our reconstruction method is agnostic vis-à-vis the arbitrarily chosen encoding-decoding scheme and other observer-dependent characteristics, such as the arbitrarily chosen computational model, probability distributions, or underlying mathematical theory. We investigate how non-random messages encode information about their intended physical properties, such as dimension and length scales of the space in which a signal or message may have been originally encoded, embedded, or generated. We focus on image data as a first illustration of the capabilities of the new method. We argue that our results have applications to life and technosignature detection, and to coding theory in general.

The wide application of digital design, the advances of digital fabrication and robotic processes have facilitated the materialization of bespoke geometries. In turn it has raised the issue of how architects can reduce design complexity using computational techniques. This paper presents a survey on complexity theory inclusive of work from the disciplines which range from cybernetics to systems and information theory. We synthesize a taxonomy of different definitions of complexity and ways of managing design complexity by decomposing its different levels as they relate to the fields of architecture, engineering and construction. Our hypothesis is that by reviewing the literature on complexity theory which appears to be highly fragmented, we can aid designers build a better understanding of the underlying principles. Thus designers can develop a more system approach towards the use of digital design tools and make use of concepts coming from the field of complexity theory such as abstraction, adaptation and self-organization in order to come up with novel computational design methods. Such methods can enable designers to deal with design problems holistically and manage design complexity in the contemporary digital design context.

It is a well-known fact that the Bayesian Networks' (BNs) use as classifiers in different fields of application has recently witnessed a noticeable growth. Yet, the Naïve Bayes' application, and even the augmented Naïve Bayes', to classifier-structure learning, has been vulnerable to certain limits, which explains the practitioners' resort to other more sophisticated types of algorithms. Consequently, the use of such algorithms has paved the way for raising the problem of super-exponential increase in computational complexity of the Bayesian classifier learning structure, with the increasing number of descriptive variables. In this context, the present work's major objective lies in setting up a further solution whereby a remedy can be conceived for the intricate algorithmic complexity imposed during the learning of Bayesian classifiers' structure with the use of sophisticated algorithms. Noteworthy, the present paper's framework is organized as follows. We start, in the first place, by to propose a novel approach designed to reduce the algorithmic complexity without engendering any loss of information when learning the structure of a Bayesian classifier. We, then, go on to test our approach on a car diagnosis and a Lymphography diagnosis databases. Ultimately, an exposition of our conducted work's interests will be a closing step to this work.

The concept of complexity as considered in terms of its algorithmic definition proposed by G. J. Chaitin and A. N. Kolmogorov is revisited for the dynamical complexity of music. When music pieces are cast in the form of time series of pitch variations, concepts of dynamical systems theory can be used to define new quantities such as the dimensionality as a measure of the global temporal dynamics of a music piece, and the Shanon entropy as an evaluation of its local dynamics. When these quantities are computed explicitly for sequences sampled in the music literature from the 18th to the 20th century, no indication is found of a systematic increase in complexity paralleling historically the evolution of classical western music, but the analysis suggests that the fractional nature of art might have an intrinsic value of more general significance.

The article is devoted to the dialectical analysis of such fundamental essences of modern organizations as complexity, emergence and management. It considers approaches to the quantitative assessment of complexity as a constructive feature of the system, while drawing attention to the fact that complexity is a direct result of the development of the system's ability to adapt. The appearance of emergent properties in an organization is accompanied by an inevitable increase in its complexity. This growth should be controlled and regulated, and the level of emergence of the organization should correspond to the complexity of the environment. The more uncertainty needs to be eliminated, the more complex the organizational system becomes. From the authors' point of view, the dialectic of entropy, complexity and emergence is as follows - strengthening the emergent properties of a set of elements requires minimizing entropy. In its turn minimizing entropy requires purposeful restri...

