Set Of Axioms Research Articles

Axioms for signatures with domain and demonic composition

Demonic composition \(*\) is an associative operation on binary relations, and demonic refinement \({\sqsubseteq }\) is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation \(\circ \) given by \(s\circ t=D(s)*t\). We show that the class of relation algebras of signature \(\{\, \sqsubseteq , D, *\, \}\), or equivalently \(\{\, \subseteq , \circ , *\, \}\), has no finite axiomatisation. A large number of other non-finite axiomatisability consequences of this result are also given, along with some further negative results for related signatures. On the positive side, a finite set of axioms is obtained for relation algebras with signature \(\{\, \sqsubseteq , \circ , *\, \}\), hence also for \(\{\, \subseteq , \circ , *\, \}\).

An axiomatic characterization of a generalized ecological footprint

ABSTRACT The purpose of this paper is to propose an axiomatic characterization of ecological footprint indices. Using an axiomatic approach, we define a set of axioms representing the properties considered appropriate to ecological footprint measures in general. It can be shown that there exists a generalized index which is given by an arbitrary affine transformation of the land area appropriated to provide a country's share of consumption on world production. The footprint of a subset of countries is given by the sum of the individual footprints. As an implication, the well-known compound-based footprint index used by the Global Footprint Network can be characterized as a specification of the generalized index by an appropriate affine transformation. With respect to empirical applications the proposition of generalized and axiomatically characterized measures for the ecological burden by human activity may be considered as the main contribution of the paper.

An extension and an alternative characterization of May’s theorem

The context of this work is a voting scenario in which each voter expresses his/her level of affinity about a proposal, by choosing a value in the set $${\mathcal {J}}=\{-j,\dots ,-1,0,1,\dots ,j\}$$ , and these individual votes produce a collective result, in the same set $${\mathcal {J}}$$ , through a decision function. The simple majority, defined for $$j=1$$ , is a widely used example of such a decision function. In this paper, a set of independent axioms is proved to uniquely characterize the j-majority decision function. The j-majority decision is defined for any positive integer j, and it coincides with the simple majority decision when $$j=1$$ . In this way, this axiomatic characterization meets two goals: it gives a new characterization of the simple majority decision when $$j=1$$ and it extends May’s theorem to this broader context.

Certain fundamental properties of generalized natural transform in generalized spaces

This paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

Nothing But Gold. Complexities in Terms of Non-difference and Identity

Beginning from some passages by Vācaspati Miśra and Bhāskararāya Makhin discussing the relationship between a crown and the gold of which it is made, this paper investigates the complex underlying connections among difference, non-difference, coreferentiality, and qualification qua relations. Methodologically, philological care is paired with formal logical analysis on the basis of ‘Navya-Nyāya Formal Language’ premises and an axiomatic set theory-based approach. This study is intended as the first step of a broader investigation dedicated to analysing causation and transformation in non-difference.

Lines and Semi-Countably Differentiable Primes

Let l(u)⊃ |G|. A central problem in higher non-linear graph theoryis the construction of projective numbers. We show that Recent developments in axiomatic set theory [6] have raised the questionof whetherEis not dominated byl. On the other hand, the work in [6, 24] did not consider the hyper-real case.

PRIORITY MERGE AND INTERSECTION MODALITIES

AbstractWe study the logic of so-called lexicographic or priority merge for multi-agent plausibility models. We start with a systematic comparison between the logical behavior of priority merge and the more standard notion of pooling through intersection, used to define, for instance, distributed knowledge. We then provide a sound and complete axiomatization of the logic of priority merge, as well as a proof theory in labeled sequents that admits cut. We finally study Moorean phenomena and define a dynamic resolution operator for priority merge for which we also provide a complete set of reduction axioms.

2-Absorbing Primary Subsemimodules Over Partial Semirings

A partial semiring is a structure possessing an infinitary partial addition and a binary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional compo- sition is a partial semiring. In this paper we obtain the characteristics of 2-absorbing primary subsemimodules and weakly 2-absorbing primary subsemimodules in partial semirings.

Decidability of a Sound Set of Inference Rules for Computational Indistinguishability

Computational indistinguishability is a key property in cryptography and verification of security protocols. Current tools for proving it rely on cryptographic game transformations. We follow Bana and Comon’s approach [7, 8], axiomatizing what an adversary cannot distinguish. We prove the decidability of a set of first-order axioms that are computationally sound, though incomplete, for protocols with a bounded number of sessions whose security is based on an <small>IND-CCA 2 </small> encryption scheme. Alternatively, our result can be viewed as the decidability of a family of cryptographic game transformations. Our proof relies on term rewriting and automated deduction techniques.

An intersection representation for a class of anisotropic vector-valued function spaces

The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg and Netrusov (2007), which includes weighted anisotropic mixed-norm Besov and Lizorkin–Triebel spaces. In the special case of the classical Lizorkin–Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted Lq-Lp-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin–Triebel spaces occur as spaces of boundary data.

