New Set Of Axioms Research Articles

Expanding and confusing space of alternatives: A case for lexicographic preferences

This paper characterizes a general class of lexicographic preferences with a new set of axioms on the product topological space of attributes where each attribute is a topological space generated by a linear order. The classes of preferences that the paper characterizes are semi-lexicographic and uniformly semi-lexicographic. Specifically, a transitive uniformly semi-lexicographic preference is lexicographic. We also provide behavioral motivations for our axioms.

Hopf algebroids with balancing subalgebra

Recently, S. Meljanac proposed a construction of a class of examples of an algebraic structure with properties very close to the Hopf algebroids H over a noncommutative base A of other authors. His examples come along with a subalgebra B of H⊗H, here called the balancing subalgebra, which contains the image of the coproduct and such that the intersection of B with the kernel of the projection H⊗H→H⊗AH is a two-sided ideal in B which is moreover well behaved with respect to the antipode. We propose a set of abstract axioms covering this construction and make a detailed comparison to the Hopf algebroids of Lu. We prove that every scalar extension Hopf algebroid can be cast into this new set of axioms. We present an observation by G. Böhm that the Hopf algebroids constructed from weak Hopf algebras fit into our framework as well. At the end we discuss the change of balancing subalgebra under Drinfeld-Xu procedure of twisting of associative bialgebroids by invertible 2-cocycles.

A new axiomatic approach to the impartial nomination problem

In this paper we introduce a new set of axioms that characterize uniform random dictatorship (URD) as a randomized impartial nomination rule. Unlike earlier work we use a variable population model which allows us to employ axioms that reflect consistency and proportionality–axioms that ensure that the rule behaves well with respect to the combinatorial structure of the nomination profile. Earlier work characterizing URD employed strong symmetry axioms and so it is surprising that our axioms characterize the same rule.

A Hodge theoretic extension of Shapley axioms

Lloyd S. Shapley \cite{Shapley1953a, Shapley1953} introduced a set of axioms in 1953, now called the {\em Shapley axioms}, and showed that the axioms characterize a natural allocation among the players who are in grand coalition of a {\em cooperative game}. Recently, \citet{StTe2019} showed that a cooperative game can be decomposed into a sum of {\em component games}, one for each player, whose value at the grand coalition coincides with the {\em Shapley value}. The component games are defined by the solutions to the naturally defined system of least squares linear equations via the framework of the {\em Hodge decomposition} on the hypercube graph. In this paper we propose a new set of axioms which characterizes the component games given by Stern and Tettenhorst, thereby suggesting that the component values for every coalition state may also serve for a valid measure of fair allocation among the players in each coalition. Our axioms may be seen as a completion of Shapley's in view of this characterization of the Hodge-theoretic component games. In addition, we provide a path integral representation of the component games which may be seen as an extension of the {\em Shapley formula}.

Methods for Domain Specialization of Verification-Oriented Process Ontologies

User-friendly formal specification and verification of concurrent and distributed systems for various domains, such as automatic control systems, telecommunications, and business processes, are active research topics due to their practical significance. In this paper, we present methods of developing verification-oriented domain-specific process ontologies used to describe concurrent systems of subject domains. One of the advantages of such ontologies is their formal semantics, which provides formal verification of the described systems. Our method is based on an abstract verification-oriented process ontology. We use two methods of domain specialization of the abstract process ontology. The declarative method relies on specializing classes of the original ontology, introducing new declarative classes, and using a new set of axioms to set restrictions on classes and relations of the abstract ontology. The constructive method uses semantic markup and pattern matching techniques to link domain concepts to classes of the abstract process ontology. We provide detailed ontological specifications for these techniques. Our methods preserve the formal semantics of the original process ontology; therefore, formal verification methods can be applied to the resulting domain-specific process ontologies. We demonstrate that the constructive method is a refined version of the declarative method. We illustrate our methods on the example of constructing an ontology for standard elements of automatic control systems: we develop declarative descriptions of the classes and restrictions of the domain-specific ontology in the Protege system in the Web Ontology Language (OWL) using inference rules written in the Semantic Web Rule Language (SWRL) and construct a system of semantic markup patterns that implements standard elements of automatic control systems.