This article reviews pixon‐based image reconstruction, which in its current formulation uses a multiresolution language to quantify an image's algorithmic information content (AIC) using Bayesian techniques. Each pixon (or its generalization, the information) represents a fundamental quanta of an image's AIC, and an image's pixon basis represents the minimum degrees of freedom necessary to describe the image within the accuracy of the noise. We demonstrate with a number of examples that pixon‐based image reconstruction yields results consistently superior to popular competing methods, including maximum likelihood and maximum entropy methods. Typical improvements include higher spatial resolution, greater sensitivity to faint sources, and immunity to the production of spurious sources and signal correlated residuals. Finally, we show how the pixon provides a generalization of the Akaike information criterion, and how it relates to concepts of “coarse graining” and the rol...

Computer-based shape grammar implementations aim to support creative design exploration by automating rule-application. This paper reviews existing shape grammar implementations in terms of their algorithmic complexity, extends the definition of shape grammars with sets of transformations for rule application, categorizes (parametric and non-parametric) sets of transformations, and analyses these categories in terms of the resulting algorithmic complexity. Specifically, it describes how different sets of transformations admit different numbers of targets (i.e., potential inputs) for rule application. In the non-parametric case, this number is quadratic or cubic, while in the parametric case, it can be non-polynomial, depending on the size of the target shape. The analysis thus yields lower bounds for the algorithmic complexity of shape grammar implementations that hold independently of the employed algorithm or data structure. Based on these bounds, we propose novel matching algorithms for nonparametric and parametric shape grammar implementation and analyze their complexity. The results provide guidance for future, general-purpose shape grammar implementations for design exploration.

Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rate...

We describe an approach to linguistic theory based on algorithmic complexity, and motivated by Rissanen's minimum description length analysis. The approach adds a critical second stage to Chomsky's original conception of generative grammar.

Thanks to the simplicity and robustness of its calculation methods, algorithmic (or Kolmogorov) complexity appears as a useful tool to reveal chaotic dynamics when experimental time series are too short and noisy to apply Takens’ reconstruction theorem. We measure the complexity in chaotic regimes, with and without extreme events (sometimes called optical rogue waves), of three different all-solid-state lasers: Kerr lens mode locking femtosecond Ti:Sapphire (“fast” saturable absorber), Nd:YVO4 [Formula: see text] Cr:YAG (“slow” saturable absorber) and Nd:YVO4 with modulated losses. We discuss how complexity characterizes the dynamics in an understandable way in all cases, and how it provides a correction factor of predictability given by Lyapunov exponents. This approach may be especially convenient to implement schemes of chaos control in real time.

The Universe is the ultimate information system. The capacity to communicate is built into every object, from electrons to cells to stars. But information is not possible without order. Order imposes a limit on the speed of information. This leads to the surprising emergence of time, money, and markets.

Linear functional analysis historically founded by Fourier and Legendre played a significant role to provide a unified vision of mathematical transformations between vector spaces. The possibility of extending this approach is explored when basis of vector spaces is built Tailored to the Problem Specificity (TPS) and not from the convenience or effectiveness of mathematical calculations. Standardized mathematical transformations, such as Fourier or polynomial transforms, could be extended toward TPS methods, on a basis, which properly encodes specific knowledge about a problem. Transition between methods is illustrated by comparing what happens in conventional Fourier transform with what happened during the development of Jewett Transform, reported in previous articles. The proper use of computational intelligence tools to perform Jewett Transform allowed complexity algorithm optimization, which encourages the search for a general TPS methodology.

Graphs considered in this paper are finite simple undirected graphs. Let G = (V (G), E(G)) be a graph with E(G) = {e 1 , e 2 ,. .. , e m }, for some positive integer m. The edge space of G, denoted by E (G), is a vector space over the field Z 2. The elements of E (G) are all the subsets of E(G). Vector addition is defined as X + Y = X Y, the symmetric difference of sets X and Y, for X, Y 2 E (G). Scalar multiplication is defined as 1 • X = X and 0 • X = ; for X 2 E (G). Let H be a subgraph of G. The uniform set of H with respect to G, denoted by E H (G), is the set of all elements of E (G) that induces a subgraph isomorphic to H. The subspace of E (G) generated by E H (G) shall be denoted by E H (G). If E H (G) is a generating set, that is E H (G) = E (G), then H is called a generator subgraph of G. This study determines the dimension of subspace generated by the set of all subsets of E(G) with even cardinality and the subspace generated by the set of all k-subsets of E(G), for some positive integer k, 1  k  m. Moreover, this paper determines all the generator subgraphs of star graphs. Furthermore, it gives a characterization for a graph G so that star is a generator subgraph of G.

