Autometrization and the Symmetric Difference

1953 ◽  
Vol 5 ◽  
pp. 324-331 ◽  
Author(s):  
J. G. Elliott
Keyword(s):  
Boolean Algebra ◽  
Distance Function ◽  
Symmetric Difference ◽  
Group Operation ◽  
Metric Distance

The fact that the symmetric difference is a group operation in a Boolean algebra is, of course, well known. Not so well known is the fact observed by Ellis [3] that it possesses some of the desirable properties of a metric distance function. Specifically, if * denotes this operation, it is easy to verify that

scholarly journals The Edit Distance Function and Symmetrization

10.37236/2262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Ryan R. Martin
Keyword(s):  
Distance Function ◽  
Edit Distance ◽  
Symmetric Difference ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets.  The distance between a graph, G, and a hereditary property, ℋ, is the minimum of the distance between G and each G'∈ℋ.  The edit distance function of ℋ is a function of p∈[0,1] and is the limit of the maximum normalized distance between a graph of density p and ℋ.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties.  For any graph H, Forb(H) denotes the property of not having an induced copy of H.  This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

Autometrized Boolean Algebras II

1951 ◽  
Vol 3 ◽  
pp. 145-147 ◽  
Author(s):  
David Ellis
Keyword(s):  
Boolean Algebra ◽  
Distance Function ◽  
Distance Geometry ◽  
Boolean Algebras ◽  
The Given ◽  
Technical Terms

The writer [1] has previously examined the fundamental concepts of distance geometry in a Boolean algebra, B, with distance defined by . Any technical terms from distance geometry which are not defined in this paper may be found in [1]. A Boolean algebra bearing the given distance function is called an autometrized Boolean algebra. It is clear that the set of motions B (congruences of B with itself) form a group under substitution. This group we denote by M(B).

Synthesis of Antennas on the Base of the Boolean Algebra Transformations and the Atomic Functions Theory

2005 ◽  
Vol 64 (9) ◽  
pp. 699-712
Author(s):  
Victor Filippovich Kravchenko ◽  
Miklhail Alekseevich Basarab
Keyword(s):  
Boolean Algebra

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

VBTree Tutorial: Automatic Completion of Group Operation on Structural Dataset

2020 ◽  
Author(s):  
Chen Zhang ◽  
Huakang Bian ◽  
Kenta Yamanaka ◽  
Akihiko Chiba
Keyword(s):  
Group Operation
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