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\begin{document}
\begin{center}
\begin{Large}
  On a Type-Based Analysis of Feature Neutrality and the Coordination
  of Unlikes\footnotetext[0]{This paper is an updated version of
    \citep{daniels02}. Thanks go to Chris Brew, Martin Jansche, Bob
    Levine, Roger Levy, Detmar Meurers, Carl Pollard, Ivan Sag, and
    Neal Whitman for detailed comments on earlier drafts.}\\[5pt]
\end{Large}
\begin{large}
Michael W. Daniels\\
\texttt{daniels@ling.osu.edu}\\
April 26, 2002\\
\end{large}
\end{center}

\section{Introduction}

All theories of syntax aspire to be able to analyze any construction
found in any language. Thus \citepos{ingria90} argument that both
feature neutrality and the coordination of unlikes pose fundamental
problems for unification-based theories presents a formidable
challenge to those theories; moreover, it has often been assumed to
automatically apply to their successors -- constraint-based theories
of grammar, exemplified by Head-driven Phrase Structure Grammar [HPSG]
\citep{pollard94}.

This paper shows that this claim does not in fact apply to HPSG.
Instead, the nature of the type hierarchy used by HPSG for ontological
specification provides a means to adequately and accurately analyze
these phenomena without recourse to novel data structures.

I will first describe the phenomena under discussion and then discuss
previous work on this topic; this will be followed by the proposed
analysis, and finally a comparison of this analysis with another
HPSG-based approach.

\section{Overview of the Phenomena}
\subsection{Feature Neutrality}\label{sec2a}

Many languages allow for a word to be neutral between two or more
values of a morphosyntactic feature; that is, the word's morphology
does not uniquely identify a value for the feature. These words may
appear in configurations where otherwise-conflicting constraints would
be imposed on that word -- in effect, the neutral form acts as though
it simultaneously possesses both feature values. The German data in
(\ref{second}) -- (\ref{first}) present an example of this phenomenon
in a coordination construction.

\begin{exe}
  \ex[*]{\label{second}\gll Er findet und hilft M\"anner.\\
    he finds and helps men.\textsc{acc}\\}
  \ex[*]{\label{third}\gll Er findet und hilft Kindern.\\
    he finds and helps children.\textsc{dat}\\}
  \ex[]{\label{first}\gll Er findet und hilft Frauen.\\
    he finds and helps women.\textsc{acc}\slash\textsc{dat}\\}
\end{exe}

The verb \emph{findet} requires its object to be accusative, while
\emph{hilft} requires a dative object. Thus the noun \emph{M\"anner}
in (\ref{second}), which cannot be dative, cannot occur with
\emph{hilft}, while the noun \emph{Kindern} in (\ref{third}), which
cannot be accusative, cannot occur with
\emph{findet}.\footnote{Switching the order of the conjuncts does not
  affect acceptability.} The noun \emph{Frauen}, however, is neutral
between accusative and dative case and is therefore acceptable in this
configuration.

Neutrality is distinguished from the familiar phenomenon of ambiguity
by precisely this property of simultaneity. The German subject-verb
agreement\footnote{In this paper, I use the terms \emph{agreement} and
  \emph{concord} interchangeably; the traditional distinction
  revolving around information flow is not relevant here.} data in
(\ref{am-1}) -- (\ref{am-3}) give an illustration of an ambiguous
form.

\begin{exe}
  \ex[]{\label{am-1}\gll Sie singt\\
    she sings.\textsc{sg}\\}
  \ex[]{\label{am-2}\gll Sie singen\\
    they sing.\textsc{pl}\\}
  \ex[*]{\label{am-3}\gll Sie singt und singen\\
    she\slash{}they sing.\textsc{sg} and sing.\textsc{pl}\\}
\end{exe}

While the form \emph{sie} can act as either singular or plural, it
cannot act as both at the same time. It is therefore considered
ambiguous rather than neutral.\footnote{Several articles, including
  \citep{zaenen84} and \citep{pullum86}, discuss other ways to
  distinguish neutrality from ambiguity; given that such a distinction
  exists, such tests are orthogonal to the analysis of neutrality
  itself.}

A disjunctive description (for instance, \savm{case}{acc $\lor$ dat})
represents a state of partial information: we know that \textsc{case}
is either \emph{acc} or \emph{dat}, but we don't know which. These
descriptions are quite suitable for ambiguous forms, as the form's
context will fill in the missing information, indicating which way the
disjunction should be resolved.

It has been recognized since \citep{zaenen84}, however, that
disjunctive descriptions cannot be used to accurately model
neutrality. If the object corresponding to \emph{Frauen} in
(\ref{first}) were specified by \savm{case}{acc$\lor$dat}, sentence
(\ref{first}) could not be licensed: such a specification would have
to describe an object with either \savm{case}{acc} (which fails to
satisfy the selectional requirements of \emph{hilft}) or
\savm{case}{dat} (which fails to satisfy the requirements of
\emph{findet}). \footnote{This problem also holds when the lexicon
  contains two words with the phonological form of \emph{Frauen}: one
  with \savm{case}{acc} and one with \savm{case}{dat}.}

In the examples analyzed so far, we have focused on a single aspect of
morphosyntax: case. In general, however, one affix or word often
encodes the value of multiple morphosyntactic features, as shown for
German in (\ref{3-a}).

\begin{exe}
  \ex[]{\label{3-a}\gll der Antrag des oder der Dozenten\\
    the petition of-the.\textsc{gen.sg} or of-the.\textsc{gen.pl}
    docent.\textsc{masc.gen.(sg\slash{}pl)}\\
    \mytrans{the petition of the docent(s)}}
\end{exe}

It is not enough to say that \emph{Dozenten} is neutral between
singular and plural: \emph{Dozenten} cannot be taken as singular in
(\ref{3-b}).

\begin{exe}
  \ex[*]{\label{3-b}\gll der Dozenten ist hier.\\
    the.\textsc{nom.sg} docent.\textsc{nom.sg} is.\textsc{sg} here.\\}
\end{exe}

Instead, any theory of neutrality needs to be able to constrain the
neutrality of one feature based on the value of another. Here, while
it is true that \emph{Dozenten} is neutral between singular and
plural, a particular token of \emph{Dozenten} can only act as singular
while its case is not nominative.

