# HOMEWORK 6

CS 224U / LING 188/288, Spring 2014, Stanford

## Lexicon

The following natural logic relational lexicon uses the relations and concepts from MacCartney and Manning 2008, 2009. It is the basis for the questions on this assignment.

NOUNSwarthogsturtlesnon-turtlesmammalsreptilespets
warthogs = | $$\sqsubset$$$$\sqsubset$$| #
turtles | = ^ | $$\sqsubset$$#
non-turtle $$\sqsupset$$^ = $$\sqsupset$$$$\smile$$ #
mammals $$\sqsupset$$| $$\sqsubset$$= | #
reptiles | $$\sqsupset$$$$\smile$$ | = #
pets # # # # # =

VERBS walkmoveswimgrowl
walk = $$\sqsubset$$| #
move $$\sqsupset$$= $$\sqsupset$$#
swim | $$\sqsubset$$= |
growl# # | =

animate = ^ $$\sqsupset$$$$\smile$$ $$\sqsupset$$
inanimate ^ = | $$\sqsubset$$|
moving $$\sqsubset$$| = ^ #
motionless $$\smile$$ $$\sqsupset$$^ = #
curious $$\sqsubset$$| # # =

some = $$\sqsupset$$$$\sqsupset$$^
all $$\sqsubset$$= # |
at least three $$\sqsubset$$# = |
no ^ | | =

## Question 1: substitutions [5 points]

For each of the following examples, select a word from the lexicon that, when substituted for the word in bold, will preserve the overall sentence-level relation. For example, for some reptiles walk $$\sqsubset$$ some reptiles move, a correct answer would be turtles. No self-substitutions allowed!

1. all mammals walk $$\sqsubset$$ all mammals move
2. all reptiles swim $$\sqsubset$$ all turtles swim
3. no reptiles move $$\sqsubset$$ no reptiles swim
4. some turtles walk | no turtles move
5. some mammals growl # no non-turtles swim

## Question 2: projectivity [3 points]

The projectivity signature for a lexical item encodes how it interacts with its arguments compositionally. To investigate the projectivity of a lexical item $$L$$, we can take any two lexical items of the same category $$a$$ and $$b$$ that are related by $$R$$ and see what relation $$R'$$ obtains for the pair of complex phrases $$L a$$ and $$L b$$. The following table summarizes this reasoning for not: the left column comes directly from the lexicon, and the middle column uses the template not _____ to test projectivity. Those judgments determine the projectivity signature given in the right column.

LexicalModifiedSignature
curious = curiousnot curious = not curious= to =
curious $$\sqsubset$$ animatenot curious $$\sqsupset$$ not animate$$\sqsubset$$ to $$\sqsupset$$
animate $$\sqsupset$$ curiousnot animate $$\sqsubset$$ not curious$$\sqsupset$$ to $$\sqsubset$$
animate ^ inanimatenot animate ^ not inanimate^ to ^
moving | inanimatenot moving $$\smile$$ not inanimate| to $$\smile$$
motionless $$\smile$$ animatenot motionless | not animate$$\smile$$ to |
moving # curiousnot moving # not curious# to #

Your task: using the lexicon given above, construct the projectivity signature for the first argument to most as in most turtles swim. To ensure you get a clear picture, you should use a template in which the verb phrase is fixed, as in most ____ swim, where ____ is filled by a single noun. Assume the following semantics, where $$A$$ and $$B$$ are sets, $$|X|$$ is the cardinality of the set $$X$$, and $$A \cap B$$ is the intersection of the sets $$A$$ and $$B$$. $most(A)(B) = \textit{True} \quad \textit{iff} \quad \frac{|A \cap B|}{|A|} > 0.5$

## Question 3: entailment assumptions [2 points]

In their work on entailment between distributed representations, Baroni et al. 2012 semi-automatically gathered a data set of vectors $$(a_{n}, n)$$ where $$n$$ is the vector for noun N and $$a_{n}$$ is the vector for N modified by an adjective. They were confident that, in the vast majority of cases, this would represent a valid case of entailment from $$a_{n}$$ to $$n$$.

Your task: Articulate the insight from MacCartney and Manning's discussion that Baroni et al.'s procedure implicitly relies on, and say which kinds of adjectives one needs to filter out when constructing data about entailment in this way. (1-2 sentence answer.)