General models, descriptions, and choice in type theory

1972 ◽  
Vol 37 (2) ◽  
pp. 385-394 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Combinatory Logic ◽  
Logical Constants ◽  
Simple Idea ◽  
Nonstandard Models ◽  
Selection Operator ◽  
Definition Of

In [4] Alonzo Church introduced an elegant and expressive formulation of type theory with λ-conversion. In [8] Henkin introduced the concept of a general model for this system, such that a sentence A is a theorem if and only if it is true in all general models. The crucial clause in Henkin's definition of a general model ℳ is that for each assignment φ of values in ℳ to variables and for each wff A, there must be an appropriate value of A in ℳ. Hintikka points out in [10, p. 3] that this constitutes a rather strong requirement concerning the structure of a general model. Henkin draws attention to the problem of constructing nonstandard models for the theory of types in [9, p. 324].We shall use a simple idea of combinatory logic to find a characterization of general models which does not directly refer to wffs, and which is easier to work with in certain contexts. This characterization can be applied, with appropriate minor and obvious modifications, to a variety of formulations of type theory with λ-conversion. We shall be concerned with a language ℒ with extensionality in which there is no description or selection operator, and in which (for convenience) the sole primitive logical constants are the equality symbols Qoαα for each type α.

General models and extensionality

1972 ◽  
Vol 37 (2) ◽  
pp. 395-397 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Axiom Schema ◽  
Nonstandard Model ◽  
Equality Relation ◽  
Definition Of

It is well known that equality is definable in type theory. Thus, in the language of [2], the equality relation between elements of type α is definable as , i.e., xα = yα iff every set which contains xα also contains yα. However, in a nonstandard model of type theory, the sets may be so sparse that the wff above does not denote the true equality relation. We shall use this observation to construct a general model in the sense of [2] in which the Axiom of Extensionality is not valid. Thus Theorem 2 of [2] is technically incorrect. However, it is easy to remedy the situation by slightly modifying the definition of general model.Our construction will show that the Axiom Schema of Extensionality is independent even if one takes as an axiom schema.We shall assume familiarity with, and use the notation of, [2] and §§2–3 of [1].

scholarly journals Logicality and Invariance

2008 ◽  
Vol 14 (1) ◽  
pp. 29-68 ◽  
Author(s):  
Denis Bonnay
Keyword(s):  
Similarity Relation ◽  
Necessary Condition ◽  
General Notion ◽  
Sufficient Condition ◽  
Logical Constants ◽  
Similarity Relations ◽  
Definition Of

AbstractThis paper deals with the problem of giving a principled characterization of the class of logical constants. According to the so-called Tarski–Sher thesis, an operation is logical iff it is invariant under permutation. In the model-theoretic tradition, this criterion has been widely accepted as giving a necessary condition for an operation to be logical. But it has been also widely criticized on the account that it counts too many operations as logical, failing thus to provide a sufficient condition.Our aim is to solve this problem of overgeneration by modifying the invariance criterion. We introduce a general notion of invariance under a similarity relation and present the connection between similarity relations and classes of invariant operations. The next task is to isolate a similarity relation well-suited for a definition of logicality. We argue that the standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified. The revised arguments are shown to support an alternative to Tarski's criterion, according to which an operation is logical iff it is invariant under potential isomorphism.

scholarly journals ESTIMATION OF GEOGRAPHICAL DATABASES CAPTURE SCALE BASED ON INTER-VERTICES DISTANCES EXPLORATION

2015 ◽  
Vol II-3/W5 ◽  
pp. 305-310
Author(s):  
J.-F. Girres
Keyword(s):  
General Model ◽  
Extraction Method ◽  
Level Of Detail ◽  
Estimation Model ◽  
Test Experiment ◽  
Length And Area ◽  
Definition Of ◽  
Error Simulation ◽  
Clear Relation

This article presents a method to estimate the capture scale of a geographical database based on the characterization of its level of detail. This contribution fits in a larger research, dealing with the development of a general model to estimate the imprecision of length and area measurements computed from the geometry of objects of weakly informed geographical databases. In order to parameterize automatically a digitizing error simulation model, the characteristic capture scale is required. Thus, after a definition of the different notions of scales in geographical databases, the proposed method is presented. The goal of the method is to model the relation between the level of detail of a geographical database, by exploring inter-vertices distances, and its characteristic capture scale. To calibrate the model, a digitizing test experiment is provided, showing a clear relation between median intervertices distance and characteristic capture scale. The proposed knowledge extraction method proves to be the usefull in order to parameterize the measurement imprecision estimation model, and more generally to inform the database user when the capture scale is unknown. Nevertheless, further experiments need to be provided to improve the method, and model the relation between level of detail and capture scale with more efficiency.

scholarly journals Experimental Evaluation of Shear Behavior of Stone Masonry Wall

