View source for Draft:Mathematical linguistics

{{Short description|Branch of applied mathematics}}
{{Draft topics|linguistics|stem}}
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{{Use American English|date=December 2024}}
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{{Multiple image|perrow = 2 / 2|total_width = 300
| header            = Example Applications of Mathematical Linguistics
| image1            = Parse-tree.svg
| alt1              = Parse tree of the string "aab" with respect to the production rules.
| caption1          = [[Parse tree]]
| image2            = Zipf_distribution_PMF.png
| alt2              = Plot of the Zipf PMF for N = 10.
| caption2          = [[Zipf's Law]]
| image3            = Semantic_Net.svg
| alt3              = Example of a semantic network
| caption3          = [[Semantic network]]
| image4            = Diagram_of_Normal_Modal_Logics.png
| alt4              = Diagram of common modal logics
| caption4          = [[Modal logic#Axiomatic_systems|Modal Logic]]
| image5            = Intervocalic_tapping_phonological_rule.svg
| alt5              = Phonological Rule for Intervocalic flapping in American English
| caption5          = [[Phonological rule]]
}}
{{Math topics TOC}}
{{Linguistics}}

'''Mathematical linguistics''' is the application of [[mathematics]] to model phenomena and solve problems in general [[linguistics]] and [[theoretical linguistics]]. Mathematical linguistics has a significant amount of overlap with [[computational linguistics]].

==Discrete Mathematics==
Discrete mathematics is used in language modeling, including formal grammars, language representation, and historical linguistic trends.

===Set Theory===
[[Semantic class]]es, [[Part of speech|word class]]es, [[natural class]]es, and the [[allophonic]] variations of each [[phoneme]] in a language are all examples of applied set theory. Set theory and [[concatenation theory]] are used extensively in phonetics and phonology.

===Combinatorics===
In [[phonotactics]], combinatorics is useful for determining which sequences of phonemes are permissible in a given language, and for calculating the total number of possible syllables or words, based on a given set of phonological constraints. [[Combinatorics on words]] can reveal patterns within words, morphemes, and sentences.

===Finite-State Transducers===
Context-sensitive rewriting rules of the form ''a'' → ''b'' / ''c'' _ ''d'', used in [[linguistics]] to model [[phonological rule]]s and [[sound change]], are computationally equivalent to finite-state transducers, provided that application is nonrecursive, i.e. the rule is not allowed to rewrite the same substring twice.<ref>{{cite web | url=http://acl.ldc.upenn.edu/J/J94/J94-3001.pdf | title=Regular Models of Phonological Rule Systems | access-date=August 25, 2012 | archive-url=https://web.archive.org/web/20101011225829/http://acl.ldc.upenn.edu/J/J94/J94-3001.pdf | archive-date=October 11, 2010 | url-status=dead }}</ref>

Weighted FSTs found applications in [[natural language processing]], including [[machine translation]], and in [[machine learning]].<ref name="DrosteKuich2009">{{cite book|editor1=Manfred Droste |editor2=Werner Kuich |editor3=Heiko Vogler |title=Handbook of Weighted Automata|year=2009|publisher=Springer Science & Business Media|isbn=978-3-642-01492-5|chapter=Applications of Weighted Automata in Natural Language Processing|author1=Kevin Knight |author2=Jonathan May }}</ref><ref>{{cite web | url=http://www.cs.nyu.edu/~mohri/pub/fsmnlp08.pdf | title=Learning with Weighted Transducers| access-date=April 29, 2017}}</ref> An implementation for [[part-of-speech tagging]] can be found as one component of the OpenGrm<ref>[http://opengrm.org/ OpenGrm]</ref> library.

