Undecidability of representability as binary relations

2012 ◽  
Vol 77 (4) ◽  
pp. 1211-1244 ◽  
Author(s):  
Robin Hirsch ◽  
Marcel Jackson
Keyword(s):  
Relation Algebra ◽  
Binary Relations ◽  
Wide Range ◽  
Finite Representability

AbstractIn this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

scholarly journals Finite representability of semigroups with demonic refinement

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Robin Hirsch ◽  
Jaš Šemrl
Keyword(s):  
Partial Order ◽  
Turing Machines ◽  
Binary Relations ◽  
Finite Representation ◽  
First Order ◽  
Finite Structures ◽  
Finite Base ◽  
Finite Set ◽  
Finite Representability ◽  
Representation Class

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

scholarly journals Discussion on Generalized-(αψ,βφ)-Contractive Mappings via Generalized Altering Distance Function and Related Fixed Point Theorems

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Maher Berzig ◽  
Erdal Karapınar ◽  
Antonio-Francisco Roldán-López-de-Hierro
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Contractive Mapping ◽  
Binary Relations ◽  
Contractive Mappings ◽  
Altering Distance ◽  
Wide Range ◽  
Generalized Altering Distance Function

We extend the notion of (αψ,βφ)-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts.

Relation algebras of every dimension

1992 ◽  
Vol 57 (4) ◽  
pp. 1213-1229 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Binary Relation ◽  
Sequent Calculus ◽  
Logical Consequence ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras

AbstractConjecture (1) of [Ma83] is confirmed here by the following result: if 3 ≤ α < ω, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form S ⊆ T (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations).

Undecidable semiassociative relation algebras

1994 ◽  
Vol 59 (2) ◽  
pp. 398-418 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Word Problem ◽  
Free Algebra ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras ◽  
Undecidable Theory ◽  
Algebraic Result

AbstractIf K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism ℒw× is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in ℒw× forms a hereditarily undecidable theory in ℒw×. These results generalize similar theorems, due to Tarski, concerning relation algebras and the formalism ℒ×.

scholarly journals Generalized Multiple Comparison Within the Framework of “Psychological Sociology of Politics”

2020 ◽  
Vol 6 (11) ◽  
pp. 356-366
Author(s):  
M. Basimov ◽  
V. Kornienko
Keyword(s):  
Political Parties ◽  
Statistical Processing ◽  
Multiple Comparison ◽  
General Distribution ◽  
Binary Relations ◽  
Interval Parameter ◽  
Political Leaders ◽  
Political Preferences ◽  
Scientific World ◽  
Wide Range

The article presents some results of the study of political preferences of young people based on the solved problem of multiple comparison (a generalized version in the author’s terminology). In statistical processing, 18 non-degenerate groups of respondents are considered for nominal answers to 4 questionnaire questions. Presentation of calculated information as a result of sorting the general distribution of binary relations “group of the nominal answer to one of the questionnaire questions — interval parameter (variable) according to the results of responses to sociological questionnaires and psychological tests” occurs primarily for two schemes. This article discusses the distribution on the levels of groups of nominal answers for each quantitative variable. Three distributions are considered in detail, and extreme results are given for 11 distributions. For the study on the basis of the available material, two questionnaires were prepared, including questions relating both to attitudes to individual political leaders, political parties, evaluation of the work of elected and governing authorities, and the respondents’ attitude to politics, their political participation, etc. In addition to sociological information, a wide range of psychological information is considered within the framework of a personal typology (six methodics widely known in the scientific world).

Groups and Algebras of Binary Relations

2002 ◽  
Vol 8 (1) ◽  
pp. 38-64 ◽  
Author(s):  
Steven Givant ◽  
Hajnal Andréka
Keyword(s):  
Positive Solution ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras ◽  
Abstract Theory ◽  
Atomic Relation ◽  
Measurability Condition ◽  
Positive Results ◽  
Representable Relation ◽  
Special Classes

AbstractIn 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jónsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jónsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms—called coset relation algebras—that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jónsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

Recent Applications of High Resolution Electron Microscopy to Crystalline Arrays of Virus Particles

1978 ◽  
Vol 36 (3) ◽  
pp. 470-482 ◽  
Author(s):  
R.W. Horne
Keyword(s):  
Electron Microscopy ◽  
Image Processing ◽  
Analytical Techniques ◽  
Staining Technique ◽  
High Resolution Electron Microscopy ◽  
Optical Diffraction ◽  
Full Potential ◽  
Virus Particles ◽  
Resolution Electron ◽  
Wide Range

The technique of surrounding virus particles with a neutralised electron dense stain was described at the Fourth International Congress on Electron Microscopy, Berlin 1958 (see Home & Brenner, 1960, p. 625). For many years the negative staining technique in one form or another, has been applied to a wide range of biological materials. However, the full potential of the method has only recently been explored following the development and applications of optical diffraction and computer image analytical techniques to electron micrographs (cf. De Hosier & Klug, 1968; Markham 1968; Crowther et al., 1970; Home & Markham, 1973; Klug & Berger, 1974; Crowther & Klug, 1975). These image processing procedures have allowed a more precise and quantitative approach to be made concerning the interpretation, measurement and reconstruction of repeating features in certain biological systems.

Microfabrication research at the National Submicron Facility

1983 ◽  
Vol 41 ◽  
pp. 86-89
Author(s):  
E.D. Wolf
Keyword(s):  
Integrated Circuits ◽  
Particle Physics ◽  
High Speed ◽  
New Technology ◽  
High Energy ◽  
High Energy Particle ◽  
New Science ◽  
Wide Range ◽  
Intermediate Domain ◽  
Circuit Function

Most microelectronics devices and circuits operate faster, consume less power, execute more functions and cost less per circuit function when the feature-sizes internal to the devices and circuits are made smaller. This is part of the stimulus for the Very High-Speed Integrated Circuits (VHSIC) program. There is also a need for smaller, more sensitive sensors in a wide range of disciplines that includes electrochemistry, neurophysiology and ultra-high pressure solid state research. There is often fundamental new science (and sometimes new technology) to be revealed (and used) when a basic parameter such as size is extended to new dimensions, as is evident at the two extremes of smallness and largeness, high energy particle physics and cosmology, respectively. However, there is also a very important intermediate domain of size that spans from the diameter of a small cluster of atoms up to near one micrometer which may also have just as profound effects on society as “big” physics.

Application of TEM to Deformation, Wear, and Fracture of Ceramics

1974 ◽  
Vol 32 ◽  
pp. 472-473
Author(s):  
B. J. Hockey
Keyword(s):  
Wear Resistance ◽  
High Temperature ◽  
Mechanical Behavior ◽  
Brittle Materials ◽  
High Temperature Strength ◽  
Wide Range ◽  
Corrosive Environments ◽  
Further Development

Ceramics, such as Al2O3 and SiC have numerous current and potential uses in applications where high temperature strength, hardness, and wear resistance are required often in corrosive environments. These materials are, however, highly anisotropic and brittle, so that their mechanical behavior is often unpredictable. The further development of these materials will require a better understanding of the basic mechanisms controlling deformation, wear, and fracture.The purpose of this talk is to describe applications of TEM to the study of the deformation, wear, and fracture of Al2O3. Similar studies are currently being conducted on SiC and the techniques involved should be applicable to a wide range of hard, brittle materials.

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