The Mathematics of Language

Marcus Kracht

• Publisher: De Gruyter Mouton
• Year: 2011
• Print ISBN:
  9783110176209
• eBook ISBN:
  9783110895667
• Topic: Languages & Linguistics
• Subtopic: Authorship
• Index: Languages & Linguistics

This book studies language(s) and linguistic theories from a mathematical point of view. Starting with ideas already contained in Montague's work, it develops the mathematical foundations of present day linguistics. It equips the reader with all the background necessary to understand and evaluate theories as diverse as Montague Grammar, Categorial Grammar, HPSG and GB. The mathematical tools are mainly from universal algebra and logic, but no particular knowledge is presupposed beyond a certain mathematical sophistication that is in any case needed in order to fruitfully work within these theories. The presentation focuses on abstract mathematical structures and their computational properties, but plenty of examples from different natural languages are provided to illustrate the main concepts and results. In contrast to books devoted to so-called formal language theory, languages are seen here as semiotic systems, that is, as systems of signs. A language sign correlates form with meaning. Using the principle of compositionality it is possible to gain substantial insight into the interaction between form and meaning in natural languages.

Table of contents

• 1 Fundamental Structures
• 1 Algebras and Structures
• 2 Semigroups and Strings
• 3 Fundamentals of Linguistics
• 4 Trees
• 5 Rewriting Systems
• 6 Grammar and Structure
• 7 Turing machines
• 2 Context Free Languages
• 1 Regular Languages
• 2 Normal Forms
• 3 Recognition and Analysis
• 4 Ambiguity, Transparency and Parsing Strategies
• 5 Semilinear Languages
• 6 Parikh’s Theorem
• 7 Are Natural Languages Context Free?
• 3 Categorial Grammar and Formal Semantics
• 1 Languages as Systems of Signs
• 2 Propositional Logic
• 3 Basics of λ-Calculus and Combinatory Logic
• 4 The Syntactic Calculus of Categories
• 5 The AB-Calculus
• 6 The Lambek-Calculus
• 7 Pentus’ Theorem
• 8 Montague Semantics I
• 4 Semantics
• 1 The Nature of Semantical Representations
• 2 Boolean Semantics
• 3 Intensionality
• 4 Binding and Quantification
• 5 Algebraization
• 6 Montague Semantics II
• 7 Partiality and Discourse Dynamics
• 5 PTIME Languages
• 1 Mildly-Context Sensitive Languages
• 2 Literal Movement Grammars
• 3 Interpreted LMGs
• 4 Discontinuity
• 5 Adjunction Grammars
• 6 Index Grammars
• 7 Compositionality and Constituent Structure
• 8 de Saussure Grammars
• 6 The Model Theory of Linguistic Structures
• 1 Categories
• 2 Axiomatic Classes I: Strings
• 3 Categorization and Phonology
• 4 Axiomatic Classes II: Exhaustively Ordered Trees
• 5 Transformational Grammar
• 6 GPSG and HPSG
• 7 Formal Structures of GB
• Bibliography
• Index