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Database Knowledge Discovery attention has been growing last decades using different approaches as part of a new era when information is multiplied in proportion and importance. Fuzzy Logic predicates approach is one of them, fundamental because of their interpretability properties. A new concept of transdisciplinary interpretability has been introduced by using a new axiomatic approach: Compensatory Fuzzy Logic. Several ways have been used as fuzzy predicates searching techniques, notably a Genetic Algorithm, part of a Data Analysis Platform called Eureka Universe. This paper presents two Genetic Programming Algorithm Approaches, with outstanding results and illustrated by a case study.

Hilbert’s axiomatic approach to the sciences was characterized by a dynamic methodology tied to scientific and mathematical fields under investigation. In particular, it is an analytic art for choosing axioms but, at the same time, it has to include dynamically synthetic procedures and meta-theoretical reflections. Axioms have to be useful, or capture something, or help as part of explanations. The Andreka-Nemeti group use several formal axiomatic theories together to re-capture, predict, recover or explain the phenomena of special relativity, general relativity and classical kinematics. Their scientific methodology is close to Hilbert’s conceptions of how science should be done ideally. In this paper, we compare Formica’s reading of Hilbert’s axiomatic method to the method used by the Andreka-Nemeti group.

We formalize soundness and completeness proofs for a number of axiomatic systems for propositional logic in the proof assistant Isabelle/HOL.

The design of modern cloud computing makes available a plethora of scalable cloud computing offerings. The cloud is increasingly becoming the backbone of the highly complex modern knowledge economy that includes social, mobile, IoT, Big Data, and AI. Knowledge-based products and services follow fat-tail distributions such as the power law that poses major opportunities and challenges for the designer. The Axiomatic Designer is uniquely positioned in designing for the de novo situations that the fat-tailed distributions expose. Also, the cloud frees up the architectural decision-making away from the legacy compatibility-burden, and toward various cloud-native (i.e., de novo/solution-neutral) as well as hybrid (on-prem/cloud and cloud/cloud) architectures. Furthermore, the competitive landscape around the cloud is not static; it is adaptive and evolving rapidly. Here again, Axiomatic Design (AD) is uniquely positioned in rising up to the various de novo challenges. This, however, requires contributions from frameworks such as knowledge as heterarchically hierarchical (KA|h|H), stigmergy, complex adaptive systems (CAS), Cynefin, Boyd’s OODA loop theory of asymmetric fast transients, axiomatic maturity diagram (AMD), as well as Weick’s loose-coupling approach to help unify and strengthen the axiomatic approach. This chapter unifies the above approaches in order to tackle the architectural challenges of cloud computing.

How do we discover and justify axioms of mathematics? In view of the long history of the axiomatic method, it is rather embarrassing that we are still lacking a standard answer to this simple question. Since the axiom of choice is arguably one of the most frequently discussed famous axioms throughout the history of mathematics, Thomas Forster’s recent identification of the axiom as an inference to the best explanation (IBE) provides us with a nice point of departure. I will argue that, by separating sharply between abduction and IBE, we can give a convincing account of both the discovery and the justification of the axioms of mathematics.

The milq approach to quantum physics for high schools focuses on the conceptual questions of quantum physics. Students should be given the opportunity to engage with the world view of modern physics. The aim is to achieve a conceptually clear formulation of quantum physics with a minimum of formulas. In order to provide students with verbal tools, they can use in discussions and argumentations we formulated four “reasoning tools.” They help to facilitate qualitative discussions of quantum physics, allow students to predict quantum mechanical effects, and help to avoid learning difficulties. They form a “beginners’ axiomatic system” for quantum physics.

Reverse constructive mathematics, based on the pioneering work of Kleene, Vesley, Kreisel, Troelstra, Bishop, Bridges and Ishihara, is currently under development. Bishop constructivists tend to emulate the classical reverse mathematics of Friedman and Simpson. Veldman’s reverse intuitionistic analysis and descriptive set theory split notions in the style of Brouwer. Kohlenbach’s proof mining uses interpretations and translations to extract computational information from classical proofs. We identify the classical content of a constructive mathematical theory with the Gentzen negative interpretation of its classically correct part. In this sense HA and PA have the same classical content but intuitionistic and classical two-sorted recursive arithmetic with quantifier-free countable choice do not; \(\varSigma ^0_1\) numerical double negation shift expresses the precise difference. Other double negation shift and weak comprehension principles clarify the classical content of stronger constructive theories. Any consistent axiomatic theory S based on intuitionistic logic has a minimum classical extension S\(^{+g}\), obtained by adding to S the negative interpretations of its classically correct consequences. Subsystems of Kleene’s intuitionistic analysis and supersystems of Bishop’s constructive analysis provide interesting examples, with the help of constructive decomposition theorems.