===Algorithms===
[[Optimality theory]] (OT) and maximum entropy (Maxent) phonotactics use algorithmic approaches when evaluating candidate forms (phoneme strings) for determining the phonotactic constraints of a language.<ref>{{cite journal |last1=Hayes |first1=Bruce |last2=Wilson |first2=Colin |date=July 1, 2008 |title=A Maximum Entropy Model of Phonotactics and Phonotactic Learning |url=https://brucehayes.org/papers/HayesAndWilsonPhonotactics2008.pdf |journal=Linguistic Inquiry |volume=39 |issue=3 |publisher=Massachusetts Institute of Technology |pages=379–440 |doi=10.1162/ling.2008.39.3.379 |access-date=February 13, 2025}}</ref>

===Graph Theory===
Trees have several applications in linguistics, including:
*[[Parse tree|Parsing trees]]
*[[Sentence diagram]]s
*[[Semantic network]]s
*[[Language family]] trees
*[[Etymology]] trees
*[[Glossary_of_graph_theory_terms#weighted_graph|Weighted graphs]] are used to model the lexical similarity between different languages (after computing lexicostatistics):
*[[Lattice graph]]s can model [[optimality theory]]

==Formal linguistics==
{{Main|Formal linguistics}}
Formal linguistics is the branch of [[linguistics]] which uses [[formal language]]s, [[formal grammar]]s and [[first-order logic]]al expressions for the analysis of [[natural language]]s. Since the 1980s, the term is often used to refer to [[Chomskyan linguistics]]. <ref>{{cite web |last1=Haspelmath |first1=Martin |title=How formal linguistics appeared and disappeared from the scene |date=2019 |doi=10.58079/nsuq |url=https://dlc.hypotheses.org/1698}}</ref>

===Logic===
{{Main|Logic}}
Logic is used to model [[syntax]], [[Formal semantics (natural language)|formal semantics]], and [[pragmatics]]. [[Modal logic]] can model syntax that employs different [[grammatical mood]]s.<ref>Kaufmann, S.; Condoravdi, C. & Harizanov, V. (2006) [https://web.archive.org/web/20181223121024/https://pdfs.semanticscholar.org/e1c0/a81b02d71384ea679037e5cdcd83c8e4eacd.pdf Formal approaches to modality]. Formal approaches to modality. In: Frawley, W. (Ed.). The Expression of Modality. Berlin, New York: Mouton de Gruyter</ref> Most [[linguistic universals]] (e.g. [[Greenberg's linguistic universals]]) employ [[propositional logic]]. [[Lexical relations]] between words can be determined based on whether a pair of words satisfies [[Material conditional|conditional]] propositions.<ref> {{cite book |last1=Atkins |first1=A. T. |last2=Rundell |first2=Michael |date=2008 |title=The Oxford Guide to Practical Lexicography |location=USA |publisher=Oxford University Press |page=132-144 |isbn=978-0-19-927771-1}}</ref>

{| class="wikitable"
|+ The Logical Relations of Lexical Relations
|-
! Lexical Relation !! Logical Relation !! Example
|-
| [[Synonym]] || <math>x \leftrightarrow y</math> || If ''pavement'' then ''sidewalk'', and if ''sidewalk'' then ''pavement''.
|-
| [[Opposite#Complementary antonyms|Complementary antonyms]] || <math>(x \rightarrow \neg y) \land (y \rightarrow \neg x)</math> || If ''alive'' then not ''dead'', and if ''dead'' then not ''alive''.
|-
| [[Opposite#Gradable_antonyms|Gradable antonyms]] || <math>(x \rightarrow \neg y) \land (y \rightarrow \neg x)</math> || If ''good'' then not ''bad'', and if ''bad'' then not ''good''.
|-
| [[Opposite#Relational_antonyms|Relational antonyms]] (Nouns) || If A is B's ''X'', then B is A's ''Y'' || If A is B's ''parent'', then B is A's ''child''.
|-
| [[Opposite#Relational_antonyms|Relational antonyms]] (Verbs) || If A ''X''s to B, then B ''Y''s from A || If A ''gives'' to B, then B ''receives'' from A.
|-
| [[Opposite#Relational_antonyms|Relational antonyms]] (Prepositions) || If A is ''X'' B, then B is ''Y'' A || If A is ''below'' B, then B is ''above'' A.
|-
| [[Hyponym]] || ''X'' is a ''Y'', but ''Y'' is not only an ''X'' || If a ''terrier'', then a ''dog''.
|-
| [[Cohyponym]] || ''X'' and ''Y'' are both ''Z''s || A ''rose'' and a ''tulip'' are both ''flowers''.
|-
| [[Meronym]] || the parts of a ''Y'' include the ''X''s || The parts of a ''wheel'' include the ''spokes''.
|-
| Quasi-[[Meronym]] || An ''X'' belongs to a Y || A ''tribesman'' belongs to a ''tribe''.
|}

