The decidability of hindley's axioms for strong reduction

1967 ◽  
Vol 32 (2) ◽  
pp. 237-239 ◽  
Author(s):  
Bruce Lercher
Keyword(s):  
Finite Number ◽  
Infinite System ◽  
Combinatory Logic ◽  
Strong Reduction

In his paper [3] Hindley shows that strong reduction in combinatory logic (see [1] for the basic discussion of this notion) cannot be axiomatized with a finite number of axiom schemes, and then he presents an infinite system of axiom schemes for strong reduction. Hindley lists one axiom and six axiom schemes, together with a method (clause (viii) below) for generating further axiom schemes from these. The results of this note are that different applications of clause (viii) yield different axiom schemes, and that the property of being an axiom is decidable.

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.

XX. On the classification of Loci

1878 ◽  
Vol 169 ◽  
pp. 663-681 ◽  
Keyword(s):  
Finite Number ◽  
Linear Transformation ◽  
Infinite System ◽  
Flat Space ◽  
One Dimensional

By a curve we mean a continuous one-dimensional aggregate of any sort of elements, and therefore not merely a curve in the ordinary geometrical sense, but also a singly infinite system of curves, surfaces, complexes, &c., such that one condition is sufficient to determine a finite number of them. The elements may be regarded as determined by k coordinates; and then, if these be connected by k —1 equations of any order, the curve is either the whole aggregate of common solutions of these equations, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. It is thus convenient to employ still further the language of geometry, and to speak of such a curve as the complete or partial intersection of k —1 loci in flat space of k dimensions, or, as we shall sometimes say, in a k -flat. If a certain number, say h , of the equations are linear, it is evidently possible by a linear transformation to make these equations equate h of the coordinates to zero ; it is then convenient to leave these coordinates out of consideration altogether, and only to regard the remaining k — h —1 equations between k — h coordinates. In this case the curve will, therefore, be regarded as a curve in flat space of k — h dimensions. And, in general, when we speak of a curve as in flat space of k dimensions, we mean that it cannot exist in flat space of k —1 dimensions.

On reduction properties

1994 ◽  
Vol 59 (3) ◽  
pp. 900-911 ◽  
Author(s):  
Hirotaka Kikyo ◽  
Akito Tsuboi
Keyword(s):  
Finite Number ◽  
Predicate Symbol ◽  
Strong Reduction ◽  
Unary Predicate ◽  
Symmetric Extension ◽  
Reduction Property ◽  
Slight Extension ◽  
Uniform Reduction

Let us consider countable languages L containing a unary predicate symbol P and L− =L\{P}. We also assume that L is relational. Then for any L-structure M, N = PM can naturally be considered as an L−-substructure of M. The main object of this paper will be the study of the following question: Under what condition does M have to be ℵ0-categorical. ℵ1-categorical, or stable if N is?Hodges and Pillay [6] proved that if M is a countable symmetric extension of N and T = Th(M) is minimal over P (they said that T is one-cardinal over P), then the total categoricity of N implies that of M. This is a solution to a problem in Ahlbrandt and Ziegler [1]. The condition that “M is a symmetric extension of N” is an interpretation of the condition “every relation on N definable in M is definable within N”. We shall give several interpretations of this phrase: They are the Ø-reduction property, the reduction property, the strong reduction property, and the uniform reduction property (Definition 1). Under the assumptions, we study the question proposed above.In §3 we treat the case that M is countable and show that if T is minimal over P and M has the strong reduction property over N, then M is ℵ0-categorical if N is (Theorem 5). This is a slight extension of the result of Hodges and Pillay mentioned above. (If M is countable and saturated, then the strong reduction property is equivalent to the condition that M will be symmetric over N if we add a finite number of appropriate constants.) A counterexample to this theorem has been obtained by Hrushovski in the case that only the Ø-reduction property is assumed. We also give a stronger result: If M has the Ø-reduction property over N and is ℵ0-categorical, M\N is infinite, and N is algebraically closed, then there is an expansion M* of M such that M* is not ℵo-categorical but M* still has the Ø-reduction property over N (Theorem 6). Moreover, we give an example such that M has the uniform reduction property over N. Th(M*) is minimal over P. N is ℵ0-categorical but M is not.

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.

scholarly journals III. On the classification of loci

1878 ◽  
Vol 27 (185-189) ◽  
pp. 370-371
Keyword(s):  
Finite Number ◽  
Infinite System ◽  
Flat Space ◽  
One Dimensional

“A curve,” is to be understood to mean a continuous one-dimensional aggregate of any sort of elements, and therefore not merely a curve in the ordinary geometrical sense, but also a singly infinite system of curves, surfaces, complexes, &c., such that one condition is sufficient to determine a finite number of them. The elements may be regarded as determined by k co-ordinates; and if these be. connected by k —1 equations of any order, the curve is either the aggregate of common solutions, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. In the paper, of which this is an abstract, theorems are established relating to the nature of the space in which such curves can exist, to the mode of representing them in flat space of lower dimensions, and to some of their properties. The following are the leading theorems:— I. Every proper curve of the nth order is in a flat space of n dimensions or less.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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