Showing papers on "Class (philosophy) published in 1969"

The Principal Type-Scheme of an Object in Combinatory Logic

Abstract: One of the aims of combinatory logic is to study the most basic properties of functions and other concepts, with as few restrictions as possible; hence in the simplest form of combinatory logic there is nothing to stop a function being applied to itself; thus XX is an ob for all obs X. However it is also natural to look at the consequences of introducing type-restrictions, formulated by assigning types to the obs according to certain rules, to be described later. Each type is a linguistic expression representing a set of functions, and for any type a the statement " X has type a" is intended to mean that the ob X represents a member of the set represented by a. Given types a and /, the set of all functions from the set a into the set 8 is represented by the type " Fa/3" (using Curry's notation). Now consider the ob l; if X has type a, then the ob I X must also have type a. Hence I represents a function from a into a, and so it must be given the type Faa, for each type a. Thus I has not just one type, but a whole class of types. This might seem strange, but it comes from the fact that I represents the abstract notion of " identity-function", rather than one particular identity-function for a particular set. The identity-function involves basically the same concept, no matter what we are applying it to. Similarly the other two basic combinators S and K (see later, or [1, ?5A]) represent certain simple operations which can be performed on almost any functions, and thus they too have an infinite class of types (see ?1 later for details, and ?5 for comment). To denote classes of types, we can use variables a, b, c; then the fact that I has every type of form Faa can be expressed by assigning to I the type-scheme

Data structure diagrams

Abstract: Successful communication of ideas has been and will continue to be a limiting factor in man's endeavors to survive and to better his life. The invention of algebra, essentially a graphic technique for communicating truths with respect to classes of arithmetic statements, broke the bond that slowed the development of mathematics. Whereas "12+ 13=25 '' and "3+7= 10" and "14+(-2) = 12" are arithmetic statements, "a+b=c '' is an algebraic statement. In particular, it is an algebraic statement controlling an entire class of arithmetic statements such as those listed.

1: Learning Studies in Science Education:

Abstract: s, and Dissertation Abstracts, shows that there is a relatively standard set of categories ("concept learning," "problem solving," "discovery learning," "critical thinking," etc.) for classifying learning studies. Each category might be thought to represent a closely bound set of ideas, but further problems emerge as one begins to read the studies. Take as an example the category "discovery learning." In a hypothetical study conducted by Mr. X and clearly labeled a discovery study in the title, second and third grade pupils were taught a series of "discovery lessons on flowers." The pupils used programed directions written in the Initial Teaching Alphabet. The "Mr. X Test of Discovery Learning" was given as a preand posttest. The reader knows little about the test other than that it consists of twenty items constructed by Mr. X. After analysis of pre-post data, Mr. X reports that discovery learning was far superior to conventially taught lessons. In another hypothetical study, by Mr. Y, college freshmen in an introductory geology class were engaged in a series of "discovery lessons" employing geological maps on field trips. An achievement test constructed by Mr. Y was given to classes using the discovery lessons and to a control class using only the text and lectures. Analysis of data revealed significant differences in achievement in favor of the experimental discovery class.

On products of structures for generalized logics

Abstract: In this paper we study structures for generalized logics and operations such as products of structures. If we consider the usual definition of products or powers of structures for the classical {0,1} ? valued logic then we see that these operations are not natural in some sense. Namely, structures which we obtain by these opera? tions have very different elementary properties. Moreover, it is also visible that even in the case of three-valued logic there is no "nice" definition of products or powers of structures. Some attempt of such construction has been given by Chang and Keis ler [2], but it can be applied only to some small class of logics. E.g. boolean-valued logics of Scott [6] do not belong to this class. The aim of this paper is to give uniform definitions of products and powers of structures for generalized logics. Our definitions seems to be natural ones in such a sense that the diagonal is an elementary substructure of a power (Theorem 4). The idea of such definition is contained implicite in the paper of Feferman and Vaught [3], see also Wojciechowska [8]. Besides, this paper contains also some model-theoretical results concerning with generalized logics. There are e.g. the Tarski?Vaught criterion for elementary in? clusion (Theorem 1) and the downward L?wenheim?Skolem Theorem (The? orem 2).

R69-19 On the Elimination of Endmarkers

Abstract: This paper is written by automata theorists for automata theorists. In numerous definitions of automata, there is an "input head" which is expected to be positioned at one square of an "input tape" at all times. There is often the possibility that the automaton might cause the input head to move right, say, from the rightmost input square, before theautomaton has had a chance to do all the computation it wanted to do. Thus, "endmarkers"—special symbols that appear on the leftmost and rightmost input squares but are not otherwise considered part of the input-are often part of the automaton definition. This paper demonstrates that in several cases the endmarkers are unnecessary in that a class of automata with end markers recognizes only languages that are recognizable by the analogous classes of automata without one or both of the endmarkers.