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2313
Author(s):  
Maria Luisa Beconcini ◽  
Pietro Croce ◽  
Paolo Formichi ◽  
Filippo Landi ◽  
Benedetta Puccini
Keyword(s):  
Shear Behavior ◽  
Stone Masonry ◽  
Masonry Walls ◽  
Mechanical Parameters ◽  
In Situ Tests ◽  
Capacity Curves ◽  
Historical Structures ◽  
Definition Of

The evaluation of the shear behavior of masonry walls is a first fundamental step for the assessment of existing masonry structures in seismic zones. However, due to the complexity of modelling experimental behavior and the wide variety of masonry types characterizing historical structures, the definition of masonry’s mechanical behavior is still a critical issue. Since the possibility to perform in situ tests is very limited and often conflicting with the needs of preservation, the characterization of shear masonry behavior is generally based on reference values of mechanical properties provided in modern structural codes for recurrent masonry categories. In the paper, a combined test procedure for the experimental characterization of masonry mechanical parameters and the assessment of the shear behavior of masonry walls is presented together with the experimental results obtained on three stone masonry walls. The procedure consists of a combination of three different in situ tests to be performed on the investigated wall. First, a single flat jack test is executed to derive the normal compressive stress acting on the wall. Then a double flat jack test is carried out to estimate the elastic modulus. Finally, the proposed shear test is performed to derive the capacity curve and to estimate the shear modulus and the shear strength. The first results obtained in the experimental campaign carried out by the authors confirm the capability of the proposed methodology to assess the masonry mechanical parameters, reducing the uncertainty affecting the definition of capacity curves of walls and consequently the evaluation of seismic vulnerability of the investigated buildings.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals Genomic Tackling of Human Satellite DNA: Breaking Barriers through Time

2021 ◽  
Vol 22 (9) ◽  
pp. 4707
Author(s):  
Mariana Lopes ◽  
Sandra Louzada ◽  
Margarida Gama-Carvalho ◽  
Raquel Chaves
Keyword(s):  
Human Genome ◽  
Satellite Dna ◽  
Repetitive Sequences ◽  
Nucleotide Composition ◽  
Genomic Component ◽  
Genomic Studies ◽  
Human Genomic ◽  
Definition Of ◽  
High Degree

(Peri)centromeric repetitive sequences and, more specifically, satellite DNA (satDNA) sequences, constitute a major human genomic component. SatDNA sequences can vary on a large number of features, including nucleotide composition, complexity, and abundance. Several satDNA families have been identified and characterized in the human genome through time, albeit at different speeds. Human satDNA families present a high degree of sub-variability, leading to the definition of various subfamilies with different organization and clustered localization. Evolution of satDNA analysis has enabled the progressive characterization of satDNA features. Despite recent advances in the sequencing of centromeric arrays, comprehensive genomic studies to assess their variability are still required to provide accurate and proportional representation of satDNA (peri)centromeric/acrocentric short arm sequences. Approaches combining multiple techniques have been successfully applied and seem to be the path to follow for generating integrated knowledge in the promising field of human satDNA biology.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

ON THE NOTION OF CANONICAL DERIVATIONS FROM OPEN ASSUMPTIONS AND ITS ROLE IN PROOF-THEORETIC SEMANTICS

2015 ◽  
Vol 8 (2) ◽  
pp. 296-305 ◽  
Author(s):  
NISSIM FRANCEZ
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Logical Constants ◽  
Local Completeness ◽  
Definition Of ◽  
The Impact

AbstractThe paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

scholarly journals Characterization of Spent Ni-MH Batteries

2012 ◽  
Vol 730-732 ◽  
pp. 569-574
Author(s):  
Marta Cabral ◽  
Fernanda Margarido ◽  
Carlos A. Nogueira
Keyword(s):  
Quantitative Analysis ◽  
Electrode Materials ◽  
Phase Identification ◽  
Recycling Process ◽  
X Ray ◽  
Leaching Experiments ◽  
Plastic Components ◽  
Definition Of ◽  
Physical And Chemical

Spent Ni-MH batteries are not considered too dangerous for the environment, but they have a considerable economical value due to the chemical composition of electrodes which are highly concentrated in metals. The present work aimed at the physical and chemical characterisation of spent cylindrical and thin prismatic Ni-MH batteries, contributing for a better definition of the recycling process of these spent products. The electrode materials correspond to more than 50% of the batteries weight and contain essentially nickel and rare earths (RE), and other secondary elements (Co, Mn, Al). The remaining components are the steel parts from the external case and supporting grids (near 30%) containing Fe and Ni, and the plastic components (<10%). Elemental quantitative analysis showed that the electrodes are highly concentrated in metals. Phase identification by X-ray powder diffraction combined with chemical analysis and leaching experiments allowed advancing the electrode materials composition. The cathode is essentially constituted by 6% metallic Ni, 66% Ni(OH)2, 4.3% Co(OH)2 and the anode consists mainly in 62% RENi5 and 17% of substitutes and/or additives such as Co, Mn and Al.

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