\section{A Compositional Semantics: A Worked Example}
%
%
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We are now going to outline a semantics for languages 
with iterated case. The basic principle is rather 
easy: variables will be identified with sequences 
of case--functions. This is just a naming convention. 
%%%
\footnote{This approach is actually not far fetched. 
In a run--off the mill logical language one usually 
assumes that variables have the form $x_i$, where $i$ is a 
natural number. But this defines an infinitary language 
to start with, since each variable is a distinct symbol. 
In fact, if one manipulates these variables in a computer, 
one will end up coding variables as sequences of the form 
$\mbox{\tt x}\alpha$, where $\alpha$ is a sequence of zeros 
and ones. This allows the computer's inbuilt arithmetical 
functions to perform substitutions, which must be explicitly 
defined. So, if one wants to implement Montague Semantics, 
one will end up programming the substitutions, for which 
$\lambda$--calculus was originally responsible. However, 
in the semantics we are going to propose here, this is already 
done. The reader is made aware of the fact that this semantics needs 
no explification in terms of $\lambda$--calculus, since it 
is already at a lower level. The substitutions, being defined 
as string substitutions (which in turn are easily implemented), 
are computable in linear time using a regular transducer.}
%%%
The semantics uses two levels: a DRS--level, which contains DRSs, 
and a referent level, which talks about the names of the referents 
used by the DRS. Recall that a DRS has the form $[V : \Delta]$, where 
$V$ is a set of variables and $\Delta$ a set of formulae 
or DRSs. The meaning of $V$ will not be of importance 
throughout this paper. 

There is one additional symbol: $\circ$. It is a 
variable over names of referents.  A simple lexical 
entry, for example for the verb {\sf to teach} looks 
like this:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf teach}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf teach}'(e); \\
\mbox{\sf act}'(e) \doteq \mbox{\sc nom}{^{\smallfrown}\circ}; \\
\mbox{\sf thm}'(e) \doteq \mbox{\sc acc}{^{\smallfrown}\circ}. \\
\hline
\end{array}$$
%%
Here, the upper part is the referent system, and the lower part 
an ordinary DRS, with a head section containing a set of 
variables and a body section, containing a set of clauses. 
This means that there is an event of teaching whose 
actor is some $x$ and whose theme or goal is $y$. 
However, it is not $x$ and $y$ that appear here, as 
is usual. Rather, instead of $x$ we find the variable 
$\mbox{\sc nom}{^{\smallfrown}\circ}$, and instead of 
$y$ we find $\mbox{\sc acc}{^{\smallfrown}\circ}$.
Notice that $x$ and $y$ do not appear here (they are 
metavariables), only the actual variables, though 
written down with the use of $\circ$. $\circ$ can be 
instantiated to any sequence of case functions.
Now the meaning of $\circ : \circ$ is as 
follows: it says that nothing is added to the 
sequence under merge, $\circ$ is simply replaced by 
$\circ$. This become clear in a minute. 

We shall assume that the phrase {\sf a doctor} in the 
nominative has the following semantics:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf a doctor}/} \\
\hline
\circ : \circ \\
\hline\hline
\{x\} \\
\hline
\mbox{\sf doctor}'(\mbox{\sc nom}{^{\smallfrown}\circ}) \\
\hline
\end{array}$$
%%
The two structures  for $/\mbox{\sf teach}/$ and $/\mbox{\sf a doctor}/$
can merge by identifying $\circ$. 
This means that they now unify $\circ$ to the same sequence. 
We will then get 
%%
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf a doctor teach}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf teach}'(e); \\
\mbox{\sf act}'(e) \doteq \mbox{\sc nom}{^{\smallfrown}\circ}; \\
\mbox{\sf thm}'(e) \doteq \mbox{\sc acc}{^{\smallfrown}\circ}; \\
\mbox{\sf doctor}'(\mbox{\sc nom}{^{\smallfrown}\circ}). \\
\hline
\end{array}$$
%%
If done in this way, we can merge any two constituents and 
get any kind of semantics. However, the outcome is constrained 
in just the right way as we shall see. 