An ontology-based approach for developing a harmonised data-validation tool for European cancer registration

BackgroundPopulation-based cancer registries constitute an important information source in cancer epidemiology. Studies collating and comparing data across regional and national boundaries have proved important for deploying and evaluating effective cancer-control strategies. A critical aspect in correctly comparing cancer indicators across regional and national boundaries lies in ensuring a good and harmonised level of data quality, which is a primary motivator for a centralised collection of pseudonymised data. The recent introduction of the European Union’s general data-protection regulation (GDPR) imposes stricter conditions on the collection, processing, and sharing of personal data. It also considers pseudonymised data as personal data. The new regulation motivates the need to find solutions that allow a continuation of the smooth processes leading to harmonised European cancer-registry data. One element in this regard would be the availability of a data-validation software tool based on a formalised depiction of the harmonised data-validation rules, allowing an eventual devolution of the data-validation process to the local level.ResultsA semantic data model was derived from the data-validation rules for harmonising cancer-data variables at European level. The data model was encapsulated in an ontology developed using the Web-Ontology Language (OWL) with the data-model entities forming the main OWL classes. The data-validation rules were added as axioms in the ontology. The reasoning function of the resulting ontology demonstrated its ability to trap registry-coding errors and in some instances to be able to correct errors.ConclusionsDescribing the European cancer-registry core data set in terms of an OWL ontology affords a tool based on a formalised set of axioms for validating a cancer-registry’s data set according to harmonised, supra-national rules. The fact that the data checks are inherently linked to the data model would lead to less maintenance overheads and also allow automatic versioning synchronisation, important for distributed data-quality checking processes.

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}.

Normative theory of visual receptive fields

This article gives an overview of a normative theory of visual receptive fields. We describe how idealized functional models of early spatial, spatio-chromatic and spatio-temporal receptive fields can be derived in a principled way, based on a set of axioms that reflect structural properties of the environment in combination with assumptions about the internal structure of a vision system to guarantee consistent handling of image representations over multiple spatial and temporal scales. Interestingly, this theory leads to predictions about visual receptive field shapes with qualitatively very good similarities to biological receptive fields measured in the retina, the LGN and the primary visual cortex (V1) of mammals.

Risk Measures Induced by Efficient Insurance Contracts

The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts.

A Framework for Measures of Risk under Uncertainty

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.

Flow-Based Attribution in Graphical Models: A Recursive Shapley Approach

We study the attribution problem in a graphical model, wherein the objective is to quantify how the effect of changes at the source nodes propagates through the graph. We develop a model-agnostic flow-based attribution method, called recursive Shapley value (RSV). RSV generalizes a number of existing node-based methods and uniquely satisfies a set of flow-based axioms. In addition to admitting a natural characterization for linear models and facilitating mediation analysis for non-linear models, RSV satisfies a mix of desirable properties discussed in the recent literature, including implementation invariance, sensitivity, monotonicity, and affine scale invariance.

The Class of Orderable Groups Is a Quasi-Variety with Undecidable Theory

Let G be a group. G is right-orderable provided it admits a total order ≤ satisfying hg1 ≤ hg2 whenever g1 ≤ g2. G is orderable provided it admits a total order ≤ satisfying both: hg1 ≤ hg2 whenever g1 ≤ g2 and g1h ≤ g2h whenever g1 ≤ g2. A classical result shows that free groups are orderable. In this paper, we prove that left-orderable groups and orderable groups are quasivarieties of groups both with undecidable theory. For orderable groups, we find an explicit set of universal axioms.

Extensional realizability for intuitionistic set theory

Abstract In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $\mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types, ${\textsf{AC}}_{{\textsf{FT}}}$, is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes ‘inner models’ of many set theories that additionally validate ${\textsf{AC}}_{{\textsf{FT}}}$, in particular it provides a self-validating semantics for ${\textsf{CZF}}$ (constructive Zermelo–Fraenkel set theory) and ${\textsf{IZF}}$ (intuitionistic Zermelo–Fraenkel set theory). One can also add large set axioms and many other principles.

A particular upper expectation as global belief model for discrete-time finite-state uncertain processes

To model discrete-time finite-state uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that is the most conservative one under a set of basic axioms. Our motivation for these axioms, which describe how local and global belief models should be related, is based on two possible interpretations for an upper expectation: a behavioural one similar to Walley's, and an interpretation in terms of upper envelopes of linear expectations. We show that the most conservative upper expectation satisfying our axioms, that is, our model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications: it guarantees that there is a unique most conservative global belief model satisfying our axioms; and it shows that Shafer and Vovk's model can be given an axiomatic characterisation and thereby provides an alternative motivation for adopting this model, even outside their game-theoretic framework. Finally, we relate our model to the upper expectation resulting from a traditional measure-theoretic approach. We show that this measure-theoretic upper expectation also satisfies the proposed axioms, which implies that it is dominated by our model or, equivalently, the game-theoretic model. Moreover, if all local models are precise, all three models coincide.

Quantitative mereology: An essay to align physics laws with a philosophical concept

Mereology stands for the philosophical concept of parthood and is based on a sound set of fundamental axioms and relations. One of these axioms relates to the “existence of a universe as a thing having part all other things.” The present article formulates this logical expression first as an algebraic inequality and eventually as an algebraic equation reading in words: “The universe equals the sum of all things.” “All things” here are quantified by a “number of things.” Eventually, this algebraic equation is normalized leading to an expression: “The whole equals the sum of all fractions.” This introduces “1” or “100%” as a quantitative—numerical—value describing the “whole.” The resulting “basic equation” can then be subjected to a number of algebraic operations. Especially squaring this equation leads to correlation terms between the things implying that the whole is more than just the sum of its parts. Multiplying the basic equation (or its square) by a scalar allows for the comparison to and aligning with physics equations like the entropy equation, the ideal gas equation, an equation for the Lorentz-factor, conservation laws for mass and energy, the energy-mass equivalence, the Boltzmann statistics, and the energy levels in a Hydrogen atom. It further leads to a “contrast equation,” which may form the basis for the definition of a length and a time scale. Multiplying the basic equation with vectors, pseudovectors, pseudoscalars, and eventually hypercomplex numbers opens up the realm of possibilities to generate many further equations.