LHL 09: Mathematical and Computational Linguistics

Mathematical and Computational Linguistics

H. Mark Hubey
Montclair State University

As Lass (1980) has remarked, "system" is something talked about constantly in linguistics but never beyond paying just lip-service to the concept. This book shows how linguistics constitutes a "system". Linguists (except those who study Formal Language Theory) are confronted with a dilemma. What they study is partially based on physics and is in many respects mathematical; yet the mathematics books are divorced from linguistics and linguistics books are divorced from mathematics and physics. There are no books that teach mathematics for linguists or linguistics with mathematics. This book goes a long way toward accomplishing the integration of mathematics, physics and linguistics into a whole, in other words "a system", just like those that are studied by others in the quantitative disciplines such as physics, engineering, computer science or economics.

The methods of mathematics which are used in the books to elucidate system concepts and others that are needed in linguistics includes boolean algebra, differential equations, and fuzzy logic.

Furthermore it also explains in an intuitive manner,those concepts are not only from mathematics but also from the underlying physics and engineering up to and including acoustic theory of speech, speech recognition, and even nonlinearity/catastrophe theory and quantality of phonemic systems.

All the mathematics needed to form the mathematical foundations of linguistics is illustrated with examples from linguistics and thus may be thought of as "theories", those that should replace the standard literary linguistics tradition in the same way that literary economics is no longer the de facto standard. Physical/acoustic theory of speech is blended naturally into the phonological and phonetic standard, and the standard works are used as springboards to the development of vector space concepts that are necessary for comprehension of new works in speech synthesis and speech recognition. It is rather easy then to show how seemingly nonrelated topics such as sonority scales, child language development, and various linguistics processes such as assimilation, metathesis, fortition/lenition can be seen to be a part of the greater whole. Historical processes are also treated in terms of sound change and also in terms of the most basic ideas which are needed for a thorough understanding of the problems such as multiple scale phenomena, distance and similarity, probability theory, and stochastic processes. A book of this length cannot possibly discuss all of the mathematics necessary in detail, however, there is sufficient material to motivate the topics, and furthermore to point in the direction of further study.