Now, a case suffix has the habit of adding something to the 
stack of cases. We shall assume that in addition to any 
semantic effect that it might have it contributes to the 
name of the variables in the following way. Their names 
are prolonged by one function. To see how this works, let 
us write down the semantics of, say, nominative:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sc nom}/} \\
\hline
\circ : \mbox{\sc nom}{^{\smallfrown}\circ} \\
\hline\hline
\varnothing \\\hline
\varnothing \\\hline
\end{array}$$
%%
Here we find the statement $\circ : \mbox{\sc nom}{^{\smallfrown}%
\circ}$. This says that when the above structure is merged with another 
one, say $\Delta(\circ)$, then the variable $\circ$ of 
$\Delta$ is instantiated to (or replaced by) $\mbox{\sc nom}{^{\smallfrown}
\circ}$. This means that the entire sequence of case suffixes 
is increased by one element. As a result of the merge 
we shall get however $\circ : \circ$. In order that 
this is  well defined, $\Delta$ need not contain exactly 
$\circ : \circ$. If it contains, say, $\circ : \mbox{\sc acc}%
{^{\smallfrown}\circ}$, then we only have to stipulate how the 
resulting sequence will be, ie whether we get 
$\circ : \mbox{\sc nom}{^{\smallfrown}\mbox{\sc acc}%
{^{\smallfrown}\circ}}$ or whether we get 
$\circ : \mbox{\sc acc}{^{\smallfrown}\mbox{\sc nom}%
{^{\smallfrown}\circ}}$.  
The choice is made so that merge is associative. However, 
throughout this paper we shall make the assumption that 
merge is successful only if one of the semantic structures 
has an upper line of the form $\circ : \circ$. 
Such a structure we shall also call {\it plain}.

To see the mechanism work, let us start with the following 
lexical entry for a simple noun: 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf doctor}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf doctor}'(\circ) \\
\hline
\end{array}$$
%%
Now we shall compute the merge with the semantic structure for 
nominative:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf doctor}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf doctor}'(\circ) \\
\hline
\end{array}
\quad\oplus\quad
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sc nom}/} \\
\hline
\circ : \mbox{\sc nom}{^{\smallfrown}\circ} \\
\hline\hline
\varnothing \\\hline
\varnothing \\\hline
\end{array}
\quad = \quad
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf doctor}+\mbox{\sc nom}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf doctor}'(\mbox{\sc nom}{^{\smallfrown}\circ}) \\
\hline
\end{array}$$
%%
To see why this is so, notice that the symbol $\circ$ in the first 
structure is replaced by $\mbox{\sc nom}{^{\smallfrown}\circ}$ 
according to the laws of merge. Now, in order to understand the 
potential of this proposal let us repeat this example with a 
relational noun, {\sf teacher}:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf teacher}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf teach}'(\circ, \mbox{\sc gen}{^{\smallfrown}\circ}) \\
\hline
\end{array}$$
%%
If we add the suffix {\sc nom}, we get 
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf teacher}+\mbox{\sc nom}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf teach}'(\mbox{\sc nom}{^{\smallfrown}\circ}, 
\mbox{\sc gen}{^{\smallfrown}\mbox{\sc nom}{^{\smallfrown}\circ}}) \\
\hline
\end{array}$$
%%
Notice that by the mechanics of replacement, it is not 
only the main variable that changes its name but also the 
variable of the complement. This is what we will make use 
of. 

