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Pregroup grammar (PG) is a
grammar formalism In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...
intimately related to
categorial grammar Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semant ...
s. Much like categorial grammar (CG), PG is a kind of
type logical grammar Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * Typ ...
. Unlike CG, however, PG does not have a distinguished function type. Rather, PG uses inverse types combined with its monoidal operation.


Definition of a pregroup

A pregroup is a
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
(A, 1, \cdot, -^l, -^r, \leq) such that (A, 1, \cdot) is a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
, satisfying the following relations: * x^l \cdot x \leq 1 \qquad x \cdot x^r \leq 1     (contraction) * 1 \leq x \cdot x^l \qquad 1 \leq x^r \cdot x     (expansion) The contraction and expansion relations are sometimes called
Ajdukiewicz Kazimierz Ajdukiewicz (12 December 1890 – 12 April 1963) was a Polish philosopher and logician, a prominent figure in the Lwów–Warsaw school of logic. He originated many novel ideas in semantics. Among these was categorial grammar, a highl ...
laws. From this, it can be proven that the following equations hold: * 1^l = 1 = 1^r * x^ = x = x^ * (x\cdot y)^l = y^l \cdot x^l \qquad (x\cdot y)^r = y^r \cdot x^r x^l and x^r are called the
left Left may refer to: Music * ''Left'' (Hope of the States album), 2006 * ''Left'' (Monkey House album), 2016 * "Left", a song by Nickelback from the album ''Curb'', 1996 Direction * Left (direction), the relative direction opposite of right * L ...
and
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
s of ''x'', respectively. The symbol \cdot and \leq are also written \otimes and \to respectively. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, pregroups are also known as autonomous categories or (non-symmetric) compact closed categories. More typically, x \cdot y will just be represented by adjacency, i.e. as xy.


Definition of a pregroup grammar

A pregroup grammar consists of a
lexicon A lexicon is the vocabulary of a language or branch of knowledge (such as nautical or medical). In linguistics, a lexicon is a language's inventory of lexemes. The word ''lexicon'' derives from Koine Greek language, Greek word (), neuter of () ...
of words (and possibly
morphemes A morpheme is the smallest meaningful constituent of a linguistic expression. The field of linguistic study dedicated to morphemes is called morphology. In English, morphemes are often but not necessarily words. Morphemes that stand alone a ...
) ''L'', a set of atomic types ''T'' which freely generates a pregroup, and a relation : that relates words to types. In simple pregroup grammars, typing is a function that maps words to only one type each.


Examples

Some simple, intuitive examples using English as the language to model demonstrate the core principles behind pregroups and their use in linguistic domains. Let ''L'' = , let ''T'' = , and let the following typing relation holds: : \textit : N \qquad \textit : N \qquad \textit : N \cdot N_0^l \qquad \textit : N_0 \qquad \textit : N_0 : \textit : N^r \cdot S \cdot N^l \qquad \textit : N^r \cdot S \qquad \textit : S^r \cdot N^ \cdot N^r \cdot S \cdot N^l A sentence ''S'' that has type ''T'' is said to be grammatical if T \leq S. We can prove this by use of a chain of \leq. For example, we can prove that \textit\ \textit\ \textit : N \cdot N^r \cdot S \cdot N^l \cdot N is grammatical by proving that N \cdot N^r \cdot S \cdot N^l \cdot N \leq S: : N \cdot N^r \cdot S \cdot N^l \cdot N ~\leq~ S \cdot N^l \cdot N ~\leq~ S by first using contraction on N \cdot N^r and then again on N^l \cdot N. A more convenient notation exists, however, that indicates contractions by connecting them with a drawn link between the contracting types (provided that the links are nested, i.e. don't cross). Words are also typically placed above their types to make the proof more intuitive. The same proof in this notation is simply A more complex example proves that ''the dog barked at the cat'' is grammatical:


Historical notes

Pregroup grammars were introduced by
Joachim Lambek Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as ...
in 1993 as a development of his syntactic calculus, replacing the quotients by adjoints. Such adjoints had already been used earlier by
Harris Harris may refer to: Places Canada * Harris, Ontario * Northland Pyrite Mine (also known as Harris Mine) * Harris, Saskatchewan * Rural Municipality of Harris No. 316, Saskatchewan Scotland * Harris, Outer Hebrides (sometimes called the Isle o ...
but without iterated adjoints and expansion rules. Adding such adjoints was interesting to handle more complex linguistic cases, where the fact that a^ \neq a is needed. It was also motivated by a more algebraic viewpoint: the definition of a pregroup is a weakening of that of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
, introducing a distinction between the left and right inverses and replacing the equality by an order. This weakening was needed because using types from a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
would not work: an adjective would get the type N \cdot N^ = 1, hence it could be inserted at any position in the sentence. Pregroup grammars have then been defined and studied for various languages (or fragments of them) including
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
,
Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance language *** Regional Ita ...
, French,
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...
and
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
. Languages with a relatively free word order such as Sanskrit required to introduce commutation relations to the pregroup, using precyclicity.


Semantics of pregroup grammars

Because of the lack of function types in PG, the usual method of giving a semantics via the
λ-calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation tha ...
or via function denotations is not available in any obvious way. Instead, two different methods exist, one purely formal method that corresponds to the λ-calculus, and one denotational method analogous to (a fragment of) the tensor mathematics of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


Purely formal semantics

The purely formal semantics for PG consists of a logical language defined according to the following rules: * Given a set of atomic terms ''T'' = and atomic function symbols ''F'' = (where subscripts are meta-notational indicating arity), and variables ''x'', ''y'', ..., all constants, variables, and well-formed function applications are basic terms (a function application is well-formed when the function symbol is applied to the appropriate number of arguments, which can be drawn from the atomic terms, variables, or can be other basic terms) * Any basic term is a term * Given any variable ''x'', 'x''is a term * Given any terms ''m'' and ''n'', m \cdot n is a term Some examples of terms are ''f''(''x''), ''g''(''a'',''h''(''x'',''y'')), g(x,b) \cdot /math>. A variable ''x'' is free in a term ''t'' if 'x''does not appear in ''t'', and a term with no free variables is a closed term. Terms can be typed with pregroup types in the obvious manner. The usual conventions regarding α conversion apply. For a given language, we give an assignment ''I'' that maps typed words to typed closed terms in a way that respects the pregroup structure of the types. For the English fragment given above we might therefore have the following assignment (with the obvious, implicit set of atomic terms and function symbols): : \begin I(\textit : N) &= j : E \\ I(\textit : N) &= m : E \\ I(the : N \cdot N_0^l) &= \iota(p) \cdot : E \cdot E_0^l \\ I(dog : N_0) &= dog : E_0 \\ I(cat : N_0) &= cat : E_0 \\ I(met : N^r \cdot S \cdot N^l) &= \cdot met(x,y) \cdot : E^r \cdot T \cdot E^l \\ I(barked : N^r \cdot S) &= \cdot barked(x) : E^r \cdot T \\ I(at : S^r \cdot N^ \cdot N^r \cdot S \cdot N^l) &= \cdot y \cdot \cdot at(x,z) \cdot : T^r \cdot E^ \cdot E^r \cdot T \cdot E^l \end where ''E'' is the type of entities in the domain, and ''T'' is the type of truth values. Together with this core definition of the semantics of PG, we also have a reduction rules that are employed in parallel with the type reductions. Placing the syntactic types at the top and semantics below, we have For example, applying this to the types and semantics for the sentence \textit\ \textit\ \textit : N \cdot (N^r \cdot S \cdot N^l) \cdot N (emphasizing the link being reduced) For the sentence \textit\ \textit\ \textit\ \textit\ \textit\ \textit : (N \cdot N_0^l) \cdot N_0 \cdot (N^r \cdot S) \cdot (S^r \cdot N^ \cdot N^r \cdot S \cdot N^l) \cdot (N \cdot N_0^l) \cdot N_0:


See also

*
Compact closed category In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the ...
*
Lambek calculus Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and sema ...


References

* * * Claudia Casadio (2004)
Pregroup Grammar. Theory and Applications
{{Reflist Grammar frameworks Semantics Type theory