One further complication arises from the fact that many languages have
NP-internal agreement (for example, determiner-noun agreement or
noun-adjective agreement) in addition to NP-external agreement (for
example, subject-verb). When multiple elements of a noun phrase are
neutral, the resulting NP acts externally as if its agreement
properties were the intersection of the individual neutralities. This
is illustrated in (\ref{dd-1}) -- (\ref{dd-3}).

\begin{exe}
  \ex[]{\label{dd-1}\gll Er hilft Dozenten.\\
    he helps docents.\textsc{dat.pl}\\}
  \ex[]{\label{dd-2}\gll Er hilft den Dozenten.\\
    he helps the.\textsc{dat.pl} docents.\textsc{dat.pl}\\}
  \ex[]{\label{dd-4}\gll Er hilft der Frau.\\
    he helps the.\textsc{dat.sg.fem} woman.\textsc{dat.sg.fem}\\}
  \ex[*]{\label{dd-3}\gll Er hilft der Dozenten.\\
    he helps the docents\\}
\end{exe}

Sentences (\ref{dd-1}) -- (\ref{dd-2}) show that \emph{Dozenten} is a
valid object of \emph{hilft} and therefore can manifest dative case.
Sentence (\ref{dd-4}) shows that the definite article \emph{der} can
also manifest dative case. In sentence (\ref{dd-3}), however, we see
that the combined noun phrase \emph{der Dozenten} cannot manifest
dative case. Thus it is insufficient for each word to individually
meet the case requirements of the verb: the NP itself may often be
less neutral than its component forms. This may seem like a trivial
conclusion, but it will be shown in section 3 that much of the
existing literature on this topic has not dealt with this aspect of
the phenomenon.

\subsection{Coordination of Unlikes}

Now consider the problem of determining the \textsc{case} value of the
coordinate noun phrase \emph{M\"anner und Kindern}. Items (\ref{gcu-3})
-- (\ref{gcu-4}) show that this phrase cannot be selected by a verb
requiring an accusative object (like \emph{findet}) or by one
requiring a dative object (like \emph{hilft}).

\begin{exe}
  \ex[*]{\label{gcu-3}\gll Er findet M\"anner und Kindern.\\
    he finds men.\textsc{acc} and children.\textsc{dat}\\}
  \ex[*]{\label{gcu-4}\gll Er hilft M\"anner und Kindern.\\
    he helps men.\textsc{acc} and children.\textsc{dat}\\}
\end{exe}

This is the case even when the verb in question could occur with
either case individually, as shown in (\ref{leh-1}) -- (\ref{leh-3}).

\begin{exe}
  \ex[]{\label{leh-1}\gll Er lehrt Kinder.\\
    he teaches children.\textsc{acc}\\}
  \ex[]{\label{leh-2}\gll Er lehrt Kindern.\\
    he teaches children.\textsc{dat}\\}
  \ex[*]{\label{leh-3}\gll Er lehrt M\"anner und Kindern.\\
    he teaches men.\textsc{acc} and children.\textsc{dat}\\}
\end{exe}

The verb \emph{lehren} can take either the accusative \emph{M\"anner}
or the dative \emph{Kindern}, but not the coordinate \emph{M\"anner
  und Kindern}. Thus the verb \emph{lehren} is said to have ambiguous
selectional requirements: it can take either of two values, but not
both simultaneously.

Ambiguous selectors like \emph{lehren} stand in contrast to those in
other languages with neutral selectional requirements. For instance,
in Russian, some verbs under certain circumstances can take either
accusative or genitive complements, as well as a coordinate complement
where one of the conjuncts is accusative and the other is genitive.
This is shown by the data from \citep{levy01} in (\ref{ru-1}).

\begin{exe}
  \ex[]{\label{ru-1}\gll V\v cera ves' den' on o\v zidal svoju
  podrugu Irinu i zvonka ot svoego brata Grigorija\\
  yesterday all day he expected self's.\textsc{acc}
  girlfriend.\textsc{acc} Irina.\textsc{acc} and call.\textsc{gen}
  from self's.\textsc{gen} brother.\textsc{gen} Gregory.\textsc{gen}\\
  \mytrans{`Yesterday he waited all day [for his girlfriend Irina] and
    [for a call from his brother Gregory].'}}
\end{exe}

We can draw a parallel between this phenomenon and that of the
coordination of unlike categories, as discussed in, for example,
\citep{sag85}. This term traditionally refers to the coordination of
distinct parts of speech, as in (\ref{ecu-2}) -- (\ref{ecu-4}).

\begin{exe}
  \ex[]{\label{ecu-2} He is [a republican] and [proud of it].}
  \ex[]{\label{ecu-3} I consider that [a rude remark] and [in very poor
    taste].}
  \ex[]{\label{ecu-4} Robin walks [slowly] and [with great care].}
\end{exe}

In (\ref{ecu-2}), the conjuncts are a noun phrase and an adjective
phrase; in (\ref{ecu-3}), a noun phrase and a prepositional phrase;
and in (\ref{ecu-4}), an adverb and a prepositional phrase. In each
case, the coordinate phrase as a whole is a dependent of the verb.

In HPSG, the concept of category (that is, the value of the
\textsc{cat} feature) allows for fine-grained distinctions: since the
conjuncts in (\ref{ecu-3}) have different values of \textsc{case} (a
subfeature of \textsc{cat}), they too can be said to be unlike
categories.

Thus, as pointed out by \citet{bayer96}, the coordination of unlikes is
effectively the dual of feature neutrality: instead of multiple
constraints needing to be true of a single object, a single constraint
needs to be true of multiple objects. These two phenomena are linked
by a single question: If we take coordinate phrases to have the same
ontological status as non-coordinate phrases, then any feature
appropriate for a normal phrase (\textsc{val}, \textsc{case}) must
also be appropriate for a coordinate phrase. What are the values of
those features?

Before answering this question, I review some of the previous and
current work in this area.

\section{Related Work}
\subsection{Ingria 1990}\nocite{ingria90}

Ingria uses data related to feature neutrality to criticize the
standard treatment of agreement in unification-based grammars, in
which objects are subject to token-identity constraints (that is, the
value of feature \textsc{x} must unify with \emph{y}). Instead, he
argues that agreement constraints should have the form of a
non-distinctness check (that is, the value of feature \textsc{x} must
not contradict \emph{y}). For example, this check would say that since
\savm{case}{acc$\lor$dat} is non-distinct from both \savm{case}{dat}
(the selectional requirement of \emph{hilft}) and \savm{case}{acc}
(the selectional requirement of \emph{findet}), it is an acceptable
\textsc{case} value for \emph{Frauen}.

