COMS W4705 – Natural Language Processing Homework 2 solution


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Analytical Component (40 pts)

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Problem 1 (10 pts) – PCFGs and HMMs

Both PCFGs and HMMs can be seen as generative models that produce a sequence of
POS tags and words with some probability (of course the PCFG will generate even
more structure, but it will also generate POS tags and words).

(a) Consider the grammar specified in Problem 2 below, and the sentence “they are
baking potatoes”. For each sequence of POS tags that is possible for this sentence
according to the grammar, what is the joint probability P(tags,words) according to the
PCFG? Hint: consider all parses for the sentence — you may want to work on problem 2

(b) Design an HMM that produces the same joint probability P(tags, words) as the
PCFG for each of the possible tag sequences for the sentence in part (a). Note: Your
HMM does not have to assign 0 probabilities to tag sequences that are not possible
according to the PCFG.

(c) In general, is it possible to translate any PCFG into an HMM that produces the
identical joint probability P(tags,words) as the PCFG (i.e. not just for a single sentence)?
Explain how or why not. No formal proof is necessary. Hint: This has nothing to do with
probabilities, but with language classes.

Problem 2 (10 pts) – Earley Parser

Consider the following probabilistic context free grammar.
S → NP VP [1.0]
NP → Adj NP [0.3]
NP → PRP [0.1]
NP → N [0.6]
VP → V NP [0.8]
VP → Aux V NP [0.2]
PRP → they [1.0]
N → potatoes [1.0]
Adj → baking [1.0]
V → baking [0.5]
V → are [0.5]
Aux → are [1.0]

(a) Using this grammar, show how the Earley algorithm would parse the following
they are baking potatoes

Write down the complete parse chart. The chart should contain n+1 entries where n is
the length of the sentence. Each entry i should contain all parser items generated by the
parser that end in position i.

You can ignore the probabilities for part (a).

(b) Write down all parse trees for the sentence and grammar from problem 2 and
compute their probabilities according to the PCFG.

Problem 3 (10 pts) – CKY parsing

(a) Convert the grammar from problem 2 into an equivalent grammar in Chomsky
Normal Form (CNF). Write down the new grammar. Also explain what the general rule
is for dealing with

1. Rules of the form A→B (i.e. a single nonterminal on the right hand side).
2. Rules with three or more nonterminals on the right hand side (e.g. A→B C D E).
You do not have to deal with the case in which terminals and non-terminals are mixed in
a rule right-hand side. You also do not have to convert the probabilities. Hint: Think
about adding new nonterminal symbols.

(b) Using your grammar, fill the CKY parse chart as shown in class and show all parse

Problem 4 (10 pts) – Transition Based Dependency Parsing

Consider the following dependency graph.
Write down the sequence of transitions that an arc-standard dependency parser would
have to take to generate this dependency tree from the initial state
([root]σ, [he, sent, her, a, funny, meme, today]β, {}A)
Also write down the state resulting from each transition.

Programming Component – Parsing with Context Free

Grammar (60 pts)

The instructions below are fairly specific and it is okay to deviate from implementation
details. However: You will be graded based on the functionality of each function.

Make sure the function signatures (function names, parameter and return
types/data structures) match exactly the description in this assignment.
Please make sure you are developing and running your code using Python 3.


In this assignment you will implement the CKY algorithm for CFG and PCFG parsing.
You will also practice retrieving parse trees from a parse chart and working with such
tree data structures.

The files for this project are inside which should be found in the same folder as
this pdf.

Python files:
To get you started, there are three Python files for this project.
1. will contain your parser and currently contains only scaffolding code.
2. contains the class Pcfg which represents a PCFG grammar (explained
below) read in from a grammar file.
3. contains a script that evaluates your parser against a test set.

Data files:
You will work with an existing PCFG grammar and a small test corpus.
The main data for this project has been extracted from the ATIS (Air Travel Information
Services) subsection of the Penn Treebank. ATIS is originally a spoken language
corpus containing user queries about air travel. These queries have been transcribed
into text and annotated with Penn-Treebank phrase structure syntax.

The data set contains sentences such as “what is the price of flights from indianapolis
to memphis .”
There were 576 sentences in total, out of which 518 were used for training (extracting
the grammar and probabilites) and 58 for test. The data set is obviously tiny compared
to the entire Penn Treebank and typically that would not be enough training data.

However, because the domain is so restricted, the extracted grammar is actually able to
generalize reasonably well to the test data.
There are 2 data files:
atis3.pcfg – contains the PCFG grammar (980 rules)
atis3_test.ptb – contains the test corpus (58 sentence).

Take a look at these files and make sure you understand the format. The tree structures
are atis3_test.ptb are a little different from what we have seen in class. Consider the
following example from the test file:
(TOP (S (NP i) (VP (WOULD would) (VP (LIKE like) (VP (TO to) (VP (TRAVEL travel) (PP
(TO to) (NP westchester))))))) (PUN .))

