Algorithmic Information Theory

On John von Neumann’s work on biology, and on open-ended models of biological evolution, life as evolving software, and the evolution of mutating software.

Twenty-minute summary of metabiology 2.0 at https://youtube.com/watch?v=W0YjaOmAjF8, a talk that Chaitin gave in Morocco on Tuesday November 12th, 2024.

Fifteen-minute summary of metabiology 1.0 at https://youtube.com/watch?v=C1uVs4ZF0bc, an August 2023 interview with Curt Jaimungal recorded while Chaitin was living in Buenos Aires.

ERRATA— PHILOSOPHICAL MATHEMATICS, Chapter 6 is now PHILOSOPHICAL COMPUTATIONS, Part I, Chapter 6, and can be downloaded below.

Texts available on the Web have been generated by human minds. We observe that simple patterns are over-represented: abcdef is more frequent than arfbxg and 1000 appears more often than 1282. We suggest that word frequency patterns can be predicted by cognitive models based on complexity minimization. Conversely, the observation of word frequencies offers an opportunity to infer particular cognitive mechanisms involved in their generation.

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.

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 Gro...

Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the reference computer. We argue that this approach can be more useful if it is refined to include an important practical case of simple binary strings. Kolmogorov complexity calculus may be developed for this case if we restrict the class of available reference computers. The interesting problem is to define a class of computers which is restricted in a natural way modeling the real-life situation where only a limited class of computers is physically available to us. We give an example of what such a natural restriction might look like mathematically, and show that under such restrictions some error terms, even logarithmic in complexity, can disappear from the standard complexity calculus.

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton's equations. The MaxEnt inference algorithm with Shannon's concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes' formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.

MaxEnt inference algorithm and information theory are relevant for the time evolution of macroscopic systems considered as problem of incomplete information. Two different MaxEnt approaches are introduced in this work, both applied to prediction of time evolution for closed Hamiltonian systems. The first one is based on Liouville equation for the conditional probability distribution, introduced as a strict microscopic constraint on time evolution in phase space. The conditional probability distribution is defined for the set of microstates associated with the set of phase space paths determined by solutions of Hamilton's equations. The MaxEnt inference algorithm with Shannon's concept of the conditional information entropy is then applied to prediction, consistently with this strict microscopic constraint on time evolution in phase space. The second approach is based on the same concepts, with a difference that Liouville equation for the conditional probability distribution is introduced as a macroscopic constraint given by a phase space average. We consider the incomplete nature of our information about microscopic dynamics in a rational way that is consistent with Jaynes' formulation of predictive statistical mechanics, and the concept of macroscopic reproducibility for time dependent processes. Maximization of the conditional information entropy subject to this macroscopic constraint leads to a loss of correlation between the initial phase space paths and final microstates. Information entropy is the theoretic upper bound on the conditional information entropy, with the upper bound attained only in case of the complete loss of correlation. In this alternative approach to prediction of macroscopic time evolution, maximization of the conditional information entropy is equivalent to the loss of statistical correlation, and leads to corresponding loss of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of information about possible microstates of the system.

First, allow me to interject that the idea that I am about to describe seems more extensive than the hyper-reals of non-standard analysis and is not to be confused with them. The hyper-reals, in my understanding (and I haven't yet thoroughly studied non-standard analysis, so this may be a mis-perception), are just an alternative description of the real number system with the same cardinality of the reals-the power set of the naturals, c := 2^(aleph-null). according to a book I have on non-standard analysis by Mark Davis published in 1977 & 2005, to the date of publication, only one theorem of real analysis has been proven by non-standard methods-not too powerful a method? Now to describe what I mean by "hyper-numbers:" H 1 is the first class of hypernumber, of cardinality of the power set of the reals, c 1 := 2^c. The "integers" of this number system are reals, with the density of the reals-no "first," "second," &c., integers & the "digits" of a hypernumber of first class are reals, with the density of the reals-no "first," "second," &c., digits. From this, we can likewise define hyper-numbers of the second class, H 2 = power set of those of the first class, whose "integers" and "digits" are H 1 numbers, with the density of H 1 numbers. This set has cardinality c 2 := 2^c 1. And, more generally, we can continue in a sequence of HN with cardinality c N := 2^c (N-1) , where N is any finite natural number. Consider N = 100, N= 1,000,000, or N = 2^100! Mind blowing! I'd like to study this theory in detail someday, maybe establishing a formal theory by proving a few seminal theorems. Right now, I'm still busy completing and writing up my theory of black holes for publication in a peer-reviewed journal (as you may know if you've been reading my prior missives)-pretty bland stuff by comparison, no?

