Categories are either atomic elements or single-argument functions that return a category as a value when provided with a desired category as argument.

Formally, we define $\mathcal{C}$ as a set of categories for a grammar, such that:
- $\mathcal{A} \subseteq \mathcal{C}$, where $\mathcal{A}$ is a given set of atomic elements
- $(X/Y), \; (X\setminus Y) \in \mathcal{C}$, if $X, Y \in \mathcal{C}$

The set of atomic categories is typically very small and includes familiar elements such as sentences and noun phrases. Functional categories include verb phrases and complex noun phrases among others.

CG Categories

The Lexicon consists of assignments of categories to words. These assignments can either be to atomic or functional categories, and due to lexical ambiguity words can be assigned to multiple categories. 

Words such as nouns and proper nouns are assigned to atomic categories, reflecting their typical role as arguments to functions. On the other hand, words such as transitive verbs are assigned to function categories.

CG Lexicon

Rules specify how functions and their arguments combine. The following two rule templates constitute the basis for all categorial grammars:
- $X/Y \; Y \implies X$
- $Y \; X \setminus Y \implies X$

The first rule, referred to as forward function application, applies a function to its argument on the right, while the second, referred to as backward function application,  looks to the left for its argument.

CG Rules

Categorical grammar is an early lexicalized grammar model which consists of three major elements: a set of categories, a lexicon that associates words with categories, and a set of rules that govern how categories combine in context.

University of Michigan - Ann Arbor

Parsing, syntax analysis, or syntactic analysis is the process of analyzing a string of symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar.



Within computational linguistics the term is used to refer to the formal analysis by a computer of a sentence or other string of words into its constituents, resulting in a parse tree showing their syntactic relation to each other, which may also contain semantic and other information (p-values). Some parsing algorithms may generate a parse forest or list of parse trees for a syntactically ambiguous input.



Simply speaking, parsing in NLP is the process of determining the syntactic structure of a text by analyzing its constituent words based on an underlying grammar (of the language).

Parsing

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Research papers, articles and examples about parsing in NLP

NLP Parsing References

A context-free grammar consists of a set of rules or productions and a lexicon of words and symbols.  Each of those rules expresses the ways that symbols of the language can be grouped and ordered together.
For example, one rule for noun phrase, like "the flight", is that a noun phrase can be composed of a determiner and a nominal.
$$ NP --> Det \quad Nominal$$



Context-Free Grammars


A parse tree is one way to represent the syntactic structure of a string according to some context-free grammar. In parsing trees, each sentence starts with the start symbol $$S$$.


Parse Tree

The syntactic structure can also be expressed by bracketed notation, which is more compact than the parse tree.
$$ [\mathit{S}[NP [Pro\,\textrm{I}]] [VP [V \,\textrm{perfer}] [NP [Det \,\textrm{a}] [Nom [N \,\textrm{morning}] [Nom [N \,\textrm{flight}]]]]]]$$



Bracketed Notation


Categorical Grammar

Unfortunately, the basic categorial approach does not give us any more expressive power than we had with traditional CFG rules; it just moves information from the grammar to the lexicon. To move beyond these limitations CCG is a modern extension of categorical grammar and includes operations that operate over functions.

Combinatory Categorical Grammar (CCG)

Constituency Parsing focuses on constituency structures, those assigned by context-free grammars. In order to know which parse tree is the right one, we need to augment CKY with scores for each possible constituent. 

Constituency Parsing

Dependency Parsing is fundamentally based on the dependency grammars in terms of grammar formalisms. In these formalisms, phrasal constituents and phrase-structure rules do not play a direct role as Constituent Parsing. Instead, the syntactic structure of a sentence is describe solely in terms of the words in a sentence and an associated set of directed binary grammatical relations that hold among the words.

Dependency Parsing

Given a sequence of sentences, automatically determine the coherence relations between them.

Discourse Structure Parsing

The Penn Discourse TreeBank (PDTB) is a second commonly used dataset that embodies model of coherence relations.

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