Motivated by algorithmic information theory, the problem of program discovery can help find candidates of underlying generative mechanisms of natural and artificial phenomena. The uncomputability of such inverse problem, however, significantly restricts a wider application of exhaustive methods. Here we present a proof of concept of an approach based on IMP, a high-level imperative programming language. Its main advantage is that conceptually complex computational routines are more succinctly expressed, unlike lowerlevel models such as Turing machines or cellular automata. We investigate if a more expressive higher-level programming language can be more efficient at generating approximations to algorithmic complexity of recursive functions, often of mathematical interest.

Background: The objective of this research is to compare the relational and non-relational (NoSQL) database systems approaches in order to store, recover, query and persist standardized medical information in the form of ISO/EN 13606 normalized Electronic Health Record XML extracts, both in isolation and concurrently. NoSQL database systems have recently attracted much attention, but few studies in the literature address their direct comparison with relational databases when applied to build the persistence layer of a standardized medical information system. Methods: One relational and two NoSQL databases (one document-based and one native XML database) of three different sizes have been created in order to evaluate and compare the response times (algorithmic complexity) of six different complexity growing queries, which have been performed on them. Similar appropriate results available in the literature have also been considered. Results: Relational and non-relational NoSQL database systems show almost linear algorithmic complexity query execution. However, they show very different linear slopes, the former being much steeper than the two latter. Document-based NoSQL databases perform better in concurrency than in isolation, and also better than relational databases in concurrency. Conclusion: Non-relational NoSQL databases seem to be more appropriate than standard relational SQL databases when database size is extremely high (secondary use, research applications). Document-based NoSQL databases perform in general better than native XML NoSQL databases. EHR extracts visualization and edition are also document-based tasks more appropriate to NoSQL database systems. However, the appropriate database solution much depends on each particular situation and specific problem.

The Robust Stable Marriage problem (RSM) is a variant of the classic Stable Marriage problem in which the robustness of a given stable matching is measured by the number of modifications required to find an alternative stable matching should some pairings break due to an unforeseen event. We focus on the complexity of finding an (a, b)-supermatch. An (a, b)-supermatch is defined as a stable matching in which if any a (non-fixed) men/women break up it is possible to find another stable matching by changing the partners of those a men/women and the partners of at most b others. We first discuss a model based on independent sets for finding (1, 1)-supermatches. Secondly, in order to show that deciding whether or not there exists a (1, b)-supermatch is N P-complete, we first introduce a SAT formulation for which the decision problem is N P-complete by using Schaefer's Dichotomy Theorem. We then show the equivalence between this SAT formulation and finding a (1, 1)supermatch on a specific family of instances. We also focus on studying the threshold between the cases in P and N P-complete for this problem.

This paper describes the theory of pixon-based image reconstruction. After a brief introduction of the basic concepts of the pixon, the paper concentrates primarily on our current implementation of the techniques along with the approximations and shortcuts that seem to provide practical software algorithms. We then present an example of application of the method to astrophysical data.

This article attempts to show how summary statistics can be misleading if not complemented by directly inspecting the data. Instead, algorithmic complexity measures can be used to uniquely identify each data element. The data science community has embraced the Datasaurus Dozen as an important example of why looking beyond statistics is necessary to truly grasp all facets of a dataset.