A novel decision-making method using R-Norm concept and VIKOR approach under picture fuzzy environment

The Picture Fuzzy Sets (PFS) are well suitable to capture inconsistent, imprecise and uncertain information in multiple-criteria decision-making problems. This communication is intended to introduce one such information measure defined on PFSs called R-norm picture fuzzy information measure. Moreover, a new set of axioms is proposed as a criteria for picture fuzzy entropy. Besides establishing the validity of proposed R-norm picture fuzzy information measure, some of its major properties are also discussed. In application part, the proposed information measure is applied in predicting the outcome of elections in a poll bound country through opinion polls and to solve an investment problem.

Integrable Floquet QFT: Elasticity and factorization under periodic driving

In (1+1)-dimensional quantum field theory, integrability is typically defined as the existence of an infinite number of local charges of different Lorentz spin, which commute with the Hamiltonian. A well known consequence of integrability is that scattering of particles is elastic and factorizable. These properties are the basis for the bootstrap program, which leads to the exact computation of S-matrices and form factors. We consider periodically-driven field theories, whose stroboscopic time-evolution is described by a Floquet Hamiltonian. It was recently proposed by Gritsev and Polkovnikov that it is possible for some form of integrability to be preserved even in driven systems. If a driving protocol exists such that the Floquet Hamiltonian is integrable (such that there is an infinite number of local and independent charges, a subset of which are parity-even, that commute with it), we show that there are strong conditions on the stroboscopic time evolution of particle trajectories, analogous to S-matrix elasticity and factorization. We propose a new set of axioms for the time evolution of particles which outline a new bootstrap program, which can be used to identify and classify integrable Floquet protocols. We present some simple examples of driving protocols where Floquet integrability is manifest; in particular, we also show that under certain conditions, some integrable protocols proposed by Gritsev and Polkovnikov are solutions of our new bootstrap equations.

Knowledge measure for interval-valued intuitionistic fuzzy sets and its application to decision making under uncertainty

A measure of knowledge should not be viewed simply as a dual measure of entropy in the context of intuitionistic fuzzy sets as there is no natural logic between these two kinds of measures with the introduction of hesitancy, nor in the context of interval-valued intuitionistic fuzzy sets (IVIFSs), for that matter. This work is devoted to the introduction of an axiomatic definition of knowledge measure for IVIFSs. In order to do that, a set of new axioms is presented with which knowledge measure should comply in the context of IVIFSs. A concrete model following these axioms is then developed to measure the amount of knowledge associated with an IVIFS. Two facets of knowledge associated with an IVIFS, i.e., the information content and the information clarity, are simultaneously taken into account in the construction of the model to truly reflect the nature of an IVIFS. In particular, the connection between knowledge measure and fuzzy entropy is investigated under this axiomatic framework. A series of tests is also provided to examine the performance of the developed measure. Finally, a concept of knowledge weight on attribute in multi-attribute decision making is presented and an illustrative example is used as a demonstration of the application of the developed technique to decision making under uncertainty.

Knowledge measure for intuitionistic fuzzy sets with attitude towards non-specificity

This paper is devoted to an entropy-independent measure of knowledge in the context of intuitionistic fuzzy sets (IFSs). We point out and justify that there are at least two facets of knowledge associated with an IFS, namely the information content and the information clarity. Having this in mind, we put forward a novel axiomatic definition of knowledge measure for IFSs. More specifically, a set of new axioms is presented with which knowledge measure should comply in the context of IFSs. A parametric model following these axioms is then developed to realize this measure. Both facets mentioned above are simultaneously taken into account in the axioms and the model so as to better capture the unique features of an IFS. In particular, we suggest the concept of the amount of potential knowledge related to the hesitancy or non-specificity of an IFS. This allows us to introduce the idea of an attitudinal-based knowledge measure for IFSs. We believe that the knowledge measure provided in this manner could truly reflect the nature of IFSs, and what people really want with different attitudes towards the unknown. Finally, a practical application of the developed technique to decision making under uncertainty is illustrated. As such, the developed measure could be considered as a safe and effective alternative to help tackle some special problems that are difficult to handle by using entropy alone, especially when dealing with the complex situation in which different attitudes of users have to be considered.