Generalmente alla scienza si chiede di spiegare la natura, ed in effetti la Fisica classica ci forniva un quadro di certezze che seppure incompleto risultava abbastanza coerente ed intuibile; gli studi dell'ultimo secolo e le recenti evidenze sperimentali hanno però fatto svanire i dogmi indiscutibili posti a fondamento del castello della conoscenza scientifica imponendo un cambiamento profondo nel nostro modo di pensare, e non solo limitato agli argomenti scientifici ma tale da coinvolgere la forma stessa della mente. "Se possiamo predire con certezza il valore di una quantità fisica, allora esiste un elemento di realtà fisica corrispondente a questa quantità" affermava Einstein che, pure essendo stato un grande iniziatore delle nuove conoscenze, ne ha sempre ostinatamente rifiutato l'estrema conseguenza consistente nel venir meno del concetto stesso di realtà oggettiva. A titolo di esempio basti pensare che a livello di particelle non si può separare l'osservatore dall'osservato, che una intrinseca indeterminazione non intuitivamente percepibile e pure matematicamente codificata da Heisenberg vanifica alla base la possibilità di previsione, facendo svanire il fondamentale principio di causalità. Riguardo la stessa misura di una grandezza fisica, questa appare emergere da uno stato di sovrapposizione quantistica in cui infinite misure sono insieme presenti; sono stati altresì sperimentati misteriosi collegamenti istantanei tra particelle comunque distanti, al di fuori dello spazio e del tempo 1 . Da tutto ciò ne consegue che per mantenere un po' di certezza nella realtà del nostro mondo occorre guardare le cose esclusivamente alla nostra scala di misura lasciando confinato il microscopico nelle formule degli specialistici, fingendo di dimenticare che l'universo è fatto di particelle, noi compresi. Si cercherà quindi in questo articolo di ripensare la realtà proponendo una visione complessiva del mondo sulla base del fenomeno, ma non a questo limitata, perché oggi la scienza ci propone più interrogativi che chiarimenti. Verranno esposti in termini facilmente comprensibili alcuni fenomeni fisici molto comuni, unitamente ad una breve riflessione critica, adatti a mostrare i limiti, le incongruenze ed i paradossi della Fisica. Si presenteranno quattro esperimenti esaminandone proprio gli aspetti solitamente sottaciuti per evidenziare le incongruenze non solo della visione classica ma anche di quella quantistica qualora venga filtrata dalla consueta forma mentale basata sullo spazio e sul tempo, mostrando infine l'inconsistenza di tale concetto. Nella conclusione poi si proporrà una possibile concezione dell'universo compatibile con una razionale interpretazione dei fenomeni, tenendo conto di alcune "scomodità" che si preferisce relegare nell'ambito delle curiosità, fuori del tavolo della discussione scientifica. Premesse. Come noto materia ed energia presentano una duplice natura corpuscolare ed ondulatoria in modo mutuamente esclusivo (nel senso che questa natura viene evidenziata con esperienze differenti e mai contemporaneamente), motivo per cui è stato appositamente coniato il principio di "dualita' onda-corpuscolo" al fine di sopire la evidente contraddizione logica 2 . Poiché il nostro universo è costituito da entità osservabili aventi caratteristiche di discontinuità per comodità possiamo parlare di "particelle" purché sia chiaro che si tratta di una pura convenzione: non esistono i mattoncini ultimi, le unità minime eterne e incorruttibili di antica memoria in perenne movimento in uno spazio vuoto preesistente. Occorre altresì considerare, come si è accennato prima, che a livello microscopico il termine "osservazione" risulta inappropriato in quanto presuppone l'esistenza di una realtà oggettiva dell'osservato indipendente dall'osservatore, l'esperienza ha dimostrato invece che si tratta di una interazione, e che l'osservazione modifica ciò che viene osservato per cui non può essere ripetuta. E' da rimarcare che questo fatto mina alla base uno dei fondamenti della scienza galileiana dove l'oggettività si basa sulla ripetibilità della misura e viene considerato non scientifico un evento irripetibile. Detto en-passant ad evitare equivoci legati a strane idee antropocentriche che circolano anche in ambienti scientifici si deve ricordare che le interazioni non dipendono ovviamente dalla osservazione da parte di esseri senzienti o comunque viventi, senza voler con ciò negare il nostro limitato potere di intervento. Avendo chiaro tutto questo possiamo comunque continuare ad utilizzare i termini correnti al fine di agevolare la comprensione del discorso, sottolineando che i fenomeni presentati si verificano a livello atomico, un po' come se le nostre cellule si comportassero stranamente ed in modo diverso rispetto all'organo che compongono. Considerazioni sull'interferenza delle onde (talvolta 1+1=0). Per ravvivare alcuni concetti forse sfocati e sperduti tra i vaghi ricordi scolastici si ritiene opportuno un breve inquadramento dei fatti. Un'onda che si propaga in linea retta può essere rappresentata graficamente in un dato istante con il disegno della fig.