Note that there are no part of speech tags. In some cases phrases like NP directly
project to the terminal symbol (NP -> westchester). In other cases, nonterminals for a
specific word were added (TRAVEL -> travel).

The start-symbol for the grammar (and therefore the root for all trees) is “TOP”. This is
the result of an automatic conversion to make the tree structure compatible with the
grammar in Chomsky Normal Form.

While you are working on your parser, you might find it helpful to additionally create a
small toy grammar (for example, the one in the lecture slides) that you can try on some
hand written test cases, so that you can verify by hand that the output is correct.

Part 1 – reading the grammar and getting started

Take a look at The class Pcfg represents a PCFG grammar in chomsky
normal form. To instantiate a Pcfg object, you need to pass a file object to the
constructor, which contains the data. For example:
with open(‘atis3.pcfg’,’r’) as grammar_file:
grammar = Pcfg(grammar_file)

You can then access the instance variables of the Pcfg instance to get information
about the rules:
>>> grammar.startsymbol
The dictionary lhs_to_rules maps left-hand-side (lhs) symbols to lists of rules. For
example, we will want to look up all rules of the form PP -> ??
>>> grammar.lhs_to_rules[‘PP’]

[(‘PP’, (‘ABOUT’, ‘NP’), 0.00133511348465), (‘PP’, (‘ADVP’, ‘PPBAR’), 0.0013351134846
5), (‘PP’, (‘AFTER’, ‘NP’), 0.0253671562083), (‘PP’, (‘AROUND’, ‘NP’), 0.006675567423
23), (‘PP’, (‘AS’, ‘ADJP’), 0.00133511348465), … ]
Each rule in the list is represented as (lhs, rhs, probability) triple. So the first rule in the
list would be
PP -> ABOUT NP with PCFG probability 0.00133511348465.

The rhs_to_rules dictionary contains the same rules as values, but indexed by righthand-side. For example:
>>> grammar.rhs_to_rules[(‘ABOUT’,’NP’)]
[(‘PP’, (‘ABOUT’, ‘NP’), 0.00133511348465)]
>>> grammar.rhs_to_rules[(‘NP’,’VP’)]
[(‘NP’, (‘NP’, ‘VP’), 0.00602409638554), (‘S’, (‘NP’, ‘VP’), 0.694915254237), (‘SBAR’
, (‘NP’, ‘VP’), 0.166666666667), (‘SQBAR’, (‘NP’, ‘VP’), 0.289156626506)]

• Write the method verify_grammar, that checks that the grammar is a valid PCFG in
CNF. You need to check that the rules have the right format (note all nonterminal
symbols are upper-case) and that all probabilities for the same lhs symbol sum to
1.0. Hint: For improved numeric accuracy, use math.fsum to compute the sum.

• Then change the main section of to read in the grammar, print out a
confirmation if the grammar is a valid PCFG in CNF or print an error message if it is
not. You should now be able to run on grammars and verify that they
are well formed for the CKY parser.

Part 2 – Membership checking with CKY

The file already contains a class CkyParser. When a CkyParser instance is
created a i instance is passed to the constructor. The instance variable grammar can
then be used to access this Pcfg object.

TODO: Write the method is_in_language(self, tokens) by implementing the CKY
algorithm. Your method should read in a list of tokens and return True if the grammar
can parse this sentence and False otherwise. For example:
>>> parser = CkyParser(grammar)
>>> toks =[‘flights’, ‘from’, ‘miami’, ‘to’, ‘cleveland’, ‘.’] // Or: toks= ‘flights
from miami to cleveland .’.split()
>>> parser.is_in_language(toks)

>>> toks =[‘miami’, ‘flights’,’cleveland’, ‘from’, ‘to’,’.’]
>>> parser.is_in_language(toks)

While parsing, you will need to access the dictionary self.grammar.rhs_to_rules. You
can use any data structure you want to represent the parse table (or read ahead to Part
3 of this assignment, where a specific data structure is prescribed).

The ATIS grammar actually overgenerates a lot, so many unintuitive sentences can be
parsed. You might want to create a small test grammar and test cases. Also make sure
that this method works for grammar with different start symbols.

Part 3 – Parsing with backpointers

The parsing method in part 2 can identify if a string is in the language of the grammar,
but it does not produce a parse tree. It also does not take probabilities into account. You
will now extend the parser so that it retrieves the most probable parse for the input
sentence, given the PCFG probabilities in the grammar. The lecture slides on parsing
with PCFG will be helpful for this step.

TODO: Write the method parse_with_backpointers(self, tokens). You should modify
your CKY implementation from part 2, but use (and return) specific data structures. The
method should take a list of tokens as input and returns a) the parse table b) a
probability table. Both objects should be constructed during parsing. They replace
whatever table data structure you used in part 2.