Edge contraction on dual ribbon graphs and 2D TQFT

We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A⁎. We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A.

A new knockout tournament seeding method and its axiomatic justification

A new set of axioms and new method (equal gap seeding) are designed. The equal gap seeding is the unique seeding that, under the deterministic domain assumption, satisfies the delayed confrontation, fairness, increasing competitive intensity and equal rank differences axioms. The equal gap seeding is the unique seeding that, under the linear domain assumption, maximizes the probability that the strongest participant is the winner, the strongest two participants are the finalists, the strongest four participants are the quarterfinalists, etc.

Optimising resolution-based rewriting algorithms for OWL ontologies

An important approach to query answering over OWL ontologies is via rewriting the input ontology (and query) into a new set of axioms that are expressed in logics for which scalable query answering algorithms exist. This approach has been studied for many important fragments of OWL like SHIQ, Horn-SHIQ, OWL 2 QL, and OWL 2 EL. An important family of rewriting algorithms is the family of resolution-based algorithms, mostly because of their ability to adapt to any ontology language (such algorithms have been proposed for all aforementioned logics) and the long years of research in resolution theorem-proving. However, this generality comes with performance prices and many approaches that implement algorithms that are tailor-made to a specific language are more efficient than the (usually) general-purposed resolution-based ones.In the current paper we revisit and refine the resolution approaches in order to design efficient rewriting algorithms for many important fragments of OWL. First, we present an algorithm for the language DL-LiteR,⊓ which is strongly related to OWL 2 QL. Our calculus is optimised in such a way that it avoids performing many unnecessary inferences, one of the main problems of typical resolution algorithms. Subsequently, we extend the algorithm to the language ELHI which is strongly related to OWL 2 EL. This is a difficult task as ELHI is a relatively expressive language, however, we show that the calculus for DL-LiteR,⊓ requires small extensions. Finally, we have implemented all algorithms and have conducted an extensive experimental evaluation using many well-known large and complex OWL ontologies. On the one hand, this is the first evaluation of rewriting algorithms of this magnitude, while, on the other hand, our results show that our system is in many cases several orders of magnitude faster than the existing systems even though it uses an additional backwards subsumption checking step.

Optimising Resolution-Based Rewriting Algorithms for OWL Ontologie

An important approach to query answering over OWL ontologies is via rewriting the input ontology (and query) into a new set of axioms that are expressed in logics for which scalable query answering algorithms exist. This approach has been studied for many important fragments of OWL like SHIQ, Horn-SHIQ, OWL 2 QL, and OWL 2 EL. An important family of rewriting algorithms is the family of resolution-based algorithms, mostly because of their ability to adapt to any ontology language (such algorithms have been proposed for all aforementioned logics) and the long years of research in resolution theorem-proving. However, this generality comes with performance prices and many approaches that implement algorithms that are tailor-made to a specific language are more efficient than the (usually) general-purposed resolution-based ones. In the current paper we revisit and refine the resolution approaches in order to design efficient rewriting algorithms for many important fragments of OWL. First, we present an algorithm for the language DL-LiteR,⊓ which is strongly related to OWL 2 QL. Our calculus is optimised in such a way that it avoids performing many unnecessary inferences, one of the main problems of typical resolution algorithms. Subsequently, we extend the algorithm to the language ELHI which is strongly related to OWL 2 EL. This is a difficult task as ELHI is a relatively expressive language, however, we show that the calculus for DL-LiteR,⊓ requires small extensions. Finally, we have implemented all algorithms and have conducted an extensive experimental evaluation using many well-known large and complex OWL ontologies. On the one hand, this is the first evaluation of rewriting algorithms of this magnitude, while, on the other hand, our results show that our system is in many cases several orders of magnitude faster than the existing systems even though it uses an additional backwards subsumption checking step.