Sources that generate symbolic sequences with algorithmic nature may differ in statistical complexity because they create structures that follow algorithmic schemes, rather than generating symbols from a probabilistic function assuming independence. In the case of Turing machines, this means that machines with the same algorithmic complexity can create tapes with different statistical complexity. In this paper, we use a compression-based approach to measure global and local statistical complexity of specific Turing machine tapes with the same number of states and alphabet. Both measures are estimated using the best-order Markov model. For the global measure, we use the Normalized Compression (NC), while, for the local measures, we define and use normal and dynamic complexity profiles to quantify and localize lower and higher regions of statistical complexity. We assessed the validity of our methodology on synthetic and real genomic data showing that it is tolerant to increasing rate...

Based on the principles of information theory, measure theory, and theoretical computer science, we introduce a signal deconvolution method with a wide range of applications to coding theory, particularly in zero-knowledge one-way communication channels, such as in deciphering messages (i.e., objects embedded into multidimensional spaces) from unknown generating sources about which no prior knowledge is available and to which no return message can be sent. Our multidimensional space reconstruction method from an arbitrary received signal is proven to be agnostic vis-à-vis the encoding-decoding scheme, computation model, programming language, formal theory, the computable (or semi-computable) method of approximation to algorithmic complexity, and any arbitrarily chosen (computable) probability measure. The method derives from the principles of an approach to Artificial General Intelligence (AGI) capable of building a general-purpose model of models independent of any arbitrarily assumed prior probability distribution. We argue that this optimal and universal method of decoding non-random data has applications to signal processing, causal deconvolution, topological and geometric properties encoding, cryptography, and bio- and technosignature detection.

Experimental studies of the cultural evolution of language have focused on how constraints on learning and communication drive emergence of linguistic structure. Yet language is typically transmitted by experts who adjust the input in ways that facilitates learning by novices, e.g. through child-directed speech. Using iterated language learning of binary auditory sequences, we explored how language change is affected by experts’ intention to teach the language to novices. Comparison between teaching chains and simple transmission chains revealed that teaching led to a greater rate of innovation which drove the emergence of more expressive languages consisting of shorter signals. This is the first study to show that during cultural transmission, teaching can modify, and potentially optimize, functional characteristics of language.

In two experiments, Friedenberg and Liby (2016) studied how a diversity of complexity estimates such as density, number of blocks, GIF compression rate and edge length impact the perception of beauty of semirandom two-dimensional patterns. They concluded that aesthetics ratings are positively linked with GIF compression metrics and edge length, but not with the number of blocks. They also found an inverse U-shaped link between aesthetic judgments and density. These mixed results originate in the variety of metrics used to estimate what is loosely called "complexity" in psychology and indeed refers to conflicting notions. Here, we reanalyze their data adding two more conventional and normative mathematical measures of complexity: entropy and algorithmic complexity. We show that their results can be interpreted as an aesthetic preference for low redundancy, balanced patterns and "crooked" figures, but not for high algorithmic complexity. We conclude that participants tend to have a preference for some types of complexity, but not for all. These findings may help understand divergent results in the study of perceived beauty and complexity, and illustrate the need to specify the notion of complexity used in psychology. The field would certainly benefit from a precise taxonomy of complexity measures.

The paper introduces the notion of the adaptive space-variant coordinate transformations and proposes a way of their implementation optically. It also discusses the applications of such coordinate transformations in image processing and pattern recognition.

A multi-cut rearrangement of a string S is a string S′ obtained from S by an operation called k-cut rearrangement, that consists of (1) cutting S at a given number k of places in S, making S the concatenated string X1·X2·X3·…·Xk·Xk+1, where X1 and Xk+1 are possibly empty, and (2) rearranging the Xis so as to obtain S′=Xπ(1)·Xπ(2)·Xπ(3)·…·Xπ(k+1), π being a permutation on 1,2,…,k+1 satisfying π(1)=1 and π(k+1)=k+1. Given two strings S and T built on the same multiset of characters from an alphabet Σ, the Sorting by Multi-Cut Rearrangements (SMCR) problem asks whether a given number ℓ of k-cut rearrangements suffices to transform S into T. The SMCR problem generalizes several classical genomic rearrangements problems, such as Sorting by Transpositions and Sorting by Block Interchanges. It may also model chromoanagenesis, a recently discovered phenomenon consisting of massive simultaneous rearrangements. In this paper, we study the SMCR problem from an algorithmic complexity viewpoint....

