Axiomatic Approach Research Articles

Analysis of Hamming and Hausdorff 3D distance measures for complex pythagorean fuzzy sets and their applications in pattern recognition and medical diagnosis

AbstractSimilarity measures are very effective and meaningful tool used for evaluating the closeness between any two attributes which are very important and valuable to manage awkward and complex information in real-life problems. Therefore, for better handing of fuzzy information in real life, Ullah et al. (Complex Intell Syst 6(1): 15–27, 2020) recently introduced the concept of complex Pythagorean fuzzy set (CPyFS) and also described valuable and dominant measures, called various types of distance measures (DisMs) based on CPyFSs. The theory of CPyFS is the essential modification of Pythagorean fuzzy set to handle awkward and complicated in real-life problems. Keeping the advantages of the CPyFS, in this paper, we first construct an example to illustrate that a DisM proposed by Ullah et al. does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Then, combining the 3D Hamming distance with the Hausdorff distance, we propose a new DisM for CPyFSs, which is proved to satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Moreover, similarly to some DisMs for intuitionistic fuzzy sets, we present some other new complex Pythagorean fuzzy DisMs. Finally, we apply our proposed DisMs to a building material recognition problem and a medical diagnosis problem to illustrate the effectiveness of our DisMs. Finally, we aim to compare the proposed work with some existing measures is to enhance the worth of the derived measures.

Reduction between Categorical Syllogisms Based on the Syllogism EIO-2

Syllogism reasoning is a common and important form of reasoning in human thinking from Aristotle onwards. To overcome the shortcomings of previous studies, this article makes full use of set theory and classical propositional logic, and deduces the remaining 23 valid syllogisms only on the basis of the syllogism EIO-2 from the perspective of mathematical structuralism, and then successfully establishes a concise formal axiom system for categorical syllogistic logic. More specifically, the article takes advantage of the trisection structure of categorical propositions such as Q(a, b), the transformation relations between an Aristotelian quantifier and its inner and outer negation, the symmetry of the two Aristotelian quantifier (that is, no and some), and some inference rules in classical propositional logic, and derives the remaining 23 valid syllogisms from the syllogism EIO-2, so as to realize the reduction between different valid categorical syllogisms.

Predictive ratio CUSUM (PRC): A Bayesian approach in online change point detection of short runs

The online quality monitoring of a process with low volume data is a very challenging task and the attention is most often placed in detecting when some of the underline (unknown) process parameter(s) experience a persistent shift. Self-starting methods, both in the frequentist and the Bayesian domain aim to offer a solution. Adopting the latter perspective, we propose a general closed-form Bayesian scheme, where the testing procedure is built on a memory-based control chart that relies on the cumulative ratios of sequentially updated predictive distributions. The theoretic framework can accommodate any likelihood from the regular exponential family and the use of conjugate analysis allows closed form modeling. Power priors will offer the axiomatic framework to incorporate into the model different sources of information, when available. A simulation study evaluates the performance against competitors and examines aspects of prior sensitivity. Technical details and algorithms are provided as supplementary material.

CONSTRUCTING NONSTANDARD HULLS AND LOEB MEASURES IN INTERNAL SET THEORIES

AbstractCurrently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

Characterizations of ordinal analysis

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question.First, we characterize ordinal analysis as a partition of Σ11-definable and Π11-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation ≡ is finer than the ordinal analysis partition if both: (1) T≡U whenever T and U prove the same Π11 sentences; (2) T≡T+U for every set U of true Σ11 sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.Second, we characterize ordinal analysis as an ordering on arithmetically-definable and Π11-sound theories, namely, the ordering wherein T⩽U if the proof-theoretic ordinal of T is less than or equal to the proof-theoretic ordinal of U. The standard ways of measuring the strength of theories are consistency strength and inclusion of Π10 theorems. We introduce analogues of these notions—Π11-reflection strength and inclusion of Π11 theorems—in the presence of an oracle for Σ11 truths, and prove that they coincide with the ordering induced by ordinal analysis.