===Semiotics===
{{Main|Semiotics}}
Methods of formal linguistics were introduced by [[semiotics|semioticians]] such as [[Charles Sanders Peirce]] and [[Louis Hjelmslev]]. Building on the work of [[David Hilbert]] and [[Rudolf Carnap]], Hjelmslev proposed the use of formal grammars to analyse, generate and explain language in his 1943 book ''Prolegomena to a Theory of Language''.<ref name=Hjelmslev_1969>{{cite book |last=Hjelmslev|first=Louis |title=Prolegomena to a Theory of Language |publisher=University of Wisconsin Press |date=1969 |orig-year=First published 1943|isbn= 0299024709| author-link=Louis Hjelmslev}}</ref><ref name=Seuren_1998>{{cite book|author=Seuren, Pieter A. M. |authorlink = Pieter Seuren |year=1998|title=Western linguistics: An historical introduction|publisher=Wiley-Blackwell|isbn=0-631-20891-7|pages=160–167}}</ref> In this view, language is regarded as arising from a mathematical relationship between meaning and form.

The formal description of language was further developed by linguists including [[J. R. Firth]] and [[Simon Dik]], giving rise to modern grammatical frameworks such as [[systemic functional linguistics]] and [[functional discourse grammar]]. Computational methods have been developed by the framework [[functional generative description]] among others.

[[Dependency grammar]], created by French [[Structural linguistics|structuralist]] [[Lucien Tesnière]],<ref name="Tesniere_1959">{{cite book |last=Tesnière|first=Lucien |year=1959|  title=Éléments de syntaxe structurale  |publisher=Klincksieck }}</ref> has been used widely in [[natural language processing]].

==Differential Equations & Multivariate Calculus==
The [[Speech_recognition#Hidden_Markov_models|Fast Fourier Transform]], [[Kalman filters]], and [[Neural_machine_translation|autoencoding]] are all used in signal processing (advanced phonetics, speech recognition).

==Statistics==
In linguistics, statistical methods are necessary to describe and validate research results, as well as to understand observations and trends within an area of study.
<!-- [[Statistical learning in language acquisition]] -->

===Corpus statistics===
{{main|Corpus linguistics}}
[[Student's t-test|Student's ''t''-test]] can be used to determine whether the occurrence of a [[collocation]] in a corpus is statistically significant.<ref>{{Cite book|title=Foundations of Statistical Natural Language Processing|url=https://archive.org/details/foundationsstati00mann_118|url-access=limited|last1=Manning|first1=Chris|last2=Schütze|first2=Hinrich|publisher=MIT Press|year=1999|isbn=0262133601|location=Cambridge, MA|pages=[https://archive.org/details/foundationsstati00mann_118/page/n202 163]–166}}</ref> For a [[bigram]] <math>w_1w_2</math>, let <math>P(w_1) = \frac{\#w_1}{N}</math> be the unconditional probability of occurrence of <math>w_1</math> in a corpus with size <math>N</math>, and let <math>P(w_2) = \frac{\#w_2}{N}</math> be the unconditional probability of occurrence of <math>w_2</math> in the corpus. The t-score for the bigram <math>w_1w_2</math> is calculated as:

: <math>t = \frac{\bar{x} - \mu}{\sqrt{\frac{s^2}{N}}}, </math>

where <math>\bar{x} = \frac{\# w_iw_j}{N}</math> is the sample mean of the occurrence of <math>w_1w_2</math>, <math>\#w_1w_2</math> is the number of occurrences of <math>w_1w_2</math>, <math>\mu = P(w_i)P(w_j)</math> is the probability of <math>w_1w_2</math> under the null-hypothesis that <math>w_1</math> and <math>w_2</math> appear independently in the text, and <math>s^2 = \bar{x}(1-\bar{x}) \approx \bar{x}</math> is the sample variance. With a large <math>N</math>, the ''t''-test is equivalent to a [[z-test|''Z''-test]].

