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LINCOM Handbooks in Linguistics (LHL)

Artikel-Nr.: ISBN 9783895866418

Beschreibung

**Mathematical Foundations of Linguistics**

H. Mark Hubey

*Montclair State University*

Only a few decades ago, only mathematicians, physicists and engineers took calculus courses, and calculus was tailored for them using examples from physics. This made it difficult for students from the life sciences including biology, economics, and psychology to learn mathematics. Recently books using examples from the life sciences and economics have become more popular for such students. Such a math book does not exist for linguists. Even the computational linguistics books (Formal Language Theory) are written for mathematicians and computer scientists.

This book is for linguists. It is intended to teach the required math for a student to be a scientific linguist and to make linguistics a science on par with economics, and computer science.

There are many concepts that are central to the sciences. Most students never see these in one place and if they do, they have to wait until graduate school to obtain them in the often-dreaded "quantitative" courses. As a result sometimes it takes years or even decades before learners are able to integrate what they have learned into a whole, if ever. We have little time and much to do.

In addition to all of these problems we are now awash in data and information. It is now that the general public should be made aware of the solution to all of these problems. The answer is obviously "knowledge compression". Knowledge is structured information; it is a system not merely a collection of interesting facts.

What this book does, and what all other math books do is teach people the tools with which they can structure and thus compress information and knowledge around them. It has also been said that mathematics is the science of patterns; it is exactly by finding such patterns that we compress knowledge. We can say that mathematics is the science of knowledge compression or information compression.

This book provides the basic tools for mathematics (even including a short and intuitive explanation of differential and integral calculus). The broad areas of linguistics, probability theory, speech synthesis, speech recognition, computational linguistics (formal languages and machines), historical linguistics require mathematics of counting/combinatorics, Bayesian theory, correlation-regression analysis, stochastic processes, differential equations, vectors/tensors. These in turn are based on set theory, logic, measurement theory, graph theory, algebra, Boolean algebra, harmonic analysis etc.

The mathematical fields introduced here are all common ideas from one which one can branch off into more advanced study in any of these fields thus this book brings together ideas from many disparate fields of mathematics which would not normally be put together into a single course. This is what makes this a book especially written for linguists.

Table of Contents:

1. Generic Building Blocks

Layering

Numerals, Multiplication, Constants and Variables

Summation--Gauss

Zeno's Paradox and Euler

Continuous Products

Decision Trees, Prisoner's Dilemma

CPM/PERT Methods

2. Symbolic Computation, Iteration and Recursion

Algorithmic Definition of Integers

Parallel and Serial Choices

Multiplicative vs. Additive Intelligence

Strong vs. Long Chain Trade-off

Recursion/Iteration and Solution of a Nonlinear Equation

Programming Charts

Learning Iteration

Frequency vs. Wavelength

3. Basic Counting and Reasoning Principles

Product, Series

Logical-AND Rule

4. Hazards of Doing Science

Dimensionless Numbers

Mass vs. Surface Area

5. Normalization

Grade Normalization, Boxing Normalization

Extensive vs. Intensive Variables

Gymnastics & Diving

Boyle's Law and Charles's Law

Color Space & Vectors

Torque

Brain and Body Mass

6. Accuracy and Precision

Significant Digits

Paleontology

7. Reliability and Validity

Ratio Scale

Distance

Hamming Distance

Phonological Distance -- Distinctive Features

Vowels and Consonants- Ordinal Cube

What's a Bird?

Interval Scale

Temperature Scale

Ordinal Scale

Likert Scale

Nominal Scale

Sets, and Categorization

8. Sets: An Introduction

Languages

Cardinality, Empty Set

Union, Intersection, Partition, Power Set, Complement, Difference

Characteristic Bitstrings (functions)

9. Graphs: An Introduction

Subgraphs, Unions & Intersections of Graphs

Graph Representation: Incidence, Adjacency, Degree, Paths, Digraphs

Hypercubes, Complete Graphs, Bipartite Graphs

Multiple Comparisons of Historical Linguistics

Representation: Incidence and Adjacency

Matrices

Euler Circuits

10. Objects & Spaces

Cartesian Products, Vectors, Matrices, Tensors

Matrix Multiplication

Zero-One Matrices, Toeplitz matrices

Markov Matrices, Leontieff Matrices, Phonotactics Matrices

Rotation Matrices of Computer Graphics

Sonority Scale and Vectors

Venn Diagrams (Set Independence?)

