Thompson's construction algorithm

Template:Underlinked Pregroup grammar (PG) is a grammar formalism intimately related to categorial grammars. Much like categorial grammar (CG), PG is a kind of type logical grammar. 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 algebra $(A,1,\cdot ,-^{l},-^{r},\leq )$ such that $(A,1,\cdot )$ is a monoid, 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 laws.

From this, it can be proven that the following equations hold:

• $1^{l}=1=1^{r}$

• $x^{lr}=x=x^{rl}$

• $(x\cdot y)^{l}=y^{l}\cdot x^{l}\qquad (x\cdot y)^{r}=y^{r}\cdot x^{r}$

$x^{l}$ and $x^{r}$ are called the left and right adjoints of x, respectively.

The symbol $\cdot$ and $\leq$ are also written $\otimes$ and $\to$ respectively. In category theory, pregroups are also known as (non-symmetric) compact closed (monoidal) 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 of words (and possibly morphemes) L, a set of atomic types T which freely generates a pregroup, and an 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 = {John, Mary, the, dog, cat, met, barked, at}, let T = {N, S, N₀}, and let the following typing relation holds:

${\text{John}}:N\qquad {\text{Mary}}:N\qquad {\text{the}}:N\cdot N_{0}^{l}\qquad {\text{dog}}:N_{0}\qquad cat:N_{0}$

$met:N^{r}\cdot S\cdot N^{l}\qquad barked:N^{r}\cdot S\qquad at:S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l}$

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 $John\ met\ Mary:N\cdot N^{r}\cdot S\cdot N^{l}\cdot N$ 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:

Semantics of pregroup grammars

Because of the lack of function types in PG, the usual method of giving a semantics via the λ-calculus 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.

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 = {a, b, ...} and atomic function symbols F = {f_m, g_n, ...} (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)), $g(x,b)\cdot [x]$. 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):

$I(John:N)=j:E$

$I(Mary:N)=m:E$

$I(the:N\cdot N_{0}^{l})=\iota (p)\cdot [p]:E\cdot E_{0}^{l}$

$I(dog:N_{0})=dog:E_{0}$

$I(cat:N_{0})=cat:E_{0}$

$I(met:N^{r}\cdot S\cdot N^{l})=[x]\cdot met(x,y)\cdot [y]:E^{r}\cdot T\cdot E^{l}$

$I(barked:N^{r}\cdot S)=[x]\cdot barked(x):E^{r}\cdot T$

$I(at:S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l})=[x]\cdot y\cdot [y]\cdot at(x,z)\cdot [z]:T^{r}\cdot E^{rr}\cdot E^{r}\cdot T\cdot E^{l}$

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 $John\ met\ Mary:N\cdot (N^{r}\cdot S\cdot N^{l})\cdot N$ (emphasizing the link being reduced)

For the sentence $the\ dog\ barked\ at\ the\ cat:(N\cdot N_{0}^{l})\cdot N_{0}\cdot (N^{r}\cdot S)\cdot (S^{r}\cdot N^{rr}\cdot N^{r}\cdot S\cdot N^{l})\cdot (N\cdot N_{0}^{l})\cdot N_{0}$:

See also

• Compact closed category
• Lambek calculus

References

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Claudia Casadio (2004), Pregroup Grammar. Theory and Applications

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