Computer-based shape grammar implementations aim to support creative design exploration by automating rule-application. This paper reviews existing shape grammar implementations in terms of their algorithmic complexity, extends the definition of shape grammars with sets of transformations for rule application, categorizes (parametric and non-parametric) sets of transformations, and analyses these categories in terms of the resulting algorithmic complexity. Specifically, it describes how different sets of transformations admit different numbers of targets (i.e., potential inputs) for rule application. In the non-parametric case, this number is quadratic or cubic, while in the parametric case, it can be non-polynomial, depending on the size of the target shape. The analysis thus yields lower bounds for the algorithmic complexity of shape grammar implementations that hold independently of the employed algorithm or data structure. Based on these bounds, we propose novel matching algorit...

Linear functional analysis historically founded by Fourier and Legendre played a significant role to provide a unified vision of mathematical transformations between vector spaces. The possibility of extending this approach is explored when basis of vector spaces is built Tailored to the Problem Specificity (TPS) and not from the convenience or effectiveness of mathematical calculations. Standardized mathematical transformations, such as Fourier or polynomial transforms, could be extended toward TPS methods, on a basis, which properly encodes specific knowledge about a problem. Transition between methods is illustrated by comparing what happens in conventional Fourier transform with what happened during the development of Jewett Transform, reported in previous articles. The proper use of computational intelligence tools to perform Jewett Transform allowed complexity algorithm optimization, which encourages the search for a general TPS methodology.

We demonstrate that the assembly pathway method underlying ``Assembly Theory" (AT) is a suboptimal restricted version of Huffman's encoding (Shannon-Fano type) for `counting copies,' the stated objective of the authors of AT, introduced in computer science in the 1960s and widely used by popular statistical and computable compression algorithms that have been applied to all sort of biosignatures before. We show how simple modular instructions can mislead AT, leading to failure to accomplish what the authors originally intended (counting the `number of copies') or to capture subtleties, beyond very trivial statistical properties of biological systems. We present cases whose low complexity can arbitrarily diverge from the random-like appearance to which the AT would assign arbitrarily high statistical significance, and show that it fails in simple cases (synthetic or natural) which the assembly theory was supposed to shed some light on. Our theoretical and empirical results imply that the assembly index, whose computable nature is not an advantage, does not offer any substantial improvement over existing concepts and methods, computable or (semi) uncomputable. No strong compression or algorithmic complexity results were required to prove that AT and MA are ill-defined and under-perform as compared to simple coding schemes. We show that despite the claims of experimental data, the assembly measure is driven mostly or only by InChI codes which had already been reported before to discriminate organic from inorganic compounds by other indexes.

A hybrid method based on the combination of generalized forward backward method (GFBM) and Green's function for the grounded dielectric slab together with the acceleration of the combination via a discrete Fourier transform (DFT) based algorithm is developed for the efficient and accurate analysis of electromagnetic radiation/scattering from electrically large, irregularly contoured two-dimensional arrays consisting of finite number of probe-fed microstrip patches. In this method, unknown current coefficients corresponding to a single patch are first solved by a conventional Galerkin type hybrid method of moments (MoM)/Green's function technique that uses the grounded dielectric slab's Green's function. Because the current distribution on the microstrip patch can be expanded using an arbitrary number of subsectional basis functions, the patch can have any shape. The solution for the array currents is then found through GFBM, where it sweeps the current computation element by element. The computational complexity of this method, which is originally 2 tot (tot being the total number of unknowns) for each iteration, is reduced to (tot) using a DFT based acceleration algorithm making use of the fact that array elements are identical and the array is periodic. Numerical results in the form of array current distribution are given for various sized arrays of probe-fed microstrip patches with elliptical and/or circular boundaries, and are compared with the conventional MoM results to illustrate the efficiency and accuracy of the method. Index Terms-Discrete Fourier transform (DFT), finite arrays, generalized forward backward method (GFBM), method of moments (MoM). I. INTRODUCTION M AJORITY of commercial and military applications require phased arrays fabricated using the integrated circuit technology. However, a great portion of available design