Cases may or may not have a semantics. This actually does not 
make much of a difference for this calculus. Take the genetive, 
which is many languages is used for marking possession:
%%
$$\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sc gen}/} \\
\hline
\circ : \mbox{\sc gen}{^{\smallfrown}\circ} \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf belong-to}'(\circ,\mbox{\sc gen}{^{\smallfrown}\circ}) \\
\hline
\end{array}$$
%%
So, when a genetive is attached, it says that the thing to which 
it attaches owns something. Here, $\circ$ represents the thing 
that is possessed, while $\mbox{\sc gen}{^{\smallfrown}\circ}$ 
is the thing that owns it. To see how this works, we shall 
turn to a real example. The following is a construction of 
Old--Georgian, taken from \cite{blake:case}, Page 103.
%%
\\[2mm]
\begin{tabular}{lll}
(\thesection.1) & {\sf sarel-ita} & {\sf man-isa-jta} \\
                & name-{\sc inst} & father-{\sc gen}-{\sc inst} \\
                & \multicolumn{2}{l}{\it with father's name} 
\end{tabular}
\\[2mm]
%%
The first part is clear: {\sf sarel-ita} is
%%
$$(A)\qquad\qquad\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf sarel-ita}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf name}'(\mbox{\sc inst}{^{\smallfrown}\circ}); \\
\mbox{\sf instrument}'(\mbox{\sc inst}{^{\smallfrown}\circ}). \\
\hline
\end{array}$$
%%
Now let us turn to {\sf man-isa-jta}. First we attach the 
genetive to {\sf man}:
%%
$$\begin{array}{l}
%%
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf man}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf father}'(\circ,\mbox{\sc gen}{^{\smallfrown}\circ}) \\
\hline
\end{array} 
%%
\; \oplus \; 
%%
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf isa}/} \\
\hline
\circ : \mbox{\sc gen}{^{\smallfrown}\circ} \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf belong-to}'(\circ,\mbox{\sc gen}{^{\smallfrown}\circ}) \\
\hline
\end{array} 
%%
\\\\
\quad = 
%%
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf man-isa}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf father}'(\mbox{\sc gen}{^{\smallfrown}\circ}, %
\mbox{\sc gen}{^{\smallfrown}\mbox{\sc gen}{^{\smallfrown}\circ}}); \\
\mbox{\sf belong-to}'(\circ, \mbox{\sc gen}{^{\smallfrown}\circ}). \\
\hline
\end{array}
%%
\end{array}$$
%%
Next we attach the nominative suffix:
%%
$$(B)\qquad\qquad
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf man-isa-jta}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf father}'(\mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
	inst}{^{\smallfrown}\circ}}; %
\mbox{\sc gen}{^{\smallfrown}\mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
	inst}{^{\smallfrown}\circ}}}); \\
\mbox{\sf belong-to}'(\mbox{\sc inst}{^{\smallfrown}\circ}, 
	\mbox{\sc gen}{^{\smallfrown}\mbox{\sc inst}{^{\smallfrown}\circ}}). 
	\\
\hline
\end{array}$$
%%
Finally, the two structures (A) and (B) are merged to derive the final 
representation (C).
%%
$$(C)\qquad\qquad 
\begin{array}{|c|}
\multicolumn{1}{c}{/\mbox{\sf sarel-ita man-isa-jta}/} \\
\hline
\circ : \circ \\
\hline\hline
\varnothing \\
\hline
\mbox{\sf name}'(\mbox{\sc inst}{^{\smallfrown}\circ}); \\
\mbox{\sf father}'(\mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
	inst}{^{\smallfrown}\circ}}, %
\mbox{\sc gen}{^{\smallfrown}\mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
	inst}{^{\smallfrown}\circ}}}); \\
\mbox{\sf instrument}'(\mbox{\sc inst}{^{\smallfrown}\circ}); \\
\mbox{\sf belong-to}'(\mbox{\sc inst}{^{\smallfrown}\circ}, 
	\mbox{\sc gen}{^{\smallfrown}\mbox{\sc inst}{^{\smallfrown}\circ}}). 
	\\
\hline
\end{array}$$
%%
(C) is true in a model under an assignment for the 
variables if $\mbox{\sc inst}{^{\smallfrown}\circ}$ 
is instantiated to a thing $x$ that is a name, and 
$\mbox{\sc gen}{^{\smallfrown}\mbox{\sc inst}{^{\smallfrown}\circ}}$ 
is instantiated to a thing $y$ that is a father, and $x$ belongs 
to $y$. And that $y$ is a father of the value of  
$\mbox{\sc gen}{^{\smallfrown}\mbox{\sc gen}{^{\smallfrown}\mbox{\sc 
inst}{^{\smallfrown}\circ}}}$.
This is exactly as it should be. 