Physics concepts have often been borrowed and independently developed by other fields of science. In this perspective, a significant example is that of the entropy in information theory. The aim of this paper is to provide a short and pedagogical introduction to the use of data compression techniques for the estimate of the entropy and other relevant quantities in information theory and algorithmic information theory. We consider in particular the LZ77 algorithm as a case study and discuss how a zipper can be used for information extraction.

Consider a joint quantum state of a system and its environment. A measurement on the environment induces a decomposition of the system state. Using algorithmic information theory, we define the preparation information of a pure or mixed state in a given decomposition. We then define an optimal decomposition as a decomposition for which the average preparation information is minimal. The average preparation information for an optimal decomposition characterizes the system-environment correlations. We discuss properties and applications of the concepts introduced above and give several examples.

We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. In the above paper, it was shown that the prefix version of information distance equals max(K(x|y), K(y|x) up to additive logarithmic terms. It was claimed by Mahmud [12] that this equality holds up to additive O(1)-precision. We show that this claim is false, but does hold if the distance is at least logarithmic. This implies that the original definition provides a metric on strings that are at superlogarithmically separated.

This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli

The dissertation – The diagrammatics of post-truth in the light of Deleuze and Guattari's transcendental materialism. Semiotics, technology and automatic rationality – presents a hypothesis that post-truth should not be conceived merely as the result of the systemic misrepresentation of reality by the algorithms of social media platforms executing preprogrammed economic and political goals, but rather as a new semiotic system with its own rationality, forms and processes of knowledge and subjectivity production. The functioning of this semiotic system is conditioned by technologies of automated computation and information processing, which are framed here as social practices. In order to explicate these conditions of the production of post-truth semiotics, the subsequent chapters of the dissertation include surveys in semiotic theory (Hjelmslev, Peirce), the notion of technics (cybernetics, Heidegger, Simondon, Stiegler, Land), automated rationality of contemporary algorithmic architectures (Parisi, Negarestani, Pasquinelli), to finally move on to an analysis of forms of knowledge and subjectivity on social platforms (Deleuze, Wark). The concept driving and integrating all the theoretical levels of this project is the diagram – understood in the light of Gilles Deleuze’s and Felix Guattari’s transcendental materialism. Thus, a specific research method, labelled here diagrammatics, is developed throughout the dissertation, which turns itself into an example of the application of this method to the chosen problem.