When presented with two fully proved formal methods-based specifications, how can a System Engineer decide which is superior when both models specify the same requirements, but in two different ways? This paper investigates and propose a methodology by which formal methods (using the specific example of the Event-B notation) can be differentiated in terms of their quality, using criteria that may be highly subjective in nature. Established complexity functions applied to software are not applicable to formal methods, thus the paper proposes a new function which quantifies the “quality” of a given model. Complexity is not the only factor involved in determining the quality of formal methods, the quality of system thinking involved also play an impactful role. We propose a quality function which uses the well-established properties of axiomatic systems in theoretical mathematics with the addition of a specifically formulated complexity function. The distinction criteria are based on evaluating how four main properties have been achieved: “Consistency”, “Completeness”, “Independence” and “Complexity”. We base our approach according to the paradigm of; “if the formal specification looks visually complicated for a set-theory novice, then it is a poorly modeled specification”. Furthermore, we explore the notion of Miller’s rule (magic No. 7) to define what “good” should look like. We conclude that we need more than Miller’s 7, we need 1, 2 and 3 to help us with defining what good quality looks like, by taking human cognitive capacity as a benchmark. This novel approach implies considerable further research, described in future work section.

Coming system - informational culture supersedes industrial era and its ergonomics incapable to find time for dynamics of professional requirements sophistication. Inter-disciplinary activity in computer networks leans on education universality necessary for meanings understanding. Subject’s thinking-consciousness exists in strain of discoordination with up-to-date rational noosphere not being trained relatively trans-fundamental speculation ability. Proto-phenomenon of man is endowed by spontaneous integrity and viability allowing adaptation to needed transcendental self-developing processes. Fundamental evolvement of mentality can be achieved in anthropogenic info-sphere of cypher galaxy only in partnership with deep-learned artificial intelligence (DL IA). To accomplish rational auto-poiesis, image of integrity mechanism is to be applied by DL IA to untwist ergo-mind subjective objectization up to functional systems of ego-mind. Cogno-ontological knowledge base and system axiomatic method are to be used. The technology displays innate universal mathematical hardware for ergo-mind speculation.

This chapter outlines an axiomatic system called L. The language of L is L−, the fragment of L whose logical constants are ∼ and ⊃. So, L may be regarded as an axiomatic version of G−, the poor cousin of G considered in Sect. 8.5.

Many-sorted theory for Information retrieval were proposed to provide a generic model of the different items of an information Retrieval system, the IR process is then seen as a proof of theorems based on a set of five axioms. The authors have proved that three of the classical IR models satisfy the theory: boolean, vector space and fuzzy-set models. However, it is not proved that the theory satisfies the modal-logic based IR models. In this paper we propose a new interpretation of the theory, this interpretation is based on possible worlds semantics where a document is a possible world and the accessibility relation is the transformation of the document. It will be proved that the interpretation proposed satisfies the many-sorted theory for IR, thus consolidating its consistency and genericity.

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, in our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

Ranking rules are a crucial component of every tournament design. This chapter argues for a better consideration of theoretical results in the construction of ranking criteria. Our axiomatic approach provides insight into three fields: points scoring systems applied to determine the outcome of championships including multiple competitions, ranking in Swiss-system tournaments where the strength of opponents has to be taken into account, and tie-breaking criteria in round-robin tournaments. A method is proposed to decide the final ranking in a round-robin league if the season is stopped and cannot resume. The previous FIFA World Ranking, used between 2006 and 2018, illustrates how many ways a rating formula can go awry.

This chapter sets out an axiomatic system in Lq called Q. The axioms of Q are all the formulas of Lq that instantiate the following schemas

Our 50-years scientific journey is debriefed with propositions for the future design of a long-term longitudinal study, comments on conceptualization and measurements, and a balance sheet of the MTSFGCLS replications. Then we look forward with the formulation of a discursive and axiomatic theory of the development of antisocial behavior.

We intend to construct these geometries using a slightly modified Hilbert’s axioms system in the same way as it is done in [7–10]. An interesting thing is related to the fact that it exists a common part for Euclidean and Non-Euclidean Geometry, the so called Absolute Geometry. Roughly speaking, the Absolute Geometry consists in all theorems that can be thought and proved using the axiomatic system before introducing a parallelism axiom.