It is unclear how, in the absence of unification, the problem of
NP-internal agreement can be dealt with. Clearly the non-distinctness
check is insufficient on its own: in example (\ref{dd-3}) above, both
\emph{der} and \emph{Dozenten} are non-distinct with \textsc{dat}, yet
\emph{der Dozenten} is an unacceptable object for \emph{hilft}.

Ingria's objection to the use of token-identity is based on the
assumption that \textsc{case} values must be disjoint: anything that can
unify with a certain \textsc{case} value cannot unify with any other
\textsc{case} value. For instance, in a sentence like (\ref{first}),
the verb \emph{hilft} requires \savm{case}{dat} to hold of its object,
while the verb \emph{findet} requires \savm{case}{acc}; Ingria would
hold that no single object could meet both of these constraints. I
will show in section 4 an analysis that falsifies this assumption.

\subsection{Bayer and Johnson 1995}

Working within the framework of Lambek categorial grammar, Bayer and
Johnson (1995)\nocite{bayer95} present an account of feature
neutrality and unlike coordination that takes advantage of a certain
type-logical theory of categories. Their analysis of (\ref{first}) is
given in (\ref{cg}).

\begin{exe}
\ex[]{\begin{small}\begin{tabular}[t]{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c}
findet&$[\smnp\land\smacc\land\smdat]^1$&&hilft&
$[\smnp\land\smacc\land\smdat]^2$\\
\cfill Lx&\cfill P&&\cfill Lx&\cfill P\\
$\smvp/\smnp\land\smacc$&$\smnp\land\smacc$&&
$\smvp/\smnp\land\smdat$&$\smnp\land\smdat$\\
\multicolumn{2}{c@{\,}}{\cfill/E}&&\multicolumn{2}{c@{\,}}{\cfill/E}\\
\multicolumn{2}{c@{\,}}{$\smvp$}&und&\multicolumn{2}{c@{\,}}{$\smvp$}\\
\multicolumn{2}{c@{\,}}{\cfill/I$^1$}&\cfill Lx&
\multicolumn{2}{c@{\,}}{\cfill/I$^2$}\\
\multicolumn{2}{c@{\,}}{$\smvp/\smnp \land \smacc \land \smdat$}&$\smconj$&
\multicolumn{2}{c@{\,}}{$\smvp/\smnp \land \smacc \land \smdat$}&Frauen\\
\multicolumn{5}{c@{\,}}{\cfill Co}&\cfill Lx\\
\multicolumn{5}{c@{\,}}{$\smvp/\smnp \land \smacc \land \smdat$}&
$\smnp \land \smacc \land \smdat$\\
\multicolumn{6}{c@{\,}}{\cfill/E}\\
\multicolumn{6}{c@{\,}}{$\smvp$}\\
\end{tabular}\end{small}\label{cg}}
\end{exe}

The derivation starts with instances of lexical access (rule Lx). The
verb \emph{findet} has category $\vp\slash{}\np\land\acc$,
representing the fact that it wants an accusative object; \emph{hilft}
has category $\vp\slash{}\np\land\dat$, as it wants a dative object;
and \emph{Frauen} has category $\np\land\acc\land\dat$, representing
its neutrality between accusative and dative case.

Then the category for \emph{Frauen} is hypothesized after each
conjunct. The first hypothesis is then weakened (rule P) from
$\np\land\acc\land\dat$ to $\np\land\acc$, and the second to
$\np\land\dat$. This allows each verb to combine (rule \slash{}E) with
the introduced argument, forming in each case the category $\vp$. The
hypothesis is then discharged (rule \slash{}I) for each conjunct,
forming categories $\vp\slash{}\np\land\acc\land\dat$.

Since the conjuncts have the same category, the conjunction rule (rule
Co) can be applied to form a single conjoined
$\vp\slash{}\np\land\acc\land\dat$ category. As the last step, this
category combines with the actual argument \emph{Frauen} (rule
\slash{}E).

In this example, the hypothesized object was weakened by meet
elimination: the change from $a\land b$ to $a$. Categories can also be
weakened by join introduction: the change from $a$ to $a\lor b$. This
is used in the analysis of unlike coordination, as illustrated in
(\ref{cg2}).

\begin{exe}
\ex[]{\begin{small}\begin{tabular}[t]{c@{\,}c@{\,}c@{\,}c}
&wealthy&&a republican\\
&\cfill Lx&&\cfill Lx\\
&$\smap$&and&$\smnp$\\
&\cfill Wk&\cfill Lx&\cfill Wk\\
became&$\smnp\lor\smap$&$\smconj$&$\smnp\lor\smap$\\
\cfill Lx&\multicolumn{3}{c@{\,}}{\cfill Co}\\
$\smvp/\smnp\lor\smap$&\multicolumn{3}{c@{\,}}{$\smnp\lor\smap$}\\
\multicolumn{4}{c@{\,}}{\cfill/E}\\
\multicolumn{4}{c@{\,}}{$\smvp$}\\
\end{tabular}\end{small}\label{cg2}}
\end{exe}

Here, the lexicon assigns \emph{wealthy} category $\ap$, \emph{a
  republican} category $\np$, and \emph{became} category
$\vp/\np\lor\ap$. Each of the conjuncts is then weakened (rule P) to
$\np\lor\ap$; these categories are then conjoined to form a single
$\np\lor\ap$ category, which can combine with the category of
\emph{became}.