Metacognition in Sebastian`s Door: mathematics, education and art

The aim is to show metacognition in how mathematics and culture are related in the sculptural work of Master Sebastián; with a deep playful character from which it emerges. Constituting a kind of artistic syncretism of apparently divergent beliefs between the plastic world, pure mathematics, abstract geometry and computational cybernetics. Methodology, study with a qualitative approach, the variables to consider were: Bach's baroque music with the canon and fugue, Gödel's theorem with the incompleteness of the axiomatic system; Eschery lithographs the reiteration of forms; Borje's literary-mathematical paradoxes; recursive programming routines in artificial intelligence systems (expert systems, mathematics) and Sebastián's sculpture. In conclusion, the convergence between Bach's baroque music can be identified, through the generation of the canon and the fugue; Gödel's theorem on the incompleteness of the axiomatic system, the reiteration of forms that invent themselves in Escher's lithographs, the literary-mathematical paradoxes in the stories of Jorge Luis Borges, and the use of recursive routines in the programming of expert systems or artificial intelligence. This game of plastic metamorphosis with which Sebastián surprised us a few years ago, turns out to predate the concept of the transformers, long before Hollywood made them a commercial success. As in the most diverse efforts of syncretism, Sebastián has traveled close to profanation as the staunch conservatives of the elite of scientists and artists can describe it.

Ethical allocation of scarce vaccine doses: The Priority-Equality protocol.

Whenever vaccines for a new pandemic or widespread epidemic are developed, demand greatly exceeds the available supply of vaccine doses in the crucial, initial phases of vaccination. Rationing protocols must then fulfill a number of ethical principles balancing equal treatment of individuals and prioritization of at-risk and instrumental subpopulations. For COVID-19, actual rationing methods used a territory-based first allocation stage based on proportionality to population size, followed by locally-implemented prioritization rules. The results of this procedure have been argued to be ethically problematic. We use a formal-analytical approach arising from the mathematical social sciences which allows to investigate whether any allocation methods (known or unknown) fulfill a combination of (ethical) desiderata and, if so, how they are formulated algorithmically. Strikingly, we find that there exists one and only one method that allows to treat people equally while giving priority to those who are worse off. We identify this method down to the algorithmic level and show that it is easily implementable and it exhibits additional, desirable properties. In contrast, we show that the procedures used during the COVID-19 pandemic violate both principles. Our research delivers an actual algorithm that is readily applicable and improves upon previous ones. Since our axiomatic approach shows that any other algorithm would either fail to treat people equally or fail to prioritize those who are worse off, we conclude that ethical principles dictate the adoption of this algorithm as a standard for the COVID-19 or any other comparable vaccination campaigns.

About Limitations of the Axiomatic and Economic Theories of Price Indices

The article considers provisions of axiomatic theory of Fisher indices, which is an important part of modern theory of price statistics and GDP international comparisons based on purchasing power parities. International recommendations concerning the computation of CPI and international comparisons of GDP, proposing different index formulas, are based on provisions of axiomatic theory although some require- ments for index formulas are not consistent with each other. The Fisher index formula meets time reversal and factor reversal requirements but fails to meet additivity and transitivity requirements and for these reasons cannot be used for multilateral comparisons of GDP. Most im- portantly, many experts in index theory believe that Fisher index does not have economic foundation. It is noted that axiomatic theory does not deal with many important problems of price statistics and represents the result of application of formal mathematical approach. Nevertheless, it is useful to take into account the provisions of this theory when interpreting recommended and published index formulas.

Consistency of Information About Type-2 Fuzzy Preference Relation

In this paper, an axiomatic approach to Pareto set reduction problem is considered. The reduction is based on accounting for preferences of a decision maker which are modeled with the use of a type-2 fuzzy binary relation. This relation is only partially known through a set of so-called information quanta. Since these quanta are provided by the decision maker, it must be verified that they are consistent with the requirements of the axiomatic approach — the axioms of rational choice. Several theorems giving necessary and sufficient conditions of quanta consistency are proved. Ideas for dealing with inconsistencies are discussed with a few examples.

Soundness and completeness results for LEA and probability semantics

Abstract In Artemov (2020, J. Logic Comput., 30, 61–76), a logical system called the logic of evidence aggregation (LEA) was introduced, along with an intended semantics for it called probability semantics. The goal was to describe probabilistic evidence aggregation in the setting of formal logic. However, as noted in that paper, LEA is not complete with respect to probability semantics. This leaves the tasks open to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. In this paper, we do both. We define a class of basic models called deductive basic models. We show LEA is sound and complete with respect to the class of deductive basic models. We also define an axiomatic system LEA$_+$ extending LEA and show it is sound and complete with respect to probability semantics. Along the way, we develop an intermediate system LEA$_-$ which, it turns out, is equivalent to propositional lax logic with a classical logic base. Sequent systems and decidability results for LEA$_-$ and LEA$_+$ are also presented.