In this article we use the idea of algorithmic complexity (AC) to study various cosmological scenarios, and as a means of quantizing the gravitational interaction. We look at 5D and 7D cosmological models where the Universe begins as a higher dimensional Planck size spacetime which fluctuates between Euclidean and Lorentzian signatures. These fluctuations are governed by the AC of the two different signatures. At some point a transition to a 4D Lorentzian signature Universe occurs, with the extra dimensions becoming "frozen" or non-dynamical. We also apply the idea of algorithmic complexity to study composite wormholes, the entropy of black holes, and the path integral for quantum gravity. Some of the physical consequences of the idea presented here are: the birth of the Universe with a fluctuating metric signature; the transition from a fluctuating metric signature to Lorentzian one; "frozen" extra dimensions as a consequence of this transition; quantum handles in the spacetime foam as regions with multidimensional gravity.

A methodological proposal to estimate a Tailored to the Problem Specificity mathematical transformation is developed. To begin, Linear Analysis is briefly visited because of its significant role providing a unified vision of mathematical transformations. Thereafter it is explored the possibilities of extending this approach when basis of vector spaces are built tailored to the specific knowledge on a problem; not only from the convenience or effectiveness of mathematical calculations. Basis becomes not necessarily orthogonal neither linear. Standardized Mathematical Transformations such as Fourier or polynomial Transforms, could be extended, towards these new transformations. This was previously done to model Auditory Brainstem Responses using Jewett Transform. The proper use of Computational Intelligence tools was critical in this extension. It allowed important Complexity Algorithm optimization, which encourages the search for generalizing the methodology. In previous works, Artificial Neural Networks trained with backpropagation performed Jewett Transform. Mean Square Error in fitting Auditory Brainstem Responses to a model built using this transform are acceptable (mean ε< 0.3%, n= 600). The complexity of the best trained neural network algorithm was reduced to evaluate 100 inner products on 65 dimension vectors, 20 inner products on 100 dimension vectors and to calculate 120 sigmoid functions. Finally, using the trained Artificial Neural Network to estimate the Transform was thousands of times faster than using numerical gradient descent methods.

We show that numerical approximations of Kolmogorov complexity (K) of graphs and networks capture some group-theoretic and topological properties of empirical networks, ranging from metabolic to social networks, and of small synthetic networks that we have produced. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) based on algorithmic probability theory.

Nowadays, Bayesian Networks (BNs) have constituted one of the most complete, self-sustained and coherent formalisms useful for knowledge acquisition, representation and application through computer systems. Yet, the learning of these BNs structures from data represents a problem classified at an NP-hard range of difficulty. As such, it has turned out to be the most exciting challenge in the learning machine area. In this context, the present work's major objective lies in setting up a further solution conceived to be a remedy for the intricate algorithmic complexity problems imposed during the learning of BNstructure through a massively-huge data backlog. Our present work has been constructed according to the following framework; on a first place, we are going to proceed by defining BNs and their related problems of structurelearning from data. We, then, go on to propose a novel heuristic designed to reduce the algorithmic complexity without engendering any loss of information. Ultimately, our conceived approach will be tested on a car diagnosis as well as on a Lymphography diagnosis databases , while our achieved results would be discussed, along with an exposition of our conducted work's interests as a closing step to this work.