The two data structures are somewhat complex. They will make sense once you
understand their purpose.

The first object is parse table containing backpointers, represented as a dictionary
(this is more convenient in Python than a 2D array). The keys of the dictionary are
spans, for example table[(0,3)] retrieves the entry for span 0 to 3 from the chart. The
values of the dictionary should be dictionaries that map nonterminal symbols to

For example: table[(0,3)][‘NP’] returns the backpointers to the table entries
that were used to create the NP phrase over the span 0 and 3. For example, the value
of table[(0,3)][‘NP’] could be ((“NP”,0,2),(“FLIGHTS”,2,3)). This means that the parser
has recognized an NP covering the span 0 to 3, consisting of another NP from 0 to 2
and FLIGHTS from 2 to 3. The split recorded in the table at table[(0,3)][‘NP’] is the one
that results in the most probable parse for the span [0,3] that is rooted in NP.

Terminal symbols in the table could just be represented as strings. For example the
table entry for table[(2,3)][“FLIGHTS”] should be “flights”.

The second object is similar, but records log probabilities instead of backpointers.
For example the value of probs[(0,3)][‘NP’] might be -12.1324. This value represents the
log probability of the best parse tree (according to the grammar) for the span 0,3 that
results in an NP.

Your parse_with_backpointers(self, tokens) method should be called like this:
>>> parser = CkyParser(grammar)
>>> toks =[‘flights’, ‘from’, ‘miami’, ‘to’, ‘cleveland’, ‘.’]
>>> table, probs = parser.parse_with_backpointers(toks)

During parsing, when you fill an entry on the backpointer parse table and iterate
throught the possible splits for a (span/nonterminal) combination, that entry on the table
will contain the back-pointers for the the current-best split you have found so far. For
each new possible split, you need to check if that split would produce a higher log
probability. If so, you update the entry in the backpointer table, as well as the entry in
the probability table.

After parsing has finished, the table entry table[0,len(toks)][grammar.startsymbol] will
contain the best backpointers for the left and right subtree under the root
node. probs[0,len(toks)][grammar.startsymbol] will contain the total log-probability for
the best parse. contains two test functions check_table_format(table)

and check_prob_format(probs) that you can use to make sure the two table data
structures are formatted correctly. Both functions should return True. Note that passing
this test does not guarantee that the content of the tables is correct, just that the data
structures are probably formatted correctly.

>>> check_table_format(table)
>>> check_prob_format(probs)

Part 4 – Retrieving a parse tree

You now have a working parser, but in order to evaluate its performance we still need to
reconstruct a parse tree from the backpointer table returned by

Write the function get_tree(chart, i,j, nt) which should return the parse-tree rooted in
non-terminal nt and covering span i,j.

For example
>>> parser = CkyParser(grammar)
>>> toks =[‘flights’, ‘from’, ‘miami’, ‘to’, ‘cleveland’, ‘.’]
>>> table, probs = parser.parse_with_backpointers(toks)

>>> get_tree(table, 0, len(toks), grammar.startsymbol)
(‘TOP’, (‘NP’, (‘NP’, ‘flights’), (‘NPBAR’, (‘PP’, (‘FROM’, ‘from’), (‘NP’, ‘miami’))
, (‘PP’, (‘TO’, ‘to’), (‘NP’, ‘cleveland’)))), (‘PUN’, ‘.’))

Note that the intended format is the same as the data in the treebank. Each tree is
represented as tuple where the first element is the parent node and the remaining
elements are children. Each child is either a tree or a terminal string.

Hint: Recursively traverse the parse chart to assemble this tree.

Part 5 – Evaluating the Parser

The program evaluates your parser by comparing the output trees
to the trees in the test data set.
You can run the program like this:
python atis3.pcfg atis3_test.ptb
Even though this program has been written for you, it might be useful to take a look at
how it works.

The program imports your CKY parser class. It then reads in the test file line-by-line. For
each test tree, it extracts the yield of the tree (i.e. the list of leaf nodes). It then feeds
this list as input to your parser, obtains a parse chart, and then retrieves the predicted
tree by calling your get_tree method.

It then compares the predicted tree against the target tree in the following way (this is a
standard approach to evaluating constiuency parsers, called PARSEVAL):
First it obtains a set of the spans in each tree (including the nonterminal label). I then
computes precision, recall, and F-score (which is the harmonic mean between precision
and recall) between these two sets.

The script finally reports the coverage (percentage
of test sentences that had any parse), the average F-score for parsed sentences, and
the average F-score for all sentences (including 0 f-scores for unparsed sentences).

My parser implementation produces the following result on the atis3 test corpus:
Coverage: 67%, Average F-score (parsed sentences): 0.95, Average F-score (all
sentences): 0.64

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