Week 6 [2023-05-09 Tue]
We generalized the link between derivations and derived structures.
We have two basic operations, Merge and Move,1 each of which can be broken up into different (here mutually exclusive) sub-operations (like mergeing a complement which will not move again vs merging a specifier which will move later).
I have chosen to present the effects on exponents (the derived structure) in a generic way, using operations \(\texttt{s}(\cdot,\cdot)\), \(\texttt{c}(\cdot,\cdot)\), and t. The first two stand for the binary operation on exponents which makes the second the specifier (resp. the complement) of the first, and the final one stands for the constant derived object trace.
Depending on what we understand derived objects to be, these operations are realized differently:
| op | strings | trees |
|---|---|---|
| \(\texttt{s}(u,v)\) | \(vu\) | \(\mathop{\hbox{\tt >}}(v,u)\) |
| \(\texttt{c}(u,v)\) | \(uv\) | \(\mathop{\hbox{\tt <}}(u,v)\) |
| \(\texttt{t}\) | \(\epsilon\) | \(\hbox{\tt t}\) |
We can interpret these operations on spans as well, which we exploit in parsing.2
| op | spans |
|---|---|
| \(\texttt{s}(\langle j,k\rangle,\langle i,j\rangle)\) | \(\langle i,k\rangle\) |
| \(\texttt{c}(\langle i,j\rangle,\langle j,k\rangle)\) | \(\langle i,k\rangle\) |
| \(\texttt{t}\) | \(\langle i,i\rangle\) (for any i) |
Figure 1: Merge a complement
Figure 2: Merge a specifier
Figure 3: Move
Derivation tree
A derivation records the steps taken, as per the rules above. To save space, I write \(\bullet\) instead of Merge, and \(\circ\) instead of Move. We could write a derivation of the sentence John laugh s as follows:
Figure 4: A derivation for John laugh -s
It is sometimes useful to write derivation trees with extra information at the internal nodes, so that we can see at a glance not only the steps taken in the derivation, but also what happened at each step. The tree above can be enriched as follows:3
Figure 5: A derivation tree with more information