Very large databases are a major opportunity for science and data analytics is a remarkable new eld of investigation in computer science. The eectiveness of these tools is used to support a philosophy against the scientic method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to give scientic meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: with enough data, the numbers speak for themselves. The end of science is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this philosophy is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in randomly generated, large enough databases, which as we will prove implies that most correlations are spurious. Too much information tends to behave like very little information. The scientic method can be enriched by computer mining in immense databases, but not replaced by it. * To appear in Foundations of Science. 1 Italian:. .. you were not born to live like brutes, but to pursue virtue and knowledge. 1 Data deluge Speaking at the Techonomy conference in Lake Tahoe, CA in August 2010, E. Schmidt, then Google CEO, estimated that every two days humanity creates a quantity of data equivalent to the entire amount produced from the dawn of time up to 2003 [46]. According to Grossman [23], every day humanity generates 500 million tweets, 70 million photos on Instagram, and 4 billion videos on Facebook. Under the title What is big data? IBM researchers estimate in [25] that Every day, we create 2.5 quintillion bytes of data so much that 90% of the data in the world today has been created in the last two years alone. This data comes from everywhere: sensors used to gather climate information, posts to social media sites, digital pictures and videos, purchase transaction records, and cell phone GPS signals to name a few. This data is big data. From scarcity and diculty to nd data (and information) we now have a deluge of data. 2 Data science is the end of science In June 2008, C. Anderson, former editor-in-chief of Wired Magazine, wrote a provocative essay titled The End of Theory: The Data Deluge Makes the Scientic Method Obsolete in which he states that with enough data, the numbers speak for themselves. Correlation supersedes causation, and science can advance even without coherent models, unied theories, he continues. Anderson is not unique in stressing the role of petabytes, stored in the cloud, in replacing the scientic method. Succinctly, George Box's maxim that all models are wrong, but

Very large databases are a major opportunity for science and data analytics is a remarkable new eld of investigation in computer science. The eectiveness of these tools is used to support a philosophy against the scientic method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to give scientic meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: with enough data, the numbers speak for themselves. The end of science is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this philosophy is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in randomly generated, large enough databases, which as we will prove implies that most correlations are spurious. Too much information tends to behave like very little information. The scientic method can be enriched by computer mining in immense databases, but not replaced by it. * To appear in Foundations of Science. 1 Italian:. .. you were not born to live like brutes, but to pursue virtue and knowledge. 1 Data deluge Speaking at the Techonomy conference in Lake Tahoe, CA in August 2010, E. Schmidt, then Google CEO, estimated that every two days humanity creates a quantity of data equivalent to the entire amount produced from the dawn of time up to 2003 [46]. According to Grossman [23], every day humanity generates 500 million tweets, 70 million photos on Instagram, and 4 billion videos on Facebook. Under the title What is big data? IBM researchers estimate in [25] that Every day, we create 2.5 quintillion bytes of data so much that 90% of the data in the world today has been created in the last two years alone. This data comes from everywhere: sensors used to gather climate information, posts to social media sites, digital pictures and videos, purchase transaction records, and cell phone GPS signals to name a few. This data is big data. From scarcity and diculty to nd data (and information) we now have a deluge of data. 2 Data science is the end of science In June 2008, C. Anderson, former editor-in-chief of Wired Magazine, wrote a provocative essay titled The End of Theory: The Data Deluge Makes the Scientic Method Obsolete in which he states that with enough data, the numbers speak for themselves. Correlation supersedes causation, and science can advance even without coherent models, unied theories, he continues. Anderson is not unique in stressing the role of petabytes, stored in the cloud, in replacing the scientic method. Succinctly, George Box's maxim that all models are wrong, but

We introduce a family of unsupervised, domain-free, and (asymptotically) model-independent algorithms based on the principles of algorithmic information theory designed to minimize the loss of algorithmic information. The method coarse-grains data in an algorithmic fashion by collapsing regions that can be procedurally regenerated from the compressed version. We show that the method can preserve the salient properties of objects and structures in the process of data dimension reduction and denoising. Using suboptimal approximations of efficient (polynomial) estimations to algorithmic complexity by recent numerical methods of algorithmic probability we demonstrate how these algorithms can preserve structure properties, outperforming other algorithms in e.g. the area of network dimension reduction. As a case study, we report that the method preserves all the graph-theoretic indices measured on a well-known set of synthetic and real-world networks of very different nature, ranging from...