Palabras clave. Teoría algorítmica de la información, teoría de la computabilidad, complejidad de largo de programa, complejidad de Kolmogorov, números normales, números absolutamente normales, aleatoriedad, número Omega de Chaitin, probabilidad de detención, jerarquía aritmética, cómputos infinitos, máquina de Turing, máquina monótona, K-trivialidad, noción de lowness (bajura), traceability (rastreabilidad). iii ASPECTS OF RANDOMNESS Summary. In this thesis we investigate some aspects of randomness and triviality defined by the theory of program-size. We first deal with randomness and absolute normality of real numbers. Both sets of reals have Lebesgue measure 1 and they are notions that imply several properties of stochasticity. Despite that fact, it has not been easy to give concrete examples in such classes. We prove that there are absolutely normal numbers which are computable and we give two algorithms for constructing such numbers. The former is a computable reformulation of a result of Sierpinski of 1916. The latter is part of our reconstruction of an unpublished manuscript of Turing on normal numbers. For examples of randomness, we generalize Chaitin's halting probability Ω and we analyze the probability that a universal machine halts and gives an output in a given set X. We study the relationship between the computability theoretic properties of X and the randomness properties of the induced probability. The second aspect of randomness that we tackle is the study a variant of the classical definition of program-size complexity which does not involve oracles, and we ask whether it leads to a stronger notion of randomness. We define our complexity function based on monotone Turing machines performing unending computations. We investigate some properties of this function and we consider the induced definitions of randomness and triviality. With this last notion we characterize the computable reals. The last aspect deals with anti-randomness and the possibility to characterize the so called K-trivial reals in terms of notions that do not directly involve the prefix-free program-size complexity. We propose and investigate two computability theoretical combinatorial lowness notions by strengthening other notions already existing in the literature. We relate the plain C and the prefix-free program-size complexity K with these notions by considering variations of K-triviality and C-triviality. We conclude with a list of the main questions that remain open.

Sarebbe bello se fosse vero…invece si scoprì più tardi che il Futuro era imprevedibile, finché non diventava Presente come dimostrato dalla Meccanica Quantistica. Dio poteva solo proporre ma era la Probabilità, un’energia potenziale sempre incerta che causava gli eventi futuri.

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 show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this new approach requires less training data and is more generalizable as it shows greater resilience to random attacks. We investigate the shape of the discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not necessary to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that (i) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; (ii) that parameter solutions can be found using an algorithmic-probability classi...

**Abstract: Truth as Process**

In this paper, we explore the Halting Problem through a novel lens, examining its implications and deconstructing its perceived paradoxes. The Halting Problem posits the impossibility of creating an algorithm that can determine whether any given process will terminate. By reframing the problem, we propose that such paradoxes arise more from the constraints of our logical tools and the historical context of mathematical thought, particularly influenced by Platonic ideals of perfection.

We argue that paradoxes like the Halting Problem reflect the limitations of human-designed tools rather than intrinsic logical impossibilities. By understanding logic as a negotiable framework akin to a game, we emphasize the flexibility of our intellectual constructs. This perspective allows for innovative approaches to seemingly intractable problems.

The concept of truth as a process is introduced, suggesting that solutions to problems like the Halting Problem can be dynamic rather than static. This approach aligns with practical scenarios in distributed computing and statistical confidence intervals, where truth evolves over time and context.

Ultimately, we advocate for a more adaptable view of mathematical truth, encouraging the development of tools that reflect the fluid nature of logical inquiry. This paradigm shift has the potential to enrich the field of mathematics and logic, fostering progress through a more pragmatic and less dogmatic approach to intellectual challenges.

Kolmogorov complexity is a theory based on the premise that the complexity of a binary string can be measured by its compressibility; that is, a string’s complexity is the length of the shortest program that produces that string. We explore applications of this measure to graph theory.