While this seems to be a quite elegant solution to the problem,
\citet{bayer96} points out problems with the use of $\land$ to
represent the conjunction of orthogonal features as well as neutral
feature values. In particular, a form neutral between \textsc{nom.pl}\
and \textsc{gen.sg}\ is represented by the meet of
$(\np\land\nom\land\plural)$ and $(\np\land\gen\land\singular)$, which
is $(\np\land\nom\land\plural\land\gen\land\singular)$. This category
can be weakened to $(\np\land\gen\land\plural)$, thus predicting that
such a neutral form could be used in a context where \textsc{gen.pl}\
was required. This is contradicted by the Finnish data in
\citep{zaenen84}, where the form \emph{kirjansa} can be used for
\textsc{nom.sg}, \textsc{gen.sg}, and \textsc{nom.pl}, but not
\textsc{gen.pl}.\footnote{The work of D\"orre and Manandhar
  (1995)\nocite{doerre95} on constraint-based Lambek calculi may
  provide an approach to features in categorial grammar that resists
  this criticism (Martin Jansche, p.c.).}

\subsection{Dalrymple and Kaplan 2000}

\citet{dalrymple00} provide an LFG-based analysis of both feature
neutralization and unlike coordination. In their account, coordinate
phrases contrast with non-coordinate phrases by corresponding to a set
of f-structures rather than a single f-structure. A constraint then
holds of a set of f-structures exactly when it holds of each member of
the set. Neutrality is represented by assigning the feature the set of
those values between which the form is neutral.

Thus in their analysis of (\ref{first}), \emph{Frauen} has
\savm{case}{\{\emph{acc}, \emph{dat}\}\,}. The verb \emph{hilft}
requires that its argument's \textsc{case} value have \emph{dat} as a
member, and similarly \emph{acc} for \emph{findet}. \emph{Frauen}
meets these conditions and (\ref{first}) is licensed.

This analysis shares some properties with Ingria's analysis and
suffers from some of the same problems. As agreement is effected
through what is in effect a subsumption check rather than a shared
object, NP-internal agreement must be treated as a separate
phenomenon. While Dalrymple and Kaplan acknowledge the existence of
this problem (p.\ 795), they do not provide an analysis for it. Nor is
any account given of neutrality across features.

This approach also strips sets of having any intrinsic meaning in the
LFG framework, increasing the possibility for unwanted interactions.
Sets can be used to represent collections of arbitrary numbers of
items, neutral values, coordinations, and feature decompositions.

\subsection{Levy 2001}

\citet{levy01} (along with \citep{levy-pollard02}) presents an
HPSG-based approach developed contemporaneously with this paper that
centers around certain sets of sets (which Levy refers to as
double-sets). Further discussion of Levy's approach will be deferred
until the analysis in section 4 has been presented; this will allow
for a detailed comparison of the two approaches.

It should be noted that Levy's theory only presents an analysis of
single-feature neutralization. He speculates, however, that his theory
can handle neutrality across features (pp.\ 27--29) and suggests some
potential avenues for further exploration.

\section{A Type-Based Approach}\label{sec4}

The analysis presented in this section differs from the proposals
discussed in the previous section in two main ways. It adds no new
data structures or mechanisms to the HPSG framework and it retains the
use of structure-sharing in effecting agreement. As such it explicitly
rejects \citepos{levy01} claim that ``the problems presented by
[feature neutrality and unlike coordination] cannot be solved solely
with the structural tools made available by HPSG's type hierarchy''.

\subsection{The Phrase Structure of Coordination}

Before the analysis can be presented, I will present some baseline
assumptions about the phrase structure of coordinate structures. I
take the type \emph{coord-phrase} to be a subtype of
\emph{non-headed-phrase}. Following \citep{sag85}, I assume two
subtypes of \emph{coord-phrase} for English:
\emph{bin(ary)-coord-ph(rase)} and \emph{iter(ated)-coord-ph(rase)}.

A \emph{bin-coord-ph} is licensed whenever the first daughter has
\savm{marking}{$\alpha_0$} and the second daughter has
\savm{marking}{$\alpha_1$} for some $\alpha$. \citep{sag85} gives the
requirement $\alpha \in \{\{$both, and$\}, \{$either, or$\},
\{$unmarked, but$\}\}$ while noting that the set of possible values
for $\alpha$ is highly subject to idioletical variation.

Similarly, an \emph{iter-coord-ph} is licensed whenever the first
daughter has \savm{marking}{$\beta_0$}, the last daughter has
\savm{marking}{$\beta_2$}, and all other daughters have
\savm{marking}{$\beta_1$} for some $\beta$. Here $\beta \in
\{\{$unmarked, and, and$\}, \{$unmarked, unmarked, and$\},
\{$unmarked, unmarked, or$\}, \{$unmarked, or, or$\}, \{$neither, nor,
nor$\}\}$; this too may vary widely among idiolects.

\subsection{Feature Neutrality}\label{sec4b}

The analysis of feature neutrality developed in this paper is based on
the \citep{levine01} treatment of case-consistency for parasitic gaps.
As is standard in GPSG- and HPSG-style analyses of parasitic gaps, the
local properties of both the host and parasitic gaps are represented
by a single object: all constraints affecting one affect the other.
When different case requirements are imposed on the two gaps, then,
the resulting sentence is only acceptable when the filler is
case-neutral. This is illustrated in (\ref{fourth}) and (\ref{fifth}).

\begin{exe}
  \ex[*]{\label{fourth} whom even friends of \gap\ believe \gap\ should
    be closely watched.}
  \ex[]{\label{fifth} who even friends of \gap\ believe \gap\
    should be closely watched.}
\end{exe}

Here the preposition \emph{of} requires its object to have accusative
case, while the verb \emph{should} requires its subject to have
nominative case. The accusative pronoun \emph{whom} cannot satisfy
both of these constraints, but the case-neutral pronoun \emph{who}
can.

To analyze this, \citet{levine01} augment the type hierarchy for
\textsc{case} values with three new types: a neutral type
\emph{p-nom-acc} and two `pure' subtypes \emph{p-nom} and
\emph{p-acc}, as given in the portion of the signature shown in
(\ref{nahier}).\footnote{Typenames have been slightly modified from
  their \citep{levine01} version in anticipation of the analysis that
  will be presented in this paper.}

\begin{exe}
\ex[]{\label{nahier}\begin{tabular}[t]{l}
\xymatrix@-1pc{
&&\ar@{-}[dl]\ar@{-}[dr]\txt{case}\\
&\ar@{-}[dl]\ar@{-}[dr]\txt{acc}&&\ar@{-}[dl]\ar@{-}[dr]\txt{nom}\\
{\txt{p-acc}}&&{\txt{p-nom-acc}}&&{\txt{p-nom}}\\}
\end{tabular}}
\end{exe}

Here, the type \emph{p-nom-acc} represents neutrality between
nominative and accusative case\footnote{When reading the type names,
  it is important to take the \emph{p-} prefix as having scope over
  the rest of the name. For instance, the type \emph{p-nom-acc} is the
  pure type corresponding to the neutralization of nominative and
  accusative case, not the neutralization of \emph{p-nom} and
  \emph{acc}.}; a type like \emph{p-nom} represents a pure
(non-neutral) case value.