11. Algebra: How many kinds are there?

Arithmetic

Language Capability

Substitutes and complements

Intelligence Theory and Testing

12. Boolean algebra

Infinity Arithmetic

Electrical Circuits and Infinity

Parallel Circuits vs. Series Circuits

XOR, EQ

Representation of Integers

Hamming Distance and XOR

Phoneme Maps

13. Propositional Logic

Implication

Hempel's Raven paradox

Paradoxes of Logic

Rules of Inference

Fallacies

Integers: Division Algorithm

Divisibility

Fundamental Theorem of Arithmetic

gcd, and lcm

Mod, Div, and All that (methods of proof)

Congruence Mod m

Pseudo-Random Number Generators

Caesar Cipher, ROT13, Comparative Method

Fuzzy Logic

Appendix: Axiomatizations of Logic

14. Quantification

Syllogisms

Continuous Products & Continuous Sums

Predicates

Quantification of Two Variables

Mathematical Induction

Time-Space Super-Liar Paradox

15. Relations

Reflexive, Symmetric, Anti-symmetric, Transitive relations

Representation of Relations

Set-theoretic representation

Matrix-representation

Graph-theoretic representation

16. Boolean Matrices and Relations

Composition of Relations--associative operation

Powers of Matrices of Relations

Equivalence Relation

Inverses

Operators and Operands (see Section 26: Operator Theory)

17. Partially Ordered Sets: partitions

Hasse Diagrams -- prerequisite structure of this book

Lattices, Subsets

18. Functions, Graphs, Vectors

One-to-One Functions

Onto Functions

One-to-one correspondence

Function Inverse

Graph Isomorphism

Minimal Spanning Trees

Family Trees

Cladistics

Genetic Tree of Indo-European Languages and Isoglosses

Vector Functions

Distances on Vectors, Weighted Distances

Intelligence Measurement

Systems of Equations -- Algebraic Modeling

19. Asymptotic Analysis and Limits

Big-O notation

20. Fuzzy Logic

Axioms

Invariants of Logic

Continuous Logics

Generalized Idempotent and Continuous Max-Min Operators

21. Counting Principles

Pigeonhole Principle

Sound Changes

Permutations

Combinations

Words, Subsets, Sentences, Constrained Sentences

Queues, Books, Phonotactics, Length constraints

Distributing Objects to Containers

Vervet Languages

Distribution of Meanings

Pascal's Identity, Vandermonde's Identity, Binomial Theorem

Inclusion-Exclusion Theorem

False Cognacy Problem

22. Induction, Recursion, Summation

23. Recurrence, Iteration, Counting

Linear homogeneous first-order difference equation

Fibonacci Series

False Cognacy Problem

Coupled Difference Equations

Bitstrings and Polynomials

Polya's Method of Counting

24. Formal Language Theory

Real Human Languages

Finite State Automata and Regular Languages

Context-Free Languages

Context-Sensitive Languages and Natural Languages

25. Simple Calculus

Rates

Integration from Summation

26. Probability Fundamentals

Addition Theorem

Multiplication Theorem

Independence And Conditional Probability

27. Discrete Probability Theory from Counting

28. Bayes Theorem

29. Operator Theory (see Chapter 17)

Linear Operators

Commutativity

Integration and Differentiation

30. Statistics

Histogram

Correlation-Regression

31. Expectation Operator & Density Functions

Expectation and Moments

32. Discrete Probability Functions (Mass Functions)

Uniform

Geometric - Bernoulli

Binomial

Hypergeometric

Poisson

Birthday Problem

33. Continuous Probability Density Functions

Uniform

Exponential

Gamma Density and Chi-Square Density

Gaussian Density and the Central Limit Theorem

34. Joint and Marginal Density Functions

35. Stochastic Processes

Stationarity

Markov Processes

Chapman-Kolmogorov Equations

Speech Recognition

Random Walk

36. Harmonic Analysis

Delta function

Fourier Series and Fourier Transform

37. Differential Equations, Green's Function and the Convolution Integral

Complete solution of the First Order Linear DE

Carbon Dating

Menzerath's Law

Altmann's Law

Damped Harmonic Oscillator

38. Time and Ensemble Moments - Stationarity and Ergodicity

Stationarity

Ensemble Correlation Functions

Time Averages and Ergodicity

39. Characteristic Functions, Moments and Cumulants

40. Stochastic Response of Linear Systems

Example of Word Production in a Language

Stochastic Excitation of the DHO

41. Fokker-Planck- Kolmogorov Methods

Generalization of the Random Walk

Replacement of a General Process by a Markov Process

Appendices; Calculation of some integrals; References.

ISBN 9783895866418. LINCOM Handbooks in Linguistics 10. 260pp.1999.

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