In a genetic algorithm, fluctuations of the entropy of a genome over time are interpreted as fluctuations of the information that the genome's organism is storing about its environment, being this reflected in more complex organisms. The computation of this entropy presents technical problems due to the small population sizes used in practice. In this work we propose and test an alternative way of measuring the entropy variation in a population by means of algorithmic information theory, where the entropy variation between two generational steps is the Kolmogorov complexity of the first step conditioned to the second one. As an example application of this technique, we report experimental differences in entropy evolution between systems in which sexual reproduction is present or absent.

This paper proposes a new musical notation with Lindenmayer grammars, and describes the use of grammar evolution for the automatic generation of music expressed in this notation, with the normalized compression distance as the fitness function. The computer music generated tries to reproduce the style of a selected pre-existent piece of music. In spite of the simplicity of the algorithm, the procedure obtains interesting results.

In this paper, we revisit a central concept in Kolmogorov complexity in which one would equate program-size complexity with information content. Despite the fact that Kolmogorov complexity has been widely accepted as an objective measure of the information content of a string, it has been the subject of many criticisms including the fundamental one directed by logicians and philosophers towards the statistical and semantical theories of information, which is about confusing an object and its name. In this paper, we clarify a number of subtle issues that are at the center of this debate.

In this paper, we revisit a central concept in Kolmogorov complexity in which one would equate program-size complexity with information content. Despite the fact that Kolmogorov complexity has been widely accepted as an objective measure of the information content of a string, it has been the subject of many criticisms including the fundamental one directed by logicians and philosophers towards the statistical and semantical theories of information, which is about confusing an object and its name. In this paper, we clarify a number of subtle issues that are at the center of this debate.

We give explanations on the differences between a number of representations that have been proposed in the literature to measure the structural complexity of an object. In particular, we propose the notion of process complexity, the notion of sophistication from a given perspective, the notion of sophisticated depth, and the notion of self-dissimilarity. We also propose the notion of an approximate process for an object and argue that for all practical purposes, an approximate process is as good as an exact process.

The artistic community is increasingly relying on automatic computational analysis for authentication and classification of artistic paintings. In this paper, we identify hidden patterns and relationships present in artistic paintings by analysing their complexity, a measure that quantifies the sum of characteristics of an object. Specifically, we apply Normalized Compression (NC) and the Block Decomposition Method (BDM) to a dataset of 4,266 paintings from 91 authors and examine the potential of these information-based measures as descriptors of artistic paintings. Both measures consistently described the equivalent types of paintings, authors, and artistic movements. Moreover, combining the NC with a measure of the roughness of the paintings creates an efficient stylistic descriptor. Furthermore, by quantifying the local information of each painting, we define a fingerprint that describes critical information regarding the artists' style, their artistic influences, and shared techniques. More fundamentally, this information describes how each author typically composes and distributes the elements across the canvas and, therefore, how their work is perceived. Finally, we demonstrate that regional complexity and two-point height difference correlation function are useful auxiliary features that improve current methodologies in style and author classification of artistic paintings. The whole study is supported by an extensive website (http://panther.web.ua.pt) for fast author characterization and authentication.

Coin tossing is traditionally viewed as a random event with equal probabilities for heads and tails. However, research suggests that under controlled conditions, the outcome can be predicted with high accuracy. This research proposal seeks to investigate the deterministic aspects of coin tosses by considering latent variables, referred to as Human Factors and Physics Factors, which if fully understood, could lead to near-certain
predictions of outcomes. We propose to develop a deep learning model to integrate these factors and predict coin toss outcomes.

