Grammar equations (2021)

Abstract: Diagrammatically speaking, grammatical calculi such as pregroups provide wires between words in order to elucidate their interactions, and this enables one to verify grammatical correctness of phrases and sentences. In this paper we also provide wirings within words. This will enable us to identify grammatical constructs that we expect to be either equal or closely related. Hence, our work paves the way for a new theory of grammar, that provides novel ‘grammatical truths’. We give a nogo-theorem for the fact that our wirings for words make no sense for preordered monoids, the form which grammatical calculi usually take. Instead, they require diagrams – or equivalently, (free) monoidal categories.

## introduction grammatical calculi @xcite @xcite enable one to verify grammatical correctness of sentences. however, there are certain grammatical constructs that we expect to be closely related, if not the same, but which grammatical calculi fail to identify. we will focus on pregroups @xcite , but the core ideas of this paper extend well beyond pregroup grammars, including ccgs @xcite , drawing on the recent work in @xcite that casts ccgs as augmented pregroups. in this paper we both modify and extend grammatical calculi, by providing so-called 'internal wirings' for a substantial portion of english. diagrammatically speaking, while grammatical calculi provide wires between words in order to elucidate their interactions, we also provide wirings within words. for example, a pregroup diagram for the phrase: additional structure for grammatical calculi was previously introduced by providing semantics to certain words, for example, quantifiers within montague semantics @xcite . this is not what we do. we strictly stay within the realm of grammar, and grammar only. hence, our work paves the way for a new theory of grammar, that provides novel 'grammatical truths'. usually grammatical calculi take the form of preordered monoids @xcite . however, the internal wirings cannot be defined at the poset level, for which we provide a nogo-theorem. hence passing to the realm of diagrammatic representations -which correspond to proper free monoidal categories -is not just a convenience, but a necessity for this work. they moreover provide a clear insight in the flow of meanings. internal wirings were proposed within the dis-cocat framework @xcite , for relative pronouns and verbs @xcite @xcite @xcite @xcite . they are made up of 'spiders' (a.k.a. certain frobenius algebras) @xcite . we point out a shortcoming of those earlier proposed internal wirings, and fix them by introducing a 'wrapping gadget', that forces certain wires to stay together. this reintroduces composite types such as sentence types. what we present here is only part of the full story. for the latter we refer to a forthcoming much longer paper (coecke and wang), which besides providing many more internal wirings than given here, also uses them to provide bureaucracyfree grammar as circuits, the equivalence classes for the equations introduced here. these circuits also have direct practical applications within natural language processing -see e.g. @xcite . ## statement of the problem for our purposes, a pregroup has a set of 'basic types' n, s, ... each of which admit left and right inverses - foot_0 n and n -1 . each grammatical type is assigned a string of these, e.g. a transitive verb in english gets: @xmath0. the inverses 'cancel out' from one direction: a sentence is grammatical if when taking the string of all of its grammatical types, the inverses cancel to leave a special, 'final', basic type s (for sentence), like here for n • tv • n: this calculation can be represented diagrammatically: the pregroup diagrams now look as in figure (1 ). without any further context the factual data con-veyed by these two sentences is the same. 1 how can we formally establish this connection between the two sentences? ## rewriting pregroup diagrams via internal wirings what is needed are 'internal wirings' of certain words, that is, not treating these words as 'black boxes', but specifying what is inside, at least to some extent. equationally speaking, they provide a congruence for pregroup diagrams, and we can establish equality by means of topological deformation. for constructing these internal wirings we make use of 'spiders' @xcite (a.k.a. frobenius algebras @xcite ). one can think of these spiders as a generalisation of wires to multi-wires, as rather than having two ends, they can have multiple ends. still, all they do, like wires, is connect stuff, and when you connect connected stuff to other connected stuff (a.k.a. 'spider-fusion'): we presented internal wiring in terms of pregroup diagrams. this is because they do not make sense in terms of symbolic pregroups presented as preordered monoids: theorem 3.1. a pregroup with spiders is trivial. concretely, given a preordered monoid (x, ≤, ⊗) with unit 1, if for @xmath1, then x 1. proof. having spiders on x means that for all j, @xmath2: hence this paper requires diagrams in a fundamental manner. 2 ## the wrapping gadget above in (5) we saw that sentence wires were decomposed into noun wires. however, for pregroup proofs it is important to know that those wires do belong together, so we need to introduce a tool that enables us to express that they belong together. definition 4.1. the wrapping gadget forces a number of wires to be treated as one, i.e. it wraps them, and is denoted as follows: by unfolding we mean dropping the restrictions imposed by the wrapping gadget. cups and spiders carry over to wrapped wires in the expected way, following the conventions of @xcite . in fact, in the case of relative pronouns simply wrapping the noun-wires making up the sentence type isn't enough, as the counterexample in figure ## some more internal wirings we now provide internal wirings for some grammatical word classes that feature in the examples of the next section. we distinguish between 'content words', like the verbs in (5), and 'functional words', like the relative pronouns in (7). ## proof-of-concept we provide a number of examples of how the internal wirings proposed above enable us to re-alice flowers gives that bob plays chess figure 4 : the deleting of the sentence type of plays belongs together with the noun-wire now connecting the relative pronoun with gives, like in figure (1 ). this is enforced by the internal wiring of the object relative pronoun in figure (7 )