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    • CommentRowNumber1.
    • CommentAuthoralexis.toumi
    • CommentTimeAug 29th 2019

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthoralexis.toumi
    • CommentTimeAug 29th 2019
    I noticed the nlab did not have a dedicated page for pregroup grammars.
    I'm starting to write down some notes, trying to connect them with what was already there on categorial grammars and linguistics in general.
    • CommentRowNumber3.
    • CommentAuthoralexis.toumi
    • CommentTimeAug 29th 2019

    removing capital letter

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 30th 2019

    It should be pointed out (somewhere, sometime) the relationship between pregroups and residuated lattices, which appears in much older work of Lambek dating back at least to the 60’s. This work of Lambek was all of a piece, connecting module theory to proof theory to linguistics. I would like to add some of this history at some point.

    • CommentRowNumber5.
    • CommentAuthoralexis.toumi
    • CommentTimeAug 30th 2019
    Great suggestion! I don't know so much of the historical details, apart from the original paper from Lambek, The Mathematics of Sentence Structure (1958). There should definitely be a page for the Lambek calculus, residuated monoids and their categorification to biclosed monoidal categories. Some of the details are in Coecke et al, Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus (2013).
    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeSep 2nd 2019
    I would like to know where did this sentence come from: "A sequence of words is grammatical whenever there exists a string diagram going from the sequence of words to the sentence type.", and if there are any references and research on the use of string diagrams to model natural languages, or, more specifically, to describe context-free grammars. I am interested in the use of pregroup grammars and Lambek's research on the development of visual programming languages. Zorkedon
    • CommentRowNumber7.
    • CommentAuthoratmacen
    • CommentTimeSep 4th 2019

    Re #6: This blog post is saying something very similar, using a monoidal category for sentence derivations. “That is, a string uV u \in V^\star is grammatical whenever there exists an arrow from the start symbol ss to uu in 𝒞 R\mathcal{C}_R.” Morphisms in monoidal categories can be notated using string diagrams.

    • CommentRowNumber8.
    • CommentAuthoralexis.toumi
    • CommentTimeSep 4th 2019
    The use of string diagrams to model natural language has become folklore in the DisCoCat (categorical compositional distributional) community, see e.g. Coecke et al. (2010) https://arxiv.org/abs/1003.4394
    As far as I'm aware there isn't any reference which spells out the details fully, which is part of the reason why I started this page, as well as the one on context-free grammars.
  1. fix typos

    Richie

    diff, v7, current

    • CommentRowNumber10.
    • CommentAuthoralexis.toumi
    • CommentTimeNov 22nd 2020

    Add linguistics context.

    diff, v8, current

    • CommentRowNumber11.
    • CommentAuthoralexis.toumi
    • CommentTimeDec 1st 2020

    Adding a mention of mildly context-sensitive pregroup grammars.

    diff, v9, current

    • CommentRowNumber12.
    • CommentAuthorThomas Holder
    • CommentTimeDec 1st 2020

    I expanded on product grammars and attributed some results here to Kobele-Kracht. Since I am unfamiliar with the Lambek and Genkin et al. source my way to weave them in is somewhat conjectural: feel free to correct me on this!

    diff, v10, current

    • CommentRowNumber13.
    • CommentAuthoralexis.toumi
    • CommentTimeDec 2nd 2020
    Is this the paper you're referring to? We should definitely add it to the references at the bottom of the page.

    Kobele, Gregory M., and Marcus Kracht. "On pregroups, freedom, and (virtual) conceptual necessity." University of Pennsylvania Working Papers in Linguistics 12.1 (2006): 16.

    When you write "by a classical result in formal language theory any type 0 language is the intersection $L=L_1\cap L_2$ of two context-free languages $L_1,L_2$", are you referring to Theorem 1 in Kobele & Kracht? If so, the statement is a bit more subtle: any r.e. language is the image of an intersection of CFLs under a non-length-increasing homomorphism $L = h(L_1 \cap L_2)$.

    I feel like that theorem is cool enough to deserve its own section somewhere, maybe in a new page on language homomorphisms?
    • CommentRowNumber14.
    • CommentAuthoralexis.toumi
    • CommentTimeDec 2nd 2020
    Maybe I'm missing something here: are context-free languages closed under non-length-increasing homomorphisms? Then obviously your statement is correct, but it would be worth spelling it out explicitly.
    • CommentRowNumber15.
    • CommentAuthorThomas Holder
    • CommentTimeDec 2nd 2020

    Very likely my bad due to a false recollection of their argument, thanks for giving it a reality check! Hopefully, I got at least the substance right, namely, that product pregroup grammars land you in type 0. I’ll attend to the paragraph as soon as I find a minute. Feel free to revise concerning the Kracht-Kobele paper as you have a better overview of the relevant literature hence how the pieces fit together.

    • CommentRowNumber16.
    • CommentAuthorThomas Holder
    • CommentTimeDec 2nd 2020

    I fixed the omission of the homomorphic image part in my previous edit and added the respective references to Kobele and Kracht.

    diff, v11, current

  2. link to posets

    DavidWhitten

    diff, v12, current

    • CommentRowNumber18.
    • CommentAuthorwhitten
    • CommentTimeDec 3rd 2020
    • (edited Dec 3rd 2020)
    I was reading
    https://ncatlab.org/nlab/show/pregroup+grammar
    and it says:

    If we drop the anti-symmetry axiom for posets, we get a quasi-pregroup,

    but when I go to
    https://ncatlab.org/nlab/show/partial+order
    it says:
    A poset is precisely a proset satisfying the extra condition that x≤y≤x implies that x=y.

    Therefore, I think a quasi-pregroup is the same as a poset.

    But this can't be right because the sub-words "group" and "set" don't mean the same thing.

    I assume this is just a matter of being more precise, as tracking down
    https://ncatlab.org/nlab/show/set
    this set-page does NOT mention groups, but groupoids.

    and

    https://ncatlab.org/nlab/show/group
    this group-page does not mention 0-groupoids, which I think are groups, right ?
    but the group-page does mention a group is tied to a coset but not linked to the set-page

    I'm wondering if I am drawing the wrong conclusions, or if some of these pages need a bit more duplication
    to show when different combinations of main ideas and adjectives describing them cause you to just
    get another name for an existing idea.

    Thanks for the info,
    Dave Whitten
    713-870-3834
    whitten@netcom.com
    • CommentRowNumber19.
    • CommentAuthoralexis.toumi
    • CommentTimeDec 4th 2020

    Adding explicit definitions for preordered monoid and pomonoid. I hope this clears some of the confusion.

    diff, v13, current

    • CommentRowNumber20.
    • CommentAuthoralexis.toumi
    • CommentTimeDec 4th 2020

    Adding the example of unbounded monotone functions.

    diff, v14, current

    • CommentRowNumber21.
    • CommentAuthorzstone
    • CommentTimeMay 28th 2021
    Sorry if inappropriate, but are you the Thomas Holder who presented work on bare grammars at MoL? I've been trying to track him/you down.