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Example Applications of Mathematical Linguistics

Parse tree

Zipf's Law

Semantic network

Modal Logic

Phonological rule

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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 is used in language modeling, including formal grammars, language representation, and historical linguistic trends.

Semantic classes, word classes, natural classes, 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.

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.

Context-sensitive rewriting rules of the form a → b / c _ d, used in linguistics to model phonological rules 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.^[1]

Weighted FSTs found applications in natural language processing, including machine translation, and in machine learning.^[2][3] An implementation for part-of-speech tagging can be found as one component of the OpenGrm^[4] library.

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.^[5]

Trees have several applications in linguistics, including:

• Parsing trees
• Sentence diagrams
• Semantic networks
• Language family trees
• Etymology trees

Other graphs that are used in linguistics include:

• Weighted graphs, which are used to model the lexical similarity between different languages (after computing lexicostatistics).
• Lattice graphs, which can model optimality theory.

Formal linguistics is the branch of linguistics which uses formal languages, formal grammars and first-order logical expressions for the analysis of natural languages. Since the 1980s, the term is often used to refer to Chomskyan linguistics.^[6]

Logic is used to model syntax, formal semantics, and pragmatics. Modal logic can model syntax that employs different grammatical moods.^[7] 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 conditional propositions.^[8]

The Logical Relations of Lexical Relations

Lexical Relation	Logical Relation	Example
Synonym	${\displaystyle x\leftrightarrow y}$	If pavement then sidewalk, and if sidewalk then pavement.
Complementary antonyms	${\displaystyle (x\rightarrow \neg y)\land (y\rightarrow \neg x)}$	If alive then not dead, and if dead then not alive.
Gradable antonyms	${\displaystyle (x\rightarrow \neg y)\land (y\rightarrow \neg x)}$	If good then not bad, and if bad then not good.
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.
Relational antonyms (Verbs)	If A Xs to B, then B Ys from A	If A gives to B, then B receives from A.
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 Zs	A rose and a tulip are both flowers.
Meronym	the parts of a Y include the Xs	The parts of a wheel include the spokes.
Quasi-Meronym	An X belongs to a Y	A tribesman belongs to a tribe.

Methods of formal linguistics were introduced by 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.^[9][10] 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 structuralist Lucien Tesnière,^[11] has been used widely in natural language processing.

The Fast Fourier Transform, Kalman filters, and autoencoding are all used in signal processing (advanced phonetics, speech recognition).

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.

Student's t-test can be used to determine whether the occurrence of a collocation in a corpus is statistically significant.^[12] For a bigram ${\displaystyle w_{1}w_{2}}$, let ${\displaystyle P(w_{1})={\frac {\#w_{1}}{N}}}$ be the unconditional probability of occurrence of ${\displaystyle w_{1}}$ in a corpus with size ${\displaystyle N}$, and let ${\displaystyle P(w_{2})={\frac {\#w_{2}}{N}}}$ be the unconditional probability of occurrence of ${\displaystyle w_{2}}$ in the corpus. The t-score for the bigram ${\displaystyle w_{1}w_{2}}$ is calculated as:

${\displaystyle t={\frac {{\bar {x}}-\mu }{\sqrt {\frac {s^{2}}{N}}}},}$

where ${\displaystyle {\bar {x}}={\frac {\#w_{i}w_{j}}{N}}}$ is the sample mean of the occurrence of ${\displaystyle w_{1}w_{2}}$, ${\displaystyle \#w_{1}w_{2}}$ is the number of occurrences of ${\displaystyle w_{1}w_{2}}$, ${\displaystyle \mu =P(w_{i})P(w_{j})}$ is the probability of ${\displaystyle w_{1}w_{2}}$ under the null-hypothesis that ${\displaystyle w_{1}}$ and ${\displaystyle w_{2}}$ appear independently in the text, and ${\displaystyle s^{2}={\bar {x}}(1-{\bar {x}})\approx {\bar {x}}}$ is the sample variance. With a large ${\displaystyle N}$, the t-test is equivalent to a Z-test.

Lexicostatistics can model the lexical similarities between languages that share a language family, sprachbund, language contact, or other historical connections.

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.^[13] Synergetic linguistics was from its very beginning specifically designed for this purpose.^[14] 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 is a subfield of quantitative linguistics which applies quantitative analysis to comparative linguistics. It makes use of lexicostatistics and glottochronology, and the borrowing of phylogenetics from biology.

• Computational linguistics
• International Linguistics Olympiad