Under this scheme, the only maximally specific types are \emph{p-nom},
\emph{p-nom-acc}, and \emph{p-acc}. Case-neutral pronouns like
\emph{who}, as well as common and proper nouns, are listed in the
lexicon as \savm{case}{p-nom-acc}, while case-specific pronouns like
\emph{he} or \emph{whom} are listed as \savm{case}{p-nom} or
\savm{case}{p-acc}, as appropriate.

The remaining types -- \emph{nom}, \emph{acc}, and \emph{case} (which
is equivalent to $\nom\lor\acc$) -- are only used in object
descriptions, such as those found in valence specifications. For
instance, a typical finite English verb selects a subject with
\savm{case}{nom} and selects complements with \savm{case}{acc}.

With this hierarchy, a description like \savm{case}{acc} abbreviates
the disjunction (\savm{case}{p-acc} $\lor$ \savm{case}{p-nom-acc}).
Thus \savm{case}{acc} and \savm{case}{nom} now unify to
\savm{case}{p-nom-acc}. In effect, we are now able to model the
intuition that \emph{who} is both nominative and accusative.

The central distinction in this hierarchy, then, is between types
which begin with \emph{p-} (the pure types) and those that do not (the
non-pure types). Pure types are maximally specific; they are the types
found on linguistic objects. Non-pure types, in contrast, are found on
linguistic descriptions (for example, subcategorization requirements).
Every non-pure type corresponds to a pure type. The non-pure type is
usually the immediate supertype of the pure type, with one exception:
the most neutral type (here, \emph{p-nom-acc}) is both pure and
non-pure.\footnote{We could avoid this awkwardness by introducing a
  type \emph{nom-acc} in the place of \emph{p-nom-acc} that would then
  only dominate \emph{p-nom-acc}, but such unary branching in a type
  hierarchy is seen as equally awkward by many.}

Just as (\ref{nahier}) serves as a case hierarchy for English, so may
a similar hierarchy be constructed for German, which has a four-case
system. I will assume that such a type hierarchy would only contain
those pure types which correspond to attested patterns of
neutralization in the language, rather than containing a pure type for
every potential subset of cases. For instance, no form in German is
neutral between nominative, accusative, and genitive cases (to the
exclusion of dative), and so there is no \emph{p-nom-acc-gen} type in
the hierarchy. (See section \ref{sec4d} for an illustration of the
effect this has on a sample case hierarchy.)

The analysis of (\ref{first}) (repeated below) is now straightforward.

\begin{exe}
  \exr{first}[]{\gll Er findet und hilft Frauen.\\
    he finds and helps women.\textsc{acc/dat}\\}
\end{exe}

Under the system described above, the verb \emph{findet} imposes the
constraint \savm{case}{acc} on its object and \emph{hilft} imposes the
constraint \savm{case}{dat} on its object. The constraint in
(\ref{cps-1}) models the fact that each verb's selectional
specifications must be satisfied by the dependents of the coordination
(an outgrowth of what is often referred to as Wasow's generalization
\citep{pullum86}).

\begin{exe}
  \ex[]{\label{cps-1} \emph{coord-phrase} $\to$
    \begin{avm}\[
      \textsc{val} & \@2 \\
      \textsc{dtrs} & \<\,\[\textsc{val} & \@2\]\,,
                        \[\textsc{val} & \@2\]\,\>\]\end{avm}}
\end{exe}

Thus the constraint that the coordination of the two verbal heads
\emph{findet und hilft} imposes on its object is the logical
conjunction of the constraints \savm{case}{acc} and \savm{case}{dat}:
namely, \savm{case}{acc-dat}.

Returning to (\ref{first}), the word \emph{Frauen}, as a word neutral
between all four cases, is listed in the lexicon as
\savm{case}{p-nom-acc-dat-gen}; this is consistent with
\savm{case}{acc-dat} and \emph{Frauen} is therefore an acceptable
object for \emph{findet und hilft}.

\subsection{Coordination of Unlikes}

Recall that the data in (\ref{gcu-3}) -- (\ref{gcu-4}) (repeated
below) show that the \textsc{case} value of the coordinate noun phrase
\emph{M\"anner und Kindern} cannot be either \emph{dat} or \emph{acc}
(or any subtype thereof), or else it would be selectable by either
\emph{findet} or \emph{hilft}.

\begin{exe}
  \exr{gcu-3}[*]{\gll Er findet M\"anner und Kindern.\\
    he finds men.\textsc{acc} and children.\textsc{dat}\\}
  \exr{gcu-4}[*]{\gll Er hilft M\"anner und Kindern.\\
    he helps men.\textsc{acc} and children.\textsc{dat}\\}
\end{exe}

Instead, it must be a type more specific than \emph{case} but less
specific than \emph{acc} or \emph{dat}: a type that in effect reifies
$\acc\lor\dat$. I therefore introduce a new type \emph{p-nom+acc} into
the hierarchy, as shown in (\ref{nachier}).

\begin{exe}
\ex[]{\label{nachier}\begin{tabular}[t]{l}
\xymatrix@-1pc{
&&\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]\txt{nom+acc}\\
&\ar@{-}[dl]\ar@{-}[dr]\txt{acc}&{\txt{p-nom+acc}}%
&\ar@{-}[dl]\ar@{-}[dr]\txt{nom}\\
{\txt{p-acc}}&&{\txt{p-nom-acc}}&&{\txt{p-nom}}\\}
\end{tabular}}
\end{exe}

Just as the hyphen mnemonically represents neutrality, the plus sign
indicates coordination. With this addition, the placement of the pure
types has become completely predictable: no pure type dominates any
other type, and every pure type is either equal to its corresponding
non-pure type or is immediately dominated by it. Thus we can
considerably simplify these diagrams by leaving out the pure types;
their presence will be implicit from now on. Under this convention,
(\ref{nachier}) appears as in (\ref{nacnphier}).