A modeloid, a certain set of partial bijections, emerges from the idea to abstract from a structure to the set of its partial automorphisms. It comes with an operation, called the derivative, which is inspired by Ehrenfeucht-Fraisse games. In this paper we develop a generalization of a modeloid first to an inverse semigroup and then to an inverse category using an axiomatic approach to category theory. We then show that this formulation enables a purely algebraic view on Ehrenfeucht-Fraisse games.

The growing explosion of ideas such as Artificial Intelligence (AI), smart environments and ubiquitous computing has led to the creation of the Ambient Intelligence (AmI) paradigm. As AmI begins to take place, moving from a futuristic idea to a reality, we are gradually witnessing the creation of an omnipresent, responsive, and intelligent atmosphere in which thousands of tiny sensors and natural user interfaces will be embedded in our natural movements and in our social and physical interactions. Hence, a key challenge in this multi-disciplinary approach is to get machines to act according to ethical priorities that make sense to human beings. In this study, we improve the capacity for machine ethics to approach human ethics by assessing the computation of transaction values and we argue that it is possible to perform such a computation using recent work that describes the effects of human decision-making using an axiomatic framework. This paper clarifies the relationship between the brain’s 3-axes of neuroscience, the 3 Plato’s Transcendentals of philosophy and the biological evolution’s 3-components, as well as the top-down vs. bottom-up approaches to machine ethics.

We characterize four types of agentive knowledge using a stit semantics over branching discrete-time structures. These are ex ante knowledge, ex interim knowledge, ex post knowledge, and know-how. The first three are notions that arose from game-theoretical analyses on the stages of information disclosure across the decision making process, and the fourth has gained prominence both in logics of action and in deontic logic as a means to formalize ability. In recent years, logicians in AI have argued that any comprehensive study of responsibility attribution and blameworthiness should include proper treatment of these kinds of knowledge. This paper intends to clarify previous attempts to formalize them in stit logic and to propose alternative interpretations that in our opinion are more akin to the study of responsibility in the stit tradition. The logic we present uses an extension with knowledge operators of the Xstit language, and formulas are evaluated with respect to branching discrete-time models. We also present an axiomatic system for this logic, and address its soundness and completeness.

Kleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is in fact inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters.

This chapter presents several techniques allowing agents to come up with an agreement. We start by discussing negotiation among two agents: after having recalled the axiomatic approach of Nash, we present a standard protocol, and point to recent advances in the field. We then discuss issues raised in the multilateral case. Finally, we conclude the chapter by describing an example of persuasion-based negotiation, where agents can put forward justifying reasons through the negotiation, so as to possibly modify preferences over offers or more generally, influence the negotiation process.

Most research on concurrency involves either communication-based approaches or accepts the shared-variable model. This paper addresses the latter approach and traces the research from Hoare’s axiomatic approach, through Sue Owicki’s work up to separation logics and rely/guarantee methods. Researchers in these last two approaches have been involved in a friendly rivalry and cooperation. The focus is on the insights that have arisen rather than technical details.

Are people intuitions regarding the conceptual notion of information in agreement with the properties of measures that are common in the information theory literature? We capture the abstract notion of “informative mechanism” by a simple model of broadcasting a ’signal’ that is associated with a single binary ’secret’. The mechanism is more informative when there is a larger distance between the two distributions of the signal conditioned by each value of the secrets. We chose the f-divergence as a common metric for this distance. We consider an axiomatic model that characterize f-divergence orders, and conduct an empirical study to test whether each of the axioms in this model are in agreement with human preferences. As the main result of this study, we identify three monotone increasing relations between someone’s scores of three psychological tests (internal, powerful others, and chance locus of control) and the number of axioms that are in agreement with his preferences.

Time-lapse microscopy is a primary experimental tool for biologists to study development: the dynamic process by which an entire organism forms from an individual cell. The domain of these cellular dynamics is quite complex, and thus, demands a conceptual and computational architecture to support the integration of knowledge obtained across experiments and theories. In previous work, we have addressed the conceptual level and developed an axiomatic theory of cellular genealogies. In this work, we will address the other fundamental part of theory formation: the experimental level, where we have to deal with actual observations and discoveries. In the case of experiments from time-lapse microscopy, we need to go from the individual images taken at discrete time points to a full conceptual description of the underlying continuous cellular processes. In this work, we take a first step to bridge the general theory T(CO) and the experimental level by investigating individual cases. Any time-lapse experiment is linked to a real spatiotemporal genealogy, and we assume that these entities are particular instances of the general theory. We will investigate how this individual experimental information can be organised and represented.

