Composition

3 Key excerpts on "Composition"

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  eBook - ePub

  Handbook of Logic and Language

  • Johan F.A.K. van Benthem, Alice ter Meulen(Authors)
  • 2010(Publication Date)
  • Elsevier

    (Publisher)

  But most limitations are, in my opinion, just temporary, and not essential. There are several methods to deal Compositionally with factors such as personal differences, linguistic context, situational context or vagueness. One may use additional parameters (as in Section 10.7.2 on ambiguity), context constants or variables (see Appendix B on genitives); the influence from discourse can be treated Compositionally (see Section 10.4.4 on DRT), and vagueness by fuzzy logic. And if for some technical terms speaker and hearer have to come to agreement, and practically nothing can be said in general about their meaning, then we have not reached the limits of Compositionality, but the limits of semantics (as is the title of Partee, 1982). 10.8. A Mathematical Model of Compositionality 10.8.1. Introduction In this section a mathematical model is developed that describes the essential aspects of Compositional meaning assignment. The assumptions leading to this model have been discussed in Section 10.3. The model is closely related to the one presented in “Universal Grammar” (Montague, 1970b). The mathematical tools used in this section are tools from Universal Algebra, a branch of mathematics that deals with general structures; a standard textbook is Graetzer (1979). For easy reference, the principle is repeated here: The meaning of a compound expression is a function of the meanings of its parts and of the syntactic rule by which they are combined. 10.8.2. algebra The first notion to be considered is parts. Since the information on how expressions are formed is given by the syntax of a language, the rules of the grammar determine what the parts of an expression are. The rules build new expressions from old expressions, so they are operators taking inputs and yielding an output. A syntax with this kind of rules is a specific example of what is called in mathematics an algebra. Informally stated, an algebra is a set with functions defined on that set

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  eBook - ePub

  Memory and the Computational Brain

  Why Cognitive Science will Transform Neuroscience

  • C. R. Gallistel, Adam Philip King(Authors)
  • 2011(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Composition of functions . It is a major aspect of the writing of computer programs, because the commands in a computer program generally invoke functions, and the results are then often operated on by functions invoked by later commands.

  Categorization schemes are functions, as f 1 in Figure 3.2 illustrates. Categorization is commonly hierarchical, and this is captured by the Composition of functions, as shown in Figure 3.2 .

  In composing functions, it usually matters which one operates first and which second. If we feed numbers to the doubling function first and then to the squaring function, we map the number 3 to 36, but if we feed first to the squaring function and then to the doubling function, we map 3 to 18. Thus, the Composition of functions is not in general commutative: it is often the case f b ° f a ≠ f a ° f b , where ° denotes Composition. In alternative notation, f b (f a (x )) ≠ f a (f b (x )). The alternative notation has the advantage of making it more apparent which function operates first (the innermost). The example of functional Composition in Figure 3.2 shows that not only is the Composition of functions commonly not commutative, it may well be the case that two functions can compose in one order but not in the reverse order. In Figure 3.2 , one could not first apply the categorization in f 2 and then apply the categorization in f 1 , because the range of f 2 is not in the domain of f 1 .

  The non-commutative property of the Composition of functions suggests that any physically realizable system that computes such functions must be capable of sequencing in time the order of the individual functions. In turn, this implies that such computing devices must be capable of carrying the values of functions forward in time such that they can be utilized by functions that are sequenced later. Most of the functions that brains routinely compute involve the Composition of functions that are determined at different points in time (numerous examples are in Chapters 11–13). In the language of computer science, one would say that a physically realized computational device, such as the brain, needs memory to carry the values forward in time, and that this memory must be capable of being written to and read from.

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  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 2 The Concept of a Function 2.1 Introduction

  The concept of a “function” is one of the most basic in all of mathematics. The meaning of the word “function ” has evolved and changed during the last three centuries . Its modern meaning is much broader and deeper than its elementary meaning from earlier days. The statement: “y is a function of x ” means something very much like “y is related to x by some formula ”. In fact, this statement gives some idea about a function, but it is incomplete. In traditional algebra, x and y stand for numbers. But today, functions can be defined that have nothing to do with numbers .

  In our study of calculus, we shall be mostly concerned with functions, which are related to numbers. Like any other mathematical concept, the concept of function is nicely expressed through the language of sets . Therefore, it is useful to revise “Elementary Set Theory ” (see Appendix “A”).

  Assuming the knowledge of Elementary Set Theory, we define two important terms: (i) ordered pairs and (ii) Cartesian product of sets . These terms are needed to define a “function” on the basis of set theory. Let us discuss:

  i. Ordered Pairs : When we wish to consider a pair of things as a whole , we may use the terms couple or just pair . If A = {1, 2, 3, 4} then the subsets {1, 2}, {1, 3}, {1, 4}, {2, 1}, {3, 1} are some examples of pairs. Here we have listed some pairs twice ; for example {1, 2} = {2, 1} and {1, 3} = {3, 1}.

  We know that the order, in which the elements of a set are written, is immaterial . If in a pair we wish to single out one element as being the first, then the other element becomes the second. Once we define the procedure of fixing the position of first element (in a pair), we have example of an ordered pair

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