===Lexicostatistics===
{{Main|Lexicostatistics}}
Lexicostatistics can model the lexical similarities between languages that share a language family, [[sprachbund]], [[language contact]], or other historical connections.

===Quantitative linguistics===
{{main|Quantitative linguistics}}
Quantitative linguistics (QL) deals with language learning, language change, and application as well as structure of natural languages. QL investigates languages using statistical methods; its most demanding objective is the formulation of language laws and, ultimately, of a general [[theory of language]] in the sense of a set of interrelated languages laws.<ref>Reinhard Köhler: ''Gegenstand und Arbeitsweise der Quantitativen Linguistik''. In: Reinhard Köhler, Gabriel Altmann, Rajmund G. Piotrowski (Hrsg.): ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch.'' de Gruyter, Berlin/ New York 2005, pp. 1–16. {{ISBN|3-11-015578-8}}.</ref> [[Synergetic linguistics]] was from its very beginning specifically designed for this purpose.<ref>Reinhard Köhler: ''Synergetic linguistics''. In: Reinhard Köhler, Gabriel Altmann, Rajmund G. Piotrowski (Hrsg.): ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch.'' de Gruyter, Berlin/ New York 2005, pp. 760–774. {{ISBN|3-11-015578-8}}.</ref>
QL is empirically based on the results of language statistics, a field which can be interpreted as statistics of languages or as statistics of any linguistic object. This field is not necessarily connected to substantial theoretical ambitions. [[Corpus linguistics]] and [[computational linguistics]] are other fields which contribute important [[empirical evidence]].

===Quantitative comparative linguistics===
{{main|Quantitative comparative linguistics}}
Quantitative comparative linguistics is a subfield of quantitative linguistics which applies [[quantitative research|quantitative analysis]] to [[comparative linguistics]]. It makes use of [[lexicostatistics]] and [[glottochronology]], and the borrowing of [[phylogenetics]] from biology.

==See Also==
*[[Computational linguistics]]
*[[International Linguistics Olympiad]]

==References==
{{Reflist|3}}

==Bibliography==
{{refbegin|30}}
*{{cite book |last=Kornai |first=András |author-link=András Kornai |date=2008 |title=Mathematical Linguistics |location=London, United Kingdom |publisher=Springer-Verlag |isbn=978-1-84628-985-9}}
*{{cite book |last=Aleksej Vsevolodovic |first=Gladkij |date=1969 |title=Elementy matematiceskoj lingvistiki |trans-title=Elements of Mathematical Linguistics |language=Russian |location=Berlin, Germany |publisher=Mouton Publishers |isbn=90-279-3118-6}}
*{{cite book |last=Kracht |first=Marcus |date=September 16, 2003 |title=The Mathematics of Language |location=PO Box 951543, 450 Hilgard Avenue, Los Angeles, CA 90095–1543 USA |url=https://linguistics.ucla.edu/people/Kracht/html/formal.pdf |access-date=February 14, 2025}}
*{{cite book |last1=Parte |first1=Barbara H. |last2=ter Meulen |first2=Alice |last3=Wall |first3=Robert E. |editor-last1=Chierchia |editor-first1=Gennaro |editor-last2=Jacobson |editor-first2=Pauline |editor-last3=Pelletier |editor-first3=Francis J. |date=1993 |title=Mathematical Methods in Linguistics |volume=30 |location=P.O. Box 17, 3300 AA Dordrecht, The Netherlands |publisher=Kluwer Academic Publishers |isbn=978-94-009-2213-6}}
{{refend}}

{{Areas of mathematics}}

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{{DEFAULTSORT:Mathematical Linguistics}}
[[Category:Mathematical linguistics| ]]
[[Category:Applied mathematics]]
[[Category:Mathematics]]
[[Category:Linguistics]]
[[Category:Formal sciences]]
[[Category:Computational linguistics]]
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