Each fencepost contains two vectors.
A span is represented by concatenating the difference of each of its fencepost vectors:  
$$v(i,j)=[y_j^{\rightarrow} {-} y_i^{\rightarrow } ;y_{j+1}^{\leftarrow }-y_{i+1}^{\leftarrow }]$$

Fencepost encoding

The span vector v is then passed through a span classifier, with two fully- connected layers and one ReLU activation function, whose output dimensionality is the number of possible non-terminal labels:
$$s(\left ( i,j \right ) = {W2}\times ReLU\left (  LayerNorm\left ( W1\times {v(i, j)} \right )\right )$$

Span Scoring

Instead of providing numerosity of all possible parsing trees, span-based neural constituency parsing only offer the best parsing tree. Its intuition is to train a neural classifier to assign a score to each constituent and find out the best-scoring parse tree based on those assigned scores.


University of Colorado at Boulder

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

Chunking, a kind of partial parsing, is the process of identifying and classifying the flat, non-overlapping segments of a sentence that constitute the basic non-recursive phrases corresponding to the major content-word parts-of-speech: noun phrases, verb phrases, adjective phrases, and prepositional phrases. The task of finding all the base noun phrases in a text is particularly common.

The two fundamental tasks involved in chunking are:
- Segmenting: finding the non-overlapping extents of the chunks
- Labelling: assigning the correct tag to the discovered chunks

Chunking in NLP

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

Speech and Language Processing (3rd ed. draft) 

Structural ambiguity occurs when the grammar can assign more than one parse
to a sentence. Two common kinds of ambiguity are attachment ambiguity and coordination ambiguity. A sentence has an attachment ambiguity if a particular constituent can be attached to the parse tree at more than one place. For instance, in the following example the gerundive-VP flying to Paris can be part of a gerundive sentence whose subject is the Eiffel Tower or it can be an adjunct modifying the VP headed by saw:
We saw the Eiffel Tower flying to Paris.
In coordination ambiguity phrases can be conjoined by a conjunction like and. For example, the phrase old men and women can be bracketed as [old [men and women]], referring to old men and old women, or as [old men] and [women], in
which case it is only the men who are old.
The fact that there are many grammatically correct but semantically unreasonable parses for naturally occurring sentences is an irksome problem that affects all parsers. Fortunately, the CKY algorithm is designed to efficiently handle structural ambiguities.

Ambiguity 

Span-Based Neural Constituency Parsing

The CKY parsing allows each non-terminal paired with entires from which it was derived and multiple versions of the same non-terminal to be entered into the same table. It can return every possible parse for a sentence, which may not be very helpful.

 CKY Parsing

Also referred to as probabilistic parsing, this is when a probabilistic component is added to constituency parsing. Probabilistic Parsers compute the probability of each interpretation of a sentence and choose the most probable interpretation.

Statistical Constituency Parsing

Many language processing tasks do not require complex, complete parse trees for all inputs. In such cases, a partial parse, or shallow parse, of input sentences may be sufficient, where not all possible information are extracted from a text.

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