Table of Contents:
0: INTRODUCTORY PRELIMINARIES
0.1. Continuous Nature of Speech
0.2. Functions and Mappings
0.3. Stochastic and Fuzzy Functions
0.4. Linear Operators, Relations, and Black Boxes
0.5. Discretization -- Numerical and Closed formSolutions
0.6. Representation, Meaning, and Definition
0.7. Significance, Precision, Accuracy, and Error
0.8. Beads on a String
0.9. Discretization of Speech
0.10. Simple Discretization
0.11. Mappings, Functions, Perception, and Excessive Mentalism
0.12. Binary, Ternary, or Infinity
0.13. Universal Distinctive Features
Appendix 0.A Sets, Classes, Relations, and Functions; Phonemes, Allophones, Semantics, Orthography; Appendix 0.B Dialogue
I: OPPOSITIONS, RELATIONS, GROUPS, AND LATTICES
I.1. Features, Binary Oppositions and Binary Relations
I.2. Simple Structures: Semigroups, Monoids, Groups, Isomorphisms, and Distances
I.3. More Complex Structures: Partial Ordering, Posets, N-cubes, Lattices, Hasse Diagram
I.4. `Lattice' of Vowels: Cardinal Vowel Diagrams, Ladefoged's Modification, Discrete Distance Metric, Trubetzkoy vowels
Appendix I.A Number Systems and Codes:The Binary System, K-maps, Gray codes; Appendix I.B Boolean Algebras
II: PRIVATE AND UNIVERSAL VOWEL SPACES
II.1. Cycles, Distance, Linear Ordering, and Hilbert Curves
II.2. Bloch and Trager Private Spaces
II.3. Chomsky & Halle Private Spaces
II.4. Complement of a Graph
II.5. Pure Vowels in 3-D
II.6. Discrete Universal Spaces
II.7. Karnaugh Maps and Finnish Vowels
II.8. American English Vowels
II.9. Other Spaces -- Stanford Phonology Archive
II.10. Binarity and Simplicity
III: COMPOUND VOWELS, DIPTHONGS, AND VECTORS
III.1. Vector Spaces and Phonemes
III.2. Time Domain Compositions -- Dipthongs and Glides
III.3. Dipthong = Vowel + Vowel
III.4. Dipthong = Vowel + Semivowel
III.5. Vectors and Dependency Phonology
III.6. Trubetzkoy and Stevens
III.7. Nonorthogonality of Features and Fant
IV: SPECTRAL DOMAIN DESCRIPTIONS
IV.1. Time-domain Signals
IV.2. Frequency Domain Descriptions
IV.3. Power Spectrum, Noise, and Autocorrelation
IV.4. Source and Filter
IV.5. Formant Functions and Approximations
IV.6 Dipthongs and Glides
IV.7. Compound Vowels
IV.8. Orthographic Projection of the Vocalic Phonemes of a Generic Language
IV.9. Formant Functions Again
IV.10. Formant Plots and Their Description
IV.11. Summary of Results
IV.12. Further Refinements of the Method
IV.13. The Formant Plots
IV.14. Nonlinearity, Quantality, and Catastrophe Theory
IV.15. Nonlinear Differential Equations and Quantality in Phonetics
Appendix IV.A: Fourier Analysis; Appendix IV.B: Convolution, Correlation, Spectral Density; Appendix IV.C Ordinary Linear Differential Equations; Appendix IV.D Orthogonal Functions; Appendix IV.E: Other Normalizations; Appendix IV.F Exponential Formant Approximations.
V: 3-D VECTOR PHASE SPACE FOR SPEECH
V.1. Properties of Consonants
V.2. Towards a Space
V.3. Consonant Vector Space
V.4. Dimensional Analysis and Buckingham Pi Theorem
V.5. Natural Groupings
V.6. Path Integrals and Minimization
V.7. Acoustic and Auditory Correlates in the Phase Space
V.8. Phones, Phonemes and Allophones
V.9. Sonority, Lenition, Fortition
V.10. Child Language Development and Aphasia
V.11. Vowels in Phase Space
V.12. Distance and Birth of New Phonemes
V.13. Experimental Evidence from Dipthongs
V.14. Implications for Phonological Space
V.15. The Ordinal Vowel Cube
V.16. Sonority Scales
V.17. Vector Representation
V.18. Dynamic Stochastic Processes and Speech Realization
V.19. Forced Binary Discrimination Tests and Probability Theory
V.20. The Ambiguity Function: Another Interpretation
V.21. Entropy, Uncertainty, and Information Theory
V.22. Fuzzy Sets and Catastrophe Theory
V.23. Fuzzy Functions for Multiple Discriminations along a Single Stimulus
V.24. Binary Discriminations for Multiple Stimuli and Stochastic Proceses
VI: UPPER-LEVEL DISTANCE METRICS
VI.1. Consonant Clusters
VI.2. Turkish Vowel Harmony
VI.3. Grammar for Transitions
VI.4. Turkish Syllabification
VI.5. Word Level Measures
VI.6. Topology of Vowel Spaces of Languages Examples from Arabic, English, Chinese, French,German, Italian, Latin, Sanskrit, Irish and Tamil
VI.7. Word Formation Rules and Borrowing
VI.8. Residues of Languages and Distance Functions--Sprachbunde and Sprachfamilien.
VI.9. Propagation, Waves and Diffusion of Innovation
VI.10. Semitic Word Formation Examples
VII: MULTIDIMENSIONAL INHERITANCE
VII.0. Introduction
VII.1. Temporal and Spatial Scaling
VII.2. Time Complexity vs Space Complexity -- Compute vs Memory Bound Processes
VII.3. Order of Magnitude and Complexity
VII.4. Intensive and Extensive Parameters
VII.5. Measurement Scales: Absolute and Relative Measures
VII.6. Stability, Relaxation Time and Correlation Time
VII.7. Process vs State
VII.8. Open Systems vs Closed Systems
VII.9. Time Scales and Linguistics
VII.10 Word Orders and Artificial Non-natural Languages
VII.11. Prehistoric Times and Language
VII.12. Change: Is it infinite ?
VII.13. Family Trees
VII.14. Distance Functions
VII.15. Matching Lexemes and Semantemes
VII.16. Dynamic Stochastic Processes and Language
VII.17. Summary
Appendix VII.A Cognates or Not; Appendix VII.B: Differential Equations Initial Value Problems, Stability and Equilibrium, Static vs Dynamic Equilibrium (Steady State) Appendix VII.C Stochastic Processes; Randomness, Mass and Density Functions, Averaging, Stochastic Differential Equations, Stationarity
VIII: PHONOLOGY, MORPHOLOGY, AND SYNTAX
VIII.0. Modularity
VIII.1. Upper Level Syntactic Structure of the World's Languages
VIII.2. Permutations, Reflections, and Rotations
VIII.3. Same Set Permutations
VIII.4. Tree Traversals and Permutation Groups
VIII.5. Phonology and Morphology
VIII.6. Postfixing Morphology and Morphophonology
VIII.7. Premorphing Languages and Phonology
VIII.8. Transformational Grammar and the H-operators
VIII.9. Infixing and Erase and Replace
VIII.10.Combined Inmorphing and Endmorphing
VII.11. Indonesian and German
VIII.12.Simplicity Metric
Appendix VIII.A: String Quasi-Algebra
IX: SYNTACTIC AND SEMANTIC STRUCTURE OF NEAR NATURAL LANGUAGES
IX.1. Prologue
IX.2. Graphs
IX.3. Binary Trees and Their Growth Patterns
IX.4. Trees and Tree Traversals
IX.5. Operands, Operators and Operations
IX.6. Formal Language Theory
IX.7. Another Kind of Space for Sentences

ISBN 9783895866395. LINCOM Handbooks in Linguistics 09. 450pp.1999.