Natural selection explains how life has evolved over millions of years from more primitive forms. The speed at which this happens, however, has sometimes defied formal explanations when based on random (uniformly distributed) mutations. Here we investigate the application of a simplicity bias based on a natural but algorithmic distribution of mutations (no recombination) in various examples, particularly binary matrices in order to compare evolutionary convergence rates. Results both on synthetic and on small biological examples indicate an accelerated rate when mutations are not statistical uniform but algorithmic uniform. We show that algorithmic distributions can evolve modularity and genetic memory by preservation of structures when they first occur sometimes leading to an accelerated production of diversity but also population extinctions, possibly explaining naturally occurring phenomena such as diversity explosions (e.g. the Cambrian) and massive extinctions (e.g. the End Triassic) whose causes are currently a cause for debate. The natural approach introduced here appears to be a better approximation to biological evolution than models based exclusively upon random uniform mutations, and it also approaches a formal version of open-ended evolution based on previous formal results. These results validate some suggestions in the direction that computation may be an equally important driver of evolution. We also show that inducing the method on problems of optimization, such as genetic algorithms, has the potential to accelerate convergence of artificial evolutionary algorithms.

We show how to determine the k-th bit of Chaitin's algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N. We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter k which has an odd number of solutions iff the k-th bit of Ω is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values iff the k-th bit of Ω is 1.

We present a new method for expressing Chaitin's random real, Ω, through Diophantine equations. Where Chaitin's method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be used to create polynomials which express the bits of Ω in the number of positive values they assume.

Kolmogorov's very first paper on algorithmic information theory (Kolmogorov, Problemy peredachi infotmatsii 1(1) (1965), 3) was entitled "Three approaches to the definition of the quantity of information". These three approaches were called combinatorial, probabilistic and algorithmic. Trying to establish formal connections between combinatorial and algorithmic approaches, we prove that every linear inequality including Kolmogorov complexities could be translated into an equivalent combinatorial statement. (Note that the same linear inequalities are true for Kolmogorov complexities and Shannon entropy, see Hammer et al., (Proceedings of CCC'97, Ulm).) Entropy (complexity) proofs of combinatorial inequalities given in Llewellyn and Radhakrishnan (Personal Communication) and Hammer and Shen (Theory Comput. Syst. 31 (1998) 1) can be considered as special cases (and a natural starting points) for this translation.

In an unpublished manuscript Turing anticipated by nearly thirty years the basic ideas behind the theory of algorithmic randomness, using a computationally constrained version of ‘measure theory’ to answer a question posed by Émile Borel in number theory: this question concerned constructing what are called ‘absolutely normal’ numbers. In this chapter we explain what these mysterious terms mean, and what Turing did. Mathematicians have always been fascinated with patterns in numbers. At an early stage in our education we learn about the special nature of decimal expansions of ‘rational numbers’, fractions that we can write in the form m/n, for some whole numbers m and n with n ≠ 0. The Greeks proved that some numbers, such as √2, 3√7 and √2 + √3 are not rational—indeed, it can be shown that ‘most’ numbers (in a precise mathematical sense) are irrational. It can be shown that a real number is rational if and only if it has a finite decimal expansion, or a decimal expansion that repea...

We prove that for any computably enumerable (c.e.) degree c, if it is cappable in the computably enumerable degrees, then there is a d.c.e. degree d such that c ∪ d = 0 and c ∩ d = 0. Consequently, a computably enumerable degree is cappable if and only if it can be complemented by a nonzero d.c.e. degree. This gives a new characterization of the cappable degrees.

The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Recent machine learning and AI advances disrupt scientific practice, technological innovation, product development, and society. As a rule, success in classification, pattern recognition, and gaming occurs whenever there are clear performance evaluation criteria and access to extensive training data sets. Yet, AI has contributed less to fundamental science, such as discovering new principled explanatory models and equations. To set the stage for a fundamental AI4Science, we explore a perspective for an AI-driven, automated, generative, closed-loop approach to scientific discovery, including self-driven hypothesis generation and open-ended autonomous exploration of the hypothesis space. Generative AI, in general, and Large Language Models (LLMs), in particular, serve here to translate and break down high-level human or machine conjectures into smaller computable modules inserted in the automated loop. Discovering fundamental explanatory models requires causality analysis while enabling unbiased efficient search across the space of putative causal explanations. In addition, integrating AI-driven automation into the practice of science would mitigate current problems, including the replication of findings, systematic production of data, and ultimately democratisation of the scientific process. These advances promise to unleash AI's potential for searching and discovering the fundamental structure of our world beyond what human scientists have achieved or can achieve. Such a vision would push the boundaries of new fundamental science beyond automatizing current workflows and unleash new possibilities to solve some of humanity's most significant challenges.