An axiomatic approach to Markov decision processes

This paper presents an axiomatic approach to finite Markov decision processes where the discount rate is zero. One of the principal difficulties in the no discounting case is that, even if attention is restricted to stationary policies, a strong overtaking optimal policy need not exists. We provide preference foundations for two criteria that do admit optimal policies: 0-discount optimality and average overtaking optimality. As a corollary of our results, we obtain conditions on a decision maker’s preferences which ensure that an optimal policy exists. These results have implications for disciplines where dynamic programming problems arise, including automatic control, dynamic games, and economic development.

A General Approach To Chain Condition in BL-Algebras

In this paper, we present a general definition of the notion of Noetherian and Artinian BL-algebra and present a more comprehensive insight at the chain conditions in BL-algebras. We derive some theorems which generalize the existence results. We give an axiomatic approach to the notion of being Noetherian and Artinian, which is also applicable to other algebraic structures. We use a theoretical approach to define arithmetic notion that is also possible for other algebraic devices. In this study, we only focus to BL-algebras.

On the axiomatic theory of bargaining: a survey of recent results

On the axiomatic theory of bargaining: a survey of recent results

Matroids over skew tracts

Matroids over tracts provide an algebraic framework simultaneously generalizing the notions of matroids, oriented matroids, and valuated matroids, presented by Baker and Bowler. Pendavingh partially extended this theory to skew hyperfields and presented a new axiom system in terms of quasi-Plücker coordinates. We present a theory of matroids over skew tracts, which generalizes both the theory of matroids over tracts and the theory of weak matroids over skew hyperfields developed by Pendavingh. We give several cryptomorphic axiom systems for such matroids in terms of circuits, quasi-Plücker coordinates and dual pairs.

Ontology-based semantic retrieval of documents using Word2vec model

Semantic retrieval of engineering knowledge is crucial in various engineering activities, such as process development and product model planning. To make a solution for this issue, previously, a few word-based semantic enabling existing approaches such as Lexical Retrieval Model with Semantic Residual Embedding’s (LR-SRE), Document Retrieval Model through Semantic Linking (DR-MTS), Semantic Term Matching in Axiomatic Approaches to Information Retrieval (STM-IR) and Hybrid Ontology for Semantic Information Retrieval Model using Keyword Matching Indexing System (HOS-IR) were utilized. But, they all have shown less query-based accuracy results than the required value. In our proposed Information Retrieval (IR) design, the semantic knowledge-based retrieval scheme has been implemented. For query, entered by a user and processed for finding the dominated word. Word is then compared for its similarity equations, and similarity values are then computed to give output. Highly similar values are obtained as the class value. From class, the respective clusters are selected. Then, documents in that cluster are retrieved and ranked according to the relevance of the user. To support the accuracy level performance of this IR system, a word2vec model has been employed with the benefits of Horse Herd Optimization (HHO), which helps to extract vectors as features for classification. These results are stored as a .csv file for further retrieval. By implementing the proposed IR-word2vec model, the results showed that it outperforms other existing techniques by improved similarity index and accuracy for query results in an execution time of 1.7 s.

Putnam’s model-theoretic argument (meta)reconstructed

In “Models and Reality”, H. Putnam formulated his model-theoretic argument against “metaphysical realism”. The article proposes a meta-reconstruction of Putnam’s model-theoretic argument in the light of two mutually compatible interpretations of it–elaborated by Manuel Garcia-Carpintero and Igor van Douven. A critical reflection on these interpretations and their adequacy for Putnam’s argument allows us to expose new theses coherent with Putnam’s reasoning and indicate new paths to improve this argument for our reconstruction task. In particular, we show that Putnam’s position may be coherent with van Douven’s versions of Global Descriptivism under some conditions, but Putnam cannot reject realism as quickly as Carpintero suggests. We show that Suszko’s canonic axiomatic system and Sneed’s concept of theory may provide valuable support for Putnam’s argument. Finally, we critically evaluate Carpintero’s theses about the genesis of unintended interpretations of our languages, adopting the machinery of the Upward Skolem–Loewenheim Theorem and Knight’s Theorem.