In the last chapter we made the assertions that the statements The number \({2}^{n-1}({2}^{n} - 1)\) is a perfect number for each positive integer n for which 2 n − 1 is a prime number. If 2 n − 1 is a prime number for some positive integer n, then n is a prime number. are true, and we promised arguments that demonstrate them without any doubt. We will provide these in this chapter.

The paper studies the major reason for the contradictions in the Kleinberg’s axiomatic system for clustering [9]. We found that the so-called consistency axiom is the single source of problems because it creates new clusters instead of preserving the existent ones. Furthermore, this axiom contradicts the practice that data to be clustered is a sample of the actual population to be clustered. We correct this axiom to fit this requirement. It turns out, however, that the axiom is then too strong and implies isomorphism. Therefore we propose to relax it by allowing for centric consistency and demonstrate that under centric consistency, the axiomatic framework is not contradictory anymore. The practical gain is the availability of true cluster preserving operators.

The end of the nineteenth century and the beginning of the twentieth century marked a turning point in the way mathematicians approach their discipline. Even more than the attempts to refound the corpus on the basis of set theory, the successes of the axiomatic method challenged the traditional conception of the meaning of mathematical statements, which had hitherto been largely based on a spontaneous belief in an adequacy between these statements and their empirical and physical correlates.

Trans-disciplinary activity in system-informational culture (SIC) caused great knowledge sophistication and necessity for any person to have synthesis of true scientific presentations. Complication of informational flows satiated with scientific meanings needs their co-processing with the help of deep-learned artificial intelligence (DL IA). Artificial intelligence (IA) will assist man to identify universalities for third world understanding. Otherwise, it will be impossible to live comfortably in computer instrumental systems and its applications. Arising intellectual difficulties will alter significantly SIC subject armed with DL IA − powerful means of learning, cognition, and world study. Trained rational consciousness allows achieving semantic level of communication in SIC. In its work, DL IA leans on system axiomatic method and personal cogno-ontological knowledge base descript in language of categories. Examples explain contributed technology.

Networks and informational technologies caused mammoth growth of knowledge volume and sophistication. Serious cognitive and educational problems of scientific big data understanding have arisen. The part and parcel of coming system-informational culture is inter-disciplinary activity based on labor with meanings ensuring knowledge integrity and obviousness. Presented by the utmost mathematical abstractions they are formed by genus in natural science knowledge. Cogno-ontological knowledge base (CogOnt) as core of these scientific presentations consists of them. Achieving personal transcendentality by means of self-reflection secures cognogenesis success. Man’s life-long partnership technology with deep-learned artificial intelligence (DL IA) is contributed to do it. In order to carry out subject’s auto-poiesis (rational self-building) it is impossible to do without continuous assistance of DL IA using CogOnt and applying axiomatic method. Elaboration of CogOnt for DL IA implantation is displayed on example of some universalities study.

In various situations, decision makers face experts that may provide conflicting advice. This advice may be in the form of probabilistic forecasts over critical future events. We consider a setting where the two forecasters provide their advice repeatedly and ask whether the decision maker can learn to compare and rank the two forecasters based on past performance. We take an axiomatic approach and propose three natural axioms that a comparison test should comply with. We propose a test that complies with our axioms. Perhaps, not surprisingly, this test is closely related to the likelihood ratio of the two forecasts over the realized sequence of events. More surprisingly, this test is essentially unique. Furthermore, using results on the rate of convergence of supermartingales, we show that whenever the two experts’ advice are sufficiently distinct, the proposed test will detect the informed expert in any desired degree of precision in some fixed finite time.

A given order type in the plane can be represented by a corresponding point set. However, it might be difficult to recognize the orientations of some point triples. Recently, Aichholzer et al. [3] introduced exit graphs for visualizing order types in the plane. We present a new class of geometric graphs, called OT-graphs, using abstract order types and their axioms described in the well-known book by Knuth [15]. Each OT-graph corresponds to a unique order type. We develop efficient algorithms for recognizing OT-graphs and computing a minimal OT-graph for a given order type in the plane. We provide experimental results on all order types of up to nine points in the plane including a comparative analysis of exit graphs and OT-graphs.

The idea that science can be automated is so deeply related to the view that the method of mathematics is the axiomatic method, that confuting the claim that mathematical knowledge can be extended by means of the axiomatic method is almost equivalent to confuting the claim that science can be automated. I argue that the axiomatic view is inadequate as a view of the method of mathematics and that the analytic view is to be preferred. But, if the method of mathematics and natural sciences is the analytic method, then the advancement of knowledge cannot be mechanized, since non-deductive reasoning plays a crucial role in the analytic method, and non-deductive reasoning cannot be fully mechanized.

