Homework 1
Greg Kobele
Answer Key
Computational Minimalism
LSA 2015
Instructions:
This assignment is intended to assist you in understanding
the workings of the minimalist grammar formalism. Please attempt at least one
of the following questions (you are welcome to do all of them); you should select
one which is appropriate to your current skill level.
Question 1
Give a grammar (i.e. specify the lexical entries) which assigns (at
least) the sentences c, acb, aacbb the following structural descriptions:
>
a
>
>
c
b
>
a
>
c
a
>
b
c
b
Make sure that your lexical entries work as advertised (i.e. show that
these expressions are in fact derivable using your grammar).
Just as was done in class, we can read off the trees the features of the lexical
items. We will assume that the start symbol is s. Then, in order to derive the
tree on the far left, we need a lexical item of the following form s.c. This lexical
item just is the desired tree.
For the tree in the middle, we see that b must first merge with something of
category s (c), and must then merge with something of an unknown category,
x (a). This gives us the following two additional lexical items:
s=.x=.s.b
x.a
These three lexical items then allow us to derive the tree on the right.
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Question 2
Give a grammar for the language am bn cdn em ; i.e. the language
consisting of sentences with as, bs, cs, d s, and es in that order
where
1. there is exactly 1 c
2. there are as many as as es (possibly 0)
3. there are as many bs as d s (possibly 0)
When defining infinite patterns like this one, it is important to break things
down into smaller parts. Here, it is important that the as and the es are
independent of the bs and the d s. The previous exercise showed us how to
enforce counting dependencies; we can use the lexical items from question 1
(renaming them appropriately: a 7→ b, c 7→ c, b 7→ d) to derive the language
bn cdn (I have changed the feature s to t):
t.c
x.b
t=.x=.t.d
We would like to treat any of these sentences as the middle symbol (the )
in something of the form am em . Instead of having a single middle symbol,
with feature bundle s, we have a way of building middle symbols from things of
category t:
t=.s.
This phonetically silent lexical item selects for a phrase of category t and builds
a phrase of category s. The rest of this exercise follows the pattern above; a
should have some category y, and e selects for an s and then a y to build an s:
y.a
s=.y=.s.e
Question 3
Give a grammar for the Dijck language over a and b; i.e. the language consisting of sentences with
1. equal numbers of as and bs
2. every prefix of the sentence has at least as many as as bs
Another way of viewing this language is as the language of balanced
parentheses, where a is the open parenthesis and b the closed parenthesis.
A short context-free grammar for the Dijck language can be given as follows:
S →
S → aSbS
2
We will be basing our MG analysis on this context-free grammar, and thus
will need lexical items for both rules. We will need three lexical items, one
corresponding to each phonetic primitive (, a, and b). The first is uninteresting,
and implements the first rule.
s.
There are many possible choices as to how to implement the second rule. We
decide to have a and b do non-trivial things. Both will select for a complement
of category s, and b will select for its specifier the result of a combining with
an s:
=s.x.a
=s.x=.s.b
Thus we end up assigning the following schematic structure to the right-hand
side of the second context-free rule:
>
<
a
S
<
S
b
Question 4
Give a grammar for the non-context-free language an bn cn ; i.e. the
language consisting of sentences with as, bs and cs in that order
with equal numbers of each. You will have to use head movement.
For simplicity, assume that the morphological component simply
concatenates strings together (so the complex head u-v is realized as
uv).
The idea will be a slight variation of the grammar for an cbn . We will treat
the bs and cs as being selected for the as, which in turn are head-moving at
each step.
x.b
y.c
=>t.=y.x=.t.-a
To start things out, we introduce an empty lexical item:
t.
A single round of derivations gives us the structure below:
>
.b
<
.c
<
.
t.-a
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The basic invariant we would like to maintain is the following:
1. all the as are in the head of the tree
2. all the bs are to the left of all the cs
3. there is one b to the left of the head, all other bs are to the right
4. there are as many as as bs as cs
One can easily see that these properties hold of the expression above. Now,
assume we had an expression T of which these properties held, and let us put
that into the position of in the tree above (i.e. let us do one round of derivation
on T ). This gives us the tree below, where w is the string of as at the head of
T.
>
.b
<
.c
<
t.w -a
T
We see that if the invariant was true of T , it is also true of the resulting
expression above. Now we just need to move the as around that last b, which
we can do by means of the expression below.
=>t.s.-
Question 5
We have been assuming that feature bundles are ordered lists of
features. Let’s drop this assumption (and now feature bundles are
just unordered lists (i.e. multi-sets) of features). Reformulate the
merge rule to take this into consideration. Can you give an account
of the English auxiliary system?
Good question! Note that this can be simulated in our current framework by
simply adding, for every lexical item δ.w, and permutation γ of δ, the lexical item
γ.w. In other words, making a small free choice at each point in the derivation
(i.e. what will my first feature be now) is equivalent to making a single big free
choice at the beginning (i.e. in what order will I check my features).
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