Nilpotent Sabinin algebras

In this paper we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of the Ado theorem for loops whose commutator–associator filtration is of finite length.

Quantum Gauge Field Theory in Cohesive Homotopy Type Theory

We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere.

Bayesian Approach to Spectral Function Reconstruction for Euclidean Quantum Field Theories

We present a novel approach to the inference of spectral functions from Euclidean time correlator data that makes close contact with modern Bayesian concepts. Our method differs significantly from the maximum entropy method (MEM). A new set of axioms is postulated for the prior probability, leading to an improved expression, which is devoid of the asymptotically flat directions present in the Shanon-Jaynes entropy. Hyperparameters are integrated out explicitly, liberating us from the Gaussian approximations underlying the evidence approach of the maximum entropy method. We present a realistic test of our method in the context of the nonperturbative extraction of the heavy quark potential. Based on hard-thermal-loop correlator mock data, we establish firm requirements in the number of data points and their accuracy for a successful extraction of the potential from lattice QCD. Finally we reinvestigate quenched lattice QCD correlators from a previous study and provide an improved potential estimation at T=2.33T(C).

A theoretical development on the entropy of interval-valued fuzzy sets based on the intuitionistic distance and its relationship with similarity measure

In this contribution, we generalize recent results on the entropy of interval-valued fuzzy sets (IVFSs) based on the intuitionistic distance and its relationship with similarity measure. The intuitionistic distance provides us with a new axiomatic definition of entropy of IVFSs. Based on the set of new axioms, we also prove some theorems that entropy and similarity measure for IVFSs can be transformed by each other in a general way. The contribution ends by deriving a formula that creates a variety of new entropies by the use of given entropies of IVFSs.

External Risk Measures and Basel Accords

Choosing a proper external risk measure is of great regulatory importance, as exemplified in the Basel II and Basel III Accords, which use value-at-risk with scenario analysis as the risk measures for setting capital requirements. We argue that a good external risk measure should be robust with respect to model misspecification and small changes in the data. A new class of data-based risk measures called natural risk statistics is proposed to incorporate robustness. Natural risk statistics are characterized by a new set of axioms. They include the Basel II and III risk measures and a subclass of robust risk measures as special cases; therefore, they provide a theoretical framework for understanding and, if necessary, extending the Basel Accords.

A new single transferable vote method and its axiomatic justification

We propose a new method for conducting Single transferable vote (STV) elections and provide a unified method for describing classic STV procedures (the Gregory Method, the inclusive Gregory method and the Weighted inclusive Gregory method) as an iterative procedure. We also propose a modification for quota definition that improves the theoretical properties of the procedures. The method is justified by utilising a new set of axioms. We show that this method extends the Weighted inclusive Gregory method with the modified definition of quota and random equiprobable selection of a winning coalition in each iteration. The results are extended to the methods, allowing fractional numbers of votes.

External Risk Measures and Basel Accords

Choosing a proper external risk measure is of great regulatory importance, as exemplified in the Basel II and Basel III Accord which use Value-at-Risk (VaR) with scenario analysis as the risk measures for setting capital requirements. We argue a good external risk measure should be robust with respect to model misspecification and small changes in the data. A new class of data-based risk measures called natural risk statistics are proposed to incorporate robustness. Natural risk statistics are characterized by a new set of axioms; they include the Basel II and III risk measures and a subclass of robust risk measures as special cases; therefore, they provide a theoretical framework for understanding and, if necessary, extending the Basel accords.