Wallis’s attempted proof of Euclid’s Parallel Postulate is an important but oft neglected event leading to the discovery of non-Euclidean geometries. Our aim here is to show Wallis’s own reliance on three non-constructive diagrammatic inferences that are not (fully) explicit in his own supplement to Euclid’s axioms. Namely, there is i- an implicit assumption concerning the possibility of motion; ii- an implicit assumption about the continuous nature of space and time; and iii- an explicit assumption about the existence of similar triangles which conceals an appeal to a combinatoric principle of reasoning that is tantamount to appealing to the Axiom of Choice.

Endowing robots with the capability of assessing risk and making risk-aware decisions is widely considered a key step toward ensuring safety for robots operating under uncertainty. But, how should a robot quantify risk? A natural and common approach is to consider the framework whereby costs are assigned to stochastic outcomes–an assignment captured by a cost random variable. Quantifying risk then corresponds to evaluating a risk metric, i.e., a mapping from the cost random variable to a real number. Yet, the question of what constitutes a “good” risk metric has received little attention within the robotics community. The goal of this paper is to explore and partially address this question by advocating axioms that risk metrics in robotics applications should satisfy in order to be employed as rational assessments of risk. We provide instantiations of the class of risk metrics that satisfy these axioms (referred to as distortion risk metrics) and also discuss pitfalls of commonly used risk metrics in robotics that do not satisfy these axioms. Our hope is that the ideas presented here will lead to a foundational framework for quantifying risk (and hence safety) in robotics applications.

The gradual transformation of the regional economy has led to the creation of innovative systems. Innovative systems are an important link in the modification of the regional economy. Firstly, innovative systems form the technological economy. Secondly, innovative systems provide the necessary level of development of the region. These circumstances shape the relevance of the topic of a scientific article. The purpose of the scientific article is to consider the ways of formation and development of regional innovation systems. The objectives of a scientific article are to examine the methodological level of research of regional innovation systems; to define the definitional gradation of the notion “regional innovation system”; to identify the features of the regional innovation system; to suggest ways of forming regional innovation systems; to form the ways of implementation of regional innovation systems. The methodical research tools based on basic and additional scientific methods. The basic scientific methods include the definition and bibliographic method, the axiomatic method, the method of species structurization, the method of implication of circumstances. The additional scientific methods include the method of conceptual gradation, the method of systematic exposure, and the method of complex energy.

System-informational culture (SIC) phenomenology impels human to work in sophisticated scientific space of computer models. Applying computer instrumental systems one has to investigate and compare different fields of knowledge suffering constant cognitive, educational, and intellectual problems. Inter-discipline activity in SIC leans on meanings understanding presented in the utmost mathematical abstractions (UMA). Work in SIC era unites cognition, education, and scientific research. SIC entelechies are to evolve rational part of consciousness. The objective is achievable by means of purposeful labor assisted by deep-learned artificial intelligence (DL IA). Technology is contributed allowing consciousness double helix auto-moulding in order to solve universalities problem. DL IA is to unwind intellectual processes and develop person’s scope of life. System axiomatic method is applied to coordinatization method and continuity property investigation.

Need for automatic creation of genealogical models rises as the amount of digitalized and transcribed record from historical sources. Usually it is necessary to distinguish what is genealogical model by an axiomatic system. Automatically created models need not to fulfil the axioms when record is wrongly interconnected with other records. If we are capable to decide whether a model is a genealogical model, we can better produce models automatically from a set of transcribed parish registers record. In this paper we introduce an algorithmic approach to creation of possible models of a parish society in the middle Europe from a set of transcribed baptism, marriage and burial records. It is shown that probabilistic estimation of person interconnection from different records could be helpful for reconstruction of a societies through several centuries.

In this note, I advocate the development of a symbiotic relationship between the theoretical world of economic design and that of quantitative data analysis. I illustrate both directions of this symbiosis by looking at how the axiomatic approach of social choice theory can provide foundations for the quantitative methodological choices and, how data science tools can help testing assumptions of formal models. For the former, I focus on the two important stages of any empirical research design: measurement and methods selection. For the latter, I assert that machine learning algorithms can be employed to test assumptions regarding to the nature of individual preferences. The spatial model of electoral competition serves as a workhorse example throughout the paper.