Alexander Meduna - MODERN LANGUAGE MODELS AND COMPUTATION: theory with applications

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Part I
Part I

Introduction

This part, consisting of Chaps. , reviews important mathematical concepts with a special focus on classical language theory. In this way, it places the entire coverage of the book into its scientific context. In addition, it guarantees that the whole text is completely selfcontainedthat is, no other book is needed to follow its complete coverage.

Chapter covers important concepts used in formal language theory. Apart from the classical rudiments, it includes several lesser-known areas of this theory, such as parallel grammars, because they are also needed to fully grasp some upcoming topics covered in this book.

Springer International Publishing AG 2017

Alexander Meduna and Ondej Soukup Modern Language Models and Computation

1. Mathematical Background

Alexander Meduna 1 and Ondej Soukup 2

(1)

Department of Computer Science, Brno University of Technology, Brno, Czech Republic

(2)

Department of Information Technology, Brno University of Technology, Brno, Czech Republic

This three-section chapter reviews rudimentary mathematical concepts, including key notions concerning sets (Sect. ). For readers having background in these areas, this chapter can be skipped and treated as a reference for terminology used later in this book.

1.1 Sets and Sequences

This section outlines rudimentary concepts concerning sets and sequences.

1.1.1 Sets

In what follows, we assume that there exist primitive objects, referred to as elements , taken from some pre-specified universe , usually denoted by . We also assume that there are objects, referred to as sets , which represent collections of objects, each of which is an element or another set. If A contains an object a , then we symbolically write a A and refer to a as a member of A . On the other hand, to express that a is not a member of A , we write a A .

If A has a finite number of members, then A is a finite set ; otherwise, it is an infinite set . The finite set that has no member is the empty set , denoted by . The cardinality of a finite set A , , is the number of members that belong to A ; note that . A finite set A is customarily specified by listing its members ; that is, A = { a 1, a 2,, a n }, where a 1 through a n are all members of A; as a special case, we have {} = . An infinite set B is usually defined by a property so that B contains all elements satisfying ; in symbols, this specification has the following general format: B = { a ( a )}. Sometimes, an infinite set is defined recursively by explicitly naming the first few values (typically, just one first value) in the set and then defining later values in the set in terms of earlier values.

In this book, we denote the set of all natural numbers by . In other words, denotes the set of all positive integers, so

Furthermore, denotes the set of all non-negative integers, and denotes the entire set of all integers throughout.

Example 1.1.1.

Take . Let X be the set of all even positive integers defined as

Let Y be the set of all even positive integers between 1 and 9. Define

Observe that . Consider the next recursive definition of the set W

1. (i)
   2 is in W ;
2. (ii)
   if i is in W , then so is 2 i , for all i 2.

By again, it includes 8 as well. Continuing in this way, we see that W contains 2,4,8,16,. In words, W consists of all positive integers that represent a power of two; mathematically,

Let A and B be two sets. A is a subset of B , symbolically written as A B , if each member of A also belongs to B . A is a proper subset of B , written as A B , if A B and B contains a member that is not in A . By A B , we express that A is not a subset of B . If A B and B A , then A equals B , denoted by A = B ; simply put, A = B means that both sets are identical. By A B , we express that A is not equal to B . To express that A B and B A we call A and B to be incomparable . The power set of A , denoted by 2 A , is the set of all subsets of A ; formally, 2 A = { B B A }.

For two sets, A and B , their union , intersection , and difference are denoted by A B , A B , and A B , respectively, and defined as A B = { a a A or a B }, A B = { a a A and a B }, and A B = { a a A and a B }. If A B = , A and B are disjoint . More generally, n sets C 1, C 2,, C n , where n 2, are pairwise disjoint if C i C j = for all 1 i , j n such that i j . If A is a set over a universe , the complement of A is denoted by and defined as .

Sets whose members are other sets are usually called classes of sets rather than sets of sets.

Example 1.1.2.

Consider the sets from Example . Observe that

Furthermore, and Y X . Set

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