\begin{exe}
\ex[]{\label{nacnphier}\begin{tabular}[t]{l}
\xymatrix@-1pc{
&&\ar@{-}[dl]\ar@{-}[dr]\txt{nom+acc}\\
&\ar@{-}[dr]\txt{acc}&&\ar@{-}[dl]\txt{nom}\\
&&{\txt{nom-acc}}\\}
\end{tabular}}
\end{exe}

To more fully illustrate the range of possibilities, it will be
helpful to consider a three-valued case system. The corresponding
simplified diagram for such a system appears as (\ref{natnphier});
note that in any expression containing both $\mathord-$ and
$\mathord+$ signs, the $\mathord-$ takes scope over the $\mathord+$.

\begin{exe}
\ex[]{\label{natnphier}\begin{tabular}[t]{l}
\xymatrix@-1pc{
&&\ar@{-}[lld]\ar@{-}[d]\ar@{-}[drr]\txt{a+n+g}\\
\ar@{-}[d]\ar@{-}[drr]\txt{a+n}%
&&\ar@{-}[lld]\ar@{-}[drr]\txt{a+g}%
&&\ar@{-}[lld]\ar@{-}[d]\txt{n+g}\\
\ar@{-}[d]\ar@{-}[drrrrr]\txt{a+g-n}%
&&\ar@{-}[drrr]\ar@{-}[d]\txt{n+a-g}%
&&\ar@{-}[d]\ar@{-}[dr]\txt{g+n-a}\\ \ar@{-}[d]\txt{a}&&%
\ar@{-}[d]\txt{n}&&%
\ar@{-}[d]\txt{g}&%
\ar@{-}[dlllll]\ar@{-}[dlll]\ar@{-}[dl]\txt{a-n+a-g+n-g}\\
\ar@{-}[d]\ar@{-}[drr]\txt{a-n+a-g}%
&&\ar@{-}[dll]\ar@{-}[drr]\txt{a-n+n-g}%
&&\ar@{-}[dll]\ar@{-}[d]\txt{a-g+n-g}\\
\ar@{-}[drr]\txt{a-n}&&\ar@{-}[d]\txt{a-g}&&\ar@{-}[dll]\txt{n-g}\\
&&{\txt{a-n-g}}\\}
\end{tabular}}
\end{exe}

In one sense, these names are purely iconic: none of the constraints
presented in this paper would fail if we replaced type names like
\emph{a-n} and \emph{a+n} with types like \emph{banana} and
\emph{pineapple}. On the other hand, if we want to talk about how
these hierarchies are developed, it helps to consider the types as
members of the free bounded distributive lattice over the basic cases
(here, \emph{a}, \emph{n}, and \emph{g}) with operators $\mathord+$
and $\mathord-$ serving as meet and join. Thus the lack of a row of
types with the form \emph{(a+g)-n} is deliberate: by distribution,
such a type would be equivalent to \emph{a-n+g-n}.

The hierarchy in (\ref{natnphier}) represents that of a fully
neutralizing language: every possible neutralization of the basic
cases has an instantiation. Not all natural languages are fully
neutralizing, however. For example, \citet{dyla84} shows that in
Polish, there are forms neutral between accusative and genitive case
exist, but there are no forms neutral between nominative and
accusative case.

A sample hierarchy for a three-case language where this particular
combination of cases (nominative and accusative) fails to neutralize
is given in (\ref{napnphier}).

\begin{exe}
\ex[]{\label{napnphier}\begin{tabular}[t]{l}
\xymatrix@-1pc{
&&\ar@{-}[lld]\ar@{-}[d]\ar@{-}[drr]\txt{a+n+g}\\
\ar@{-}[d]\ar@{-}[drr]\txt{a+n}%
&&\ar@{-}[lld]\ar@{-}[ddrr]\txt{a+g}%
&&\ar@{-}[lld]\ar@{-}[dd]\txt{n+g}\\
\ar@{-}[d]\ar@/_10pt/@{-}[ddrrrr]\txt{a+g-n}%
&&\ar@{-}[ddrr]\ar@{-}[d]\txt{n+a-g}\\
\ar@{-}[ddrr]\txt{a}&&%
\ar@{-}[drr]\txt{n}&&%
\ar@{-}[d]\txt{g}\\
&&&&\ar@{-}[dll]\ar@{-}[d]\txt{a-g+n-g}\\
&&\ar@{-}[d]\txt{a-g}&&\ar@{-}[dll]\txt{n-g}\\
&&{\txt{a-n-g}}\\}
\end{tabular}}
\end{exe}

This is a sublattice of (\ref{natnphier}) in which \emph{a-n} and all
coordinations involving \emph{a-n} have been removed.

The constraint that relates the \textsc{case} value of a coordination
to the \textsc{case} value of its conjuncts can now be stated. Since
the conjuncts, being phrases, are linguistic objects, each has a pure
\textsc{case} value. Let the \textsc{case} values of the conjuncts be
$x$ and $y$, and let the corresponding non-pure types be $x'$ and
$y'$. Finally, let the \textsc{case} value of the coordination be $z$
with corresponding non-pure type $z'$. Then it must be the case that
$z'$ is the lowest type such that $z'$ is identical to or a supertype
of $x'$ and $y'$\footnote{In formal terms, $z'$ is constrained to be
  the least upper bound of $x'$ and $y'$ with respect to the subtype
  relation.}. The practical effect of this is that a coordination is
at most as valent as its least-valent component. For example, if $x$
is neutral between cases A, B, and C, while $y$ is neutral between B,
C, and D, then the coordination of $x$ and $y$ is only neutral between
B and C.

This principle allows the analysis of all of the phenomena presented
so far. Envision a hierarchy similar to (\ref{natnphier}) in which all
the \emph{n}s have been changed to \emph{d}s and take this as an
approximation of German. Then the result of coordinating \emph{p-acc}
and \emph{p-acc-dat} is \emph{p-acc}. To see this, take $x
=\emph{p-acc}$ and $y = \emph{p-acc-dat}$. Then $x'=\acc$ and
$y'=\emph{acc-dat}$. Find \emph{acc} and \emph{acc-dat} in
(\ref{natnphier}) and notice that \emph{acc} is a supertype of
\emph{acc-dat}. Thus $z'=\acc$ and $z=\emph{p-acc}$. Therefore, the
coordination of accusative \emph{M\"anner} and syncretic \emph{Frauen}
is assigned \savm{case}{p-acc}. As such, this coordination can occur
anywhere a simple accusative NP can occur but nowhere that requires a
dative NP.