forms of Kolmogoroff’s complexity, Chaitin and G_del’s theorems are stated. They are used to analyze numerous methodological issues: Kant’s Third antinomy, Parkinson’s law of committee, cooperative creative activity, multilanguage programming, benevolence to other’s views, dilemma of deism–atheism.

This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently computable. We propose a variant to the original speed prior (Schmidhuber, 2002), and show that our prior can predict sequences drawn from probability measures that are estimable in polynomial time. Our speed prior is computable in doubly-exponential time, but not in polynomial time. On a polynomial time computable sequence our speed prior is computable in exponential time. We show better upper complexity bounds for Schmidhuber's speed prior under the same conditions, and that it predicts deterministic sequences that are computable in polynomial time; however, we also show that it is not computable in polynomial time, and the question of its predictive properties for stochastic sequences remains open.

We show how to determine the-th bit of Chaitin's algorithmically random real number by solving instances of the halting problem. From this we then reduce the problem of determining the-th bit of to determining whether a certain Diophantine equation with two parameters, and , has solutions for an odd or an even number of values of. We also demonstrate two further examples of in number theory: an exponential Diophantine equation with a parameter which has an odd number of solutions iff the-th bit of is 1, and a polynomial of positive integer variables and a parameter that takes on an odd number of positive values iff the-th bit of is 1.

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.

Abstract-We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where |S| ≤ 2 L(1−δ) , for sufficiently small δ there exists a polynomial (in 1/δ) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence t ∈ S, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[δ/ ln(1/δ)](1 − 2 ln ln(1/δ)/ ln(1/δ)). DOI: 10.1134/S0032946007010061 This paper answers a question stated in [1, Section 2.4]. Let us recall basic definitions and problem settings from [1] related to the considered problem. The problem itself will be formulated below. An arbitrary frequency test for binary sequences of length L is any set S of such sequences. The specif...

We describe the generation of sequences of random bits from the parity of photon counts produced by polarization measurements on a polarization-entangled state. The resulting sequences are bias free, pass the applicable tests in the NIST battery of statistical randomness tests, and are shown to be Borel normal, without the need for experimental calibration stages or postprocessing of the output. Because the photon counts are produced in the course of a measurement of the violation of the Clauser–Horne–Shimony–Holt inequality, we are able to concurrently verify the nonclassical nature of the photon statistics and estimate a lower bound on the min-entropy of the bit-generating source. The rate of bit production in our experiment is around 13 bits/s.

The artistic community is increasingly relying on automatic computational analysis for authentication and classification of artistic paintings. In this paper, we identify hidden patterns and relationships present in artistic paintings by analysing their complexity, a measure that quantifies the sum of characteristics of an object. Specifically, we apply Normalized Compression (NC) and the Block Decomposition Method (BDM) to a dataset of 4,266 paintings from 91 authors and examine the potential of these information-based measures as descriptors of artistic paintings. Both measures consistently described the equivalent types of paintings, authors, and artistic movements. Moreover, combining the NC with a measure of the roughness of the paintings creates an efficient stylistic descriptor. Furthermore, by quantifying the local information of each painting, we define a fingerprint that describes critical information regarding the artists' style, their artistic influences, and shared techniques. More fundamentally, this information describes how each author typically composes and distributes the elements across the canvas and, therefore, how their work is perceived. Finally, we demonstrate that regional complexity and two-point height difference correlation function are useful auxiliary features that improve current methodologies in style and author classification of artistic paintings. The whole study is supported by an extensive website (http://panther.web.ua.pt) for fast author characterization and authentication.

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $\mu$-recursive functions and give a certified compiler from $\mu$-recursive functions to Minsky machines.