Axioms for strong reduction in combinatory logic

1967 ◽  
Vol 32 (2) ◽  
pp. 224-236 ◽  
Author(s):  
Roger Hindley
Keyword(s):  
Binary Relation ◽  
Equivalence Relation ◽  
Combinatory Logic ◽  
Strong Reduction ◽  
Image Position ◽  
Weak Reduction

In combinatory logic there is a system of objects which intuitively represent functions, and a binary relation between these objects, which represents the process of evaluating the result of applying a function to an argument. (This is explained fully in [1].) From this relation called weak reduction, “≥,” an equivalence relation is defined by saying that X is weakly equivalent to Y if and only if there exist n (with 0 ≤ n) and X0,…,Xη such that It turns out that equivalent objects represent the same function, but two objects representing the same function need not be equivalent.

Tree Rewriting Systems, Combinatory Weak Reduction Systems and Type Free Languages1

1982 ◽  
Vol 5 (3-4) ◽  
pp. 279-299
Author(s):  
Alberto Pettorossi
Keyword(s):  
Combinatory Logic ◽  
Rewriting Systems ◽  
Tree Transducers ◽  
Tree Languages ◽  
One To One ◽  
The One ◽  
Weak Reduction

In this paper we consider combinators as tree transducers: this approach is based on the one-to-one correspondence between terms of Combinatory Logic and trees, and on the fact that combinators may be considered as transformers of terms. Since combinators are terms themselves, we will deal with trees as objects to be transformed and tree transformers as well. Methods for defining and studying tree rewriting systems inside Combinatory Weak Reduction Systems and Weak Combinatory Logic are also analyzed and particular attention is devoted to the problem of finiteness and infinity of the generated tree languages (here defined). This implies the study of the termination of the rewriting process (i.e. reduction) for combinators.

scholarly journals Relations on topological spaces: Urysohn's lemma

1968 ◽  
Vol 8 (1) ◽  
pp. 37-42
Author(s):  
Y.-F. Lin
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Order Relation ◽  
Topological Spaces ◽  
Topological Closure ◽  
Image Position

Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

scholarly journals FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1225-1254 ◽  
Author(s):  
RUSSELL MILLER ◽  
KENG MENG NG
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Arithmetical Hierarchy ◽  
Natural Equivalence ◽  
Image Position ◽  
The Way

AbstractWe introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

Completeness in Semi-Lattices

1957 ◽  
Vol 9 ◽  
pp. 578-582 ◽  
Author(s):  
L. E. Ward
Keyword(s):  
Binary Relation ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Image Position ◽  
Transitive Binary Relation

Let (X, ≤) be a partially ordered set, that is, X is a set and ≤ is a reflexive, anti-symmetric, transitive, binary relation on X.We write,for each x ∈ X. If, moreover,exists for each x and y in X, then (X, ≤) is said to be a semi-lattice.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals An amenable equivalence relation is generated by a single transformation

1981 ◽  
Vol 1 (4) ◽  
pp. 431-450 ◽  
Author(s):  
A. Connes ◽  
J. Feldman ◽  
B. Weiss
Keyword(s):  
Equivalence Relation ◽  
Cartan Subalgebras ◽  
Singular Transformation ◽  
Image Position ◽  
Null Set

AbstractWe prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.

CANONICAL MODELS FOR FRAGMENTS OF THE AXIOM OF CHOICE

2017 ◽  
Vol 82 (2) ◽  
pp. 489-509
Author(s):  
PAUL LARSON ◽  
JINDŘICH ZAPLETAL
Keyword(s):  
Equivalence Relation ◽  
Axiom Of Choice ◽  
Canonical Models ◽  
Develop Technology ◽  
Image Position ◽  
The Axiom Of Choice

AbstractWe develop technology for investigation of natural forcing extensions of the model $L\left( \mathbb{R} \right)$ which satisfy such statements as “there is an ultrafilter” or “there is a total selector for the Vitali equivalence relation”. The technology reduces many questions about ZF implications between consequences of the Axiom of Choice to natural ZFC forcing problems.

Strong reduction and normal form in combinatory logic

1967 ◽  
Vol 32 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Bruce Lercher
Keyword(s):  
Normal Form ◽  
Combinatory Logic ◽  
Strong Reduction ◽  
The Church

The notion of strong reduction is introduced in Curry and Feys' book Combinatory logic [1] as an analogue, in the theory of combinatore, to reduction (more exactly, βη-reduction) in the theory of λ-conversion. The existence of an analogue and its possible importance are suggested by an equivalence between the theory of combinatore and λ-conversion, and the Church-Rosser theorem in λ-conversion. This theorem implies that if a formula X is convertible to a formula X* which cannot be further reduced—is irreducible, or in normal form—then X is convertible to X* by a reduction alone. Moreover, the reduction may be performed in a certain prescribed order.

Products of distribution functions attracted to extreme value laws

1971 ◽  
Vol 8 (04) ◽  
pp. 781-793 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Equivalence Relation ◽  
Complete Convergence ◽  
Convex Sets ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Distribution Functions ◽  
Image Position

When is the product of the d.f.'s H 1(·), ···, Hm (·) attracted to an extreme value law φ(x)? We associate with each Hi (·) its A-function Hi (x) is attracted to φ(x) if each Hi (x) is in the domain of attraction of φ(x) and Ai (z) ~ Aj (z), 1 ≦ i, j ≦ m. Equivalence of A-functions determines an equivalence relation which partitions the domain of attraction of φ(x)into one or more convex sets. These sets fail to be closed under passages to the limit (complete convergence).

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