Similarly, as neither \emph{acc} nor \emph{dat} are supertypes of the
other, the coordination of accusative \emph{M\"anner} and dative
\emph{Kindern} receives \savm{case}{p-acc+dat}. Here, $x =
\emph{p-acc}$ and $y=\emph{p-dat}$, so $x'=\emph{acc}$ and
$y'=\emph{dat}$. Then $z'=\emph{acc+dat}$ and $z=\emph{p-acc+dat}$.
This type cannot satisfy either of the constraints \savm{case}{acc} or
\savm{case}{dat}, as desired.

We can now return to the Russian data in (\ref{ru-1}) (repeated
below), which illustrate selectional neutrality.

\begin{exe}
  \exr{ru-1}[]{\gll V\v cera ves' den' on o\v{}zidal svoju
    podrugu Irinu i zvonka ot svoego brata Grigorija\\
    yesterday all day he expected self's.\textsc{acc}
    girlfriend.\textsc{acc} Irina.\textsc{acc} and call.\textsc{gen}
    from self's.\textsc{gen} brother.\textsc{gen} Gregory.\textsc{gen}\\
    \mytrans{`Yesterday he waited all day [for his girlfriend Irina]
      and [for a call from his brother Gregory].'}}
\end{exe}

Here, the coordination \emph{podrugu i zvonka} receives
\savm{case}{p-gen+acc}, and as such it can be selected by the
constraint \savm{case}{gen+acc} (which must therefore be part of the
valence specification of \emph{o\v{}zidal}).

With the contrasting case of selectional ambiguity (recall
(\ref{leh-3}), shown below), the verb \emph{lehren} must be analyzed
as having \savm{comps}{\<\[case & $acc\lor\dat$\]\>} (rather than
\savm{comps}{\<\[case & acc+dat\]\>}). Thus the coordination
\emph{M\"anner und Kindern}, with \savm{case}{acc+dat}, is not an
acceptable object for this verb.

\begin{exe}
\exr{leh-3}[*]{\gll Er lehrt M\"anner und Kindern.\\
 he teaches men.\textsc{acc} and children.\textsc{dat}\\}
\end{exe}

The acceptability of (\ref{ecu-2}) (repeated below) can now be
explained.

\begin{exe}
\exr{ecu-2}[]{He is [a republican] and [proud of it].}
\end{exe}

Assume that the features appropriate for a type \emph{p-a+b} are
exactly those appropriate for both \emph{a} and \emph{b}. Since the
feature \textsc{pred} is appropriate for both adjectives like
\emph{proud of it} and nouns like \emph{a republican}, it is
appropriate for the coordination type \emph{p-noun+adj}. Furthermore,
since both \emph{proud of it} and \emph{a republican} are
\savm{pred}{$\mathord+$}, the coordination is also
\savm{pred}{$\mathord+$}. The copula \emph{is} merely constrains its
argument to be \savm{pred}{$\mathord+$}, so the sentence is licensed.

To take another example, the English verb \emph{become} is like
\emph{be} in that its arguments can be of varying categories; unlike
\emph{be}, its arguments cannot be prepositional or verbal. Under this
system, \emph{become} would subcategorize for \savm{head}{noun+adj};
just as with \emph{be}, the coordination \emph{a republican and proud
  of it} meets this constraint.

For any given feature, the relationship between its value for the
conjuncts and its value for the coordination depends on the feature in
question. As presented above, \textsc{case} in English and German
illustrates one type of combination -- a join on the corresponding
non-pure types. For valence requirements, such as those from conjoined
verbs as in (\ref{first}), the value of the mother's feature is the
unification of the daughter features' values. Among other agreement
features, like \textsc{person}, \textsc{number}, and \textsc{gender},
more elaborate constraints might relate mother values to daughter
values; see \citep{sag85} for a discussion of these features in English
conjoined NPs and \citep{corbett83} for a cross-linguistic discussion
of such relationships.

\subsection{Neutrality Across Features}\label{sec4d}

We now return to the last set of data presented in section
\ref{sec2a}. As given in section \ref{sec4b}, the analysis of feature
neutralization only holds when values of a single feature are being
neutralized. The phenomenon of neutralization across features must
still be accounted for.

Recent proposals \cite[for example,][]{kathol99} represent
morphosyntactic agreement information in HPSG as a bundle of features
(\textsc{case}, \textsc{person}, \textsc{number}, and
\textsc{gender}). If the values appropriate for these features were
augmented as above, adding new types to represent neutralizations,
there would still be no way to replace a constraint like
(\savm{case}{$\neg$nom} $\land$ \savm{num}{sg}) $\lor$
(\savm{num}{pl}) in the same way that \savm{case}{nom} $\lor$
\savm{case}{acc} was replaced with\hfil\break \savm{case}{p-nom-acc}.

For illustration, consider a variation on the Russian verb in
(\ref{ru-1}) that may take either \textsc{nom.sg} or \textsc{acc.pl}
objects (but not \textsc{nom.pl} or \textsc{acc.sg}). That verb cannot
be selecting \savm{case}{p-nom-acc}, as that would improperly exclude
non-neutral forms. Neither can it select for
\savm{case}{$\nom\lor\acc$}, as discussed in section \ref{sec2a}.
There needs to be a way to account for this interdependence.

In the mathematical tradition of reducing new problems to ones already
solved, multiple inheritance can reduce the problem of neutrality
across features to that of single feature neutrality. Just as phrases
are classified along dimensions of clausality and headedness in
\citep{sag97}, morphosyntactic information can be represented in terms
of one feature \textsc{uniagr} and one type \emph{uniagr} which is
cross-classified in terms of person, number, gender, and case.

\begin{exe}\ex[]{\label{agrhier}
    \begin{psTree}{\TR{uniagr}}
      \begin{psTree}{\TR{\psframebox{\textsc{gender}}}}
        \TR{masc}\TR{fem}\TR{neut}
      \end{psTree}
      \begin{psTree}{\TR{\psframebox{\textsc{person}}}}
        \TR{1}\TR{2}\TR{3}
      \end{psTree}
      \begin{psTree}{\TR{\psframebox{\textsc{number}}}}
        \TR{sg}\TR{pl}
      \end{psTree}
      \begin{psTree}{\TR{\psframebox{\textsc{case}}}}
        \TR{nom}\TR{acc}\TR{dat}\TR{gen}
      \end{psTree}
    \end{psTree}}
\end{exe}

Each value of \emph{uniagr} is an element of the cross-product over
the possible values of each of the component features. (For example,
one such value would be \emph{masc.3.sg.acc}, corresponding to the
German pronoun \emph{ihn}.)