Distances and Similarity Measures of Q-Rung Orthopair Fuzzy Sets Based on the Hausdorff Metric with the Construction of Orthopair Fuzzy TODIM

In recent years, q-rung orthopair fuzzy sets (q-ROFSs), a novel and rigorous generalization of the fuzzy set (FS) coined by Yager in 2017, have been used to manage inexplicit and indefinite information in daily life with a high precision and greater accuracy than intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). The characterization of a measure of similarity between q-ROFSs is important, as they have applications in different areas, including pattern recognition, clustering, image segmentation and decision making. Therefore, this article is dedicated to the construction of a measure of similarity between q-ROFSs based on the Hausdorff metric. This is a very useful tool for establishing the similarity between two objects. Furthermore, some axiomatic definitions of the distances and similarity measures of q-ROFSs are also presented. In this article, we first present a novel method to calculate the distance between q-ROFSs based on the Hausdorff metric. We then utilize our proposed distance measure to construct the degree of similarity between q-ROFSs. We provide some properties for the proposed similarity measures. We offer several numerical examples related to pattern recognition and characterization linguistic variables to demonstrate the usefulness of the proposed similarity measures. We construct an algorithm for orthopair fuzzy TODIM (interactive and multi-criteria decision making, in Portuguese) based on our proposed methods. Finally, we use the constructed orthopair fuzzy TODIM method to address problems related to daily life settings involving multi-criteria decision making (MCDM). The numerical results show that the proposed similarity measures are suitable, applicable and well-suited to the contexts of pattern recognition, queries with fuzzy linguistic variables and MCDM.

A Novel Software Trustworthiness Evaluation Strategy via Relationships between Criteria

Software trustworthiness evaluation is regarded as a multi-criteria decision-making problem. However, most current software trustworthiness evaluation methods do not consider the relationships between criteria. In this paper, we present a software trustworthiness evaluation strategy via the relationships between criteria. Because the trustworthy attribute degree is evaluated by a criterion, a trustworthy attribute measurement method based on fuzzy theory is first proposed, and the relationships between criteria are described by cooperative and conflicting degrees between criteria. Then, a measure formula for the symmetric substitutivity between criteria is proposed, and the cooperative degree between criteria is taken as the approximation of the symmetric substitutivity between criteria. With the help of the symmetric substitutivity between criteria, the software trustworthiness measurement model obtained by axiomatic approaches is applied to aggregate the degree to which each optional software product meets each objective. Finally, the candidate software products are sorted according to the trustworthiness aggregation results, and the optimal product is obtained from the alternative software products on the basis of the sorting results. The theoretical rationality of the measurement model is validated by proving that it satisfies the desirable properties of software trustworthiness measures, and its effectiveness is demonstrated through a case study.

An MDCM approach for risk assessment at the polar regions: analysing the antarctic route ice navigation risk parameters

ABSTRACT Maritime sector-based accidents are the source of major environmental disasters that place the sea environment at risk. It is critical to identify hazards and risk factors to avoid accidents and minimize losses. The importance of the Polar Regions in protecting the global ecosystem and nature has been identified as crucial. An environmental problem in the polar regions will affect the entire ecosystem. In this study, The Fuzzy-AHP approach was utilized to investigate the causes of the accidents and failures in polar regions. According to the findings, human error is the most crucial risk element, followed by technical, propulsion, radar detection failure, and improper watchkeeping technique risk factors. In the light of the findings, a suggested training course via an axiomatic design approach will reduce the ice navigation risk factors.

New constructions of decision evaluation functions in three-way decision spaces based on uninorms.

In 2014, Hu introduced the concept of three-way decision spaces and axiomatic definition of decision evaluation functions. In three-way decision spaces, decision evaluation function satisfies minimum element axiom, monotonicity axiom and complement axiom. Since then, the research on construction method of decision evaluation functions from commonly used binary aggregation functions becomes a research hotspot. Meanwhile, uninorms, as one class of binary aggregation functions, have been successfully applied in various application problems, such as in decision making, image processing, data mining, etc. This paper continues to consider this research topic and mainly explores the new construction methods of decision evaluation functions based on uninorms. Firstly, we show two novel transformation methods from semi-decision evaluation functions to decision evaluation functions based on uninorms. Secondly, using known semi-decision evaluation functions, we give some new construction methods of semi-decision evaluation functions. Thirdly, we give some novel construction methods of decision evaluation functions and semi-decision evaluation functions related to fuzzy sets, interval-valued fuzzy sets, fuzzy relations and hesitant fuzzy sets. Based on them, decision maker can obtain more useful decision evaluation functions, thereby more choices can be used for realistic decision-making problems. Finally, we consider two real evaluation problems to illustrate the results obtained in this paper. The three-way decisions results of evaluation problem show that the construction method proposed in this paper is superior to some existing construction methodsunder some conditions.