Only those feature combinations actually attested on some lexical item
are part of the type hierarchy. Just as \emph{nom} and \emph{acc} form
the basis of the case hierarchy for English in (\ref{nahier}), so
could these values of \textsc{uniagr} form the basis of a unified
agreement hierarchy for German.

Every constraint expressable in the old system has an equivalent
constraint in this system. The full hierarchy of these types is too
large to provide in full, so I will use some naming conventions: each
type component will consist of four fields separated by dots. Within
each field, the initial letter\slash{}digit of the analogous type in
(\ref{agrhier}) will indicate that dimension of the \textsc{uniagr}
value. For example, the constraint (\savm{gend}{masc} $\land$
\savm{per}{3rd} $\land$ \savm{num}{sg} $\land$ \savm{case}{dat}) is
equivalent to \savm{uniagr}{m.3.s.d} under this system. I will omit
the dots whenever a field has just one member, so that the
aforementioned type would appear as \emph{m3sd}.

For notating types that are neutral between several of these values, I
will put both letters in a single field: thus \emph{m3s.da} is a type
neutral between \emph{m3sd} and \emph{m3sa}. If more than one field
has multiple letters, then that type is neutral between the
cross-product of the variants: that is, \emph{mf.3.sp.n} is neutral
between the four types \emph{m3sn, m3pn, f3sn}, and \emph{f3pn}.

In general, by removing any character (except the last in a field),
one moves from subtype to immediate supertype. Thus \emph{mf.3.sp.n}
is an immediate subtype of \emph{m3.sp.n}, \emph{f3.sp.n},
\emph{mf.3sn}, and \emph{mf.3pn}.

For neutralizations where such a full product is not appropriate, I
will continue to use the dash notation: \emph{m3sd-f3sa} is a type
neutral between \emph{m3sd} and \emph{f3sa} that (unlike
\emph{mf.3s.da}) excludes \emph{f3sd} and \emph{m3sa}. As before,
removing a dashed component yields an immediate supertype of the type
in question: thus \emph{m3sd} is an immediate supertype of
\emph{m3sd-f3sa}.

Finally, the omission of a field abbreviates the disjunction of all
possible values for that field. So \emph{msn} abbreviates (\emph{m1sn}
$\lor$ \emph{m2sn} $\lor$ \emph{m3sn}). This is different from
\emph{m.123.sn}, which is neutral between all three persons.

These rules form the names of the non-pure types; the corresponding
pure typename is again formed by prefixing with \emph{p-}.

The acceptability of (\ref{3-a}), repeated below, can now be accounted
for.

\begin{exe}
  \exr{3-a}[]{\gll der Antrag des oder der Dozenten\\
    the petition of-the.\textsc{gen.sg} or of-the.\textsc{gen.pl}
    docent.\textsc{masc.gen.(sg/pl)}\\
    \mytrans{the petition of the docent(s)}}
\end{exe}

Using the naming conventions just described, we can write the lexical
entries in (\ref{le1}) -- (\ref{le3}).

\begin{exe}
\ex[]{\label{le1}\begin{avm}\[\textsc{phon} & der \\ \textsc{uniagr} &
\emph{p-m.123.s.n-f.123.s.dg-mfn.123.p.g} \\ \textsc{val} &
\<\[\textsc{uniagr} & \emph{msn} $\lor$ \emph{fsd} $\lor$ \emph{fsg}
$\lor$ \emph{pg}\]\>\]\end{avm}}
\ex[]{\label{le2}\begin{avm}\[\textsc{phon} & des \\ \textsc{uniagr} &
\emph{p-mn.123.s.g} \\ \textsc{val} & \<\[\textsc{uniagr} & \emph{msg}
$\lor$ \emph{nsg}\]\>\]\end{avm}}
\ex[]{\label{le3}\begin{avm}\[\textsc{phon} & Dozenten \\ \textsc{uniagr} &
\emph{p-mfn.123.s.adg-mfn.123.p.nadg}\]\end{avm}}
\end{exe}

This encodes the fact that the German article \emph{der} is neutral
between \textsc{masc.nom.sg}, \textsc{fem.dat.sg},
\textsc{fem.gen.sg}, and \textsc{gen.pl}; the article \emph{des},
between \textsc{masc.gen.sg}{} and \textsc{neut.gen.sg}; and the noun
\emph{Dozenten}, between \textsc{masc.gen}, \textsc{masc.acc},
\textsc{masc.dat}, and \textsc{masc.nom.pl}.

Then if we assume as before that coordination involves the unification
of the individual conjuncts' agreement constraints, the coordination
\emph{des oder der} will be looking for a noun consistent with
\savm{uniagr}{(\emph{msg} $\lor$ \emph{nsg}) $\land$ (\emph{msn}
  $\lor$ \emph{fsd} $\lor$ \emph{fsg} $\lor$ \emph{pg})}.

The type \emph{p-mfn.123.s.adg-mfn.123.p.nadg} is a subtype of
\emph{msg}, so the first conjunct is satisfied, and it is also a
subtype of \emph{fsd}, so the second conjunct is satisfied. Hence
\emph{Dozenten} is compatible with the selectional requirements of
\emph{des oder der} and (\ref{3-a}) is acceptable.

One side effect of this approach is that it becomes less
straightforward to account for phenomena in which elements are
constrained to agree in one aspect of \textsc{uniagr} but not the
others. For example, in German there are examples where two phrases
must have the same case but may vary in person, number, and\slash{}or
gender (see \citep{mueller01} for one example of this). Since it is
not possible to structure-share part of a type, such a relationship
cannot be directly notated.

Given the existence of relational constraints, however, one could
define a \textsf{same-case} relation such that
\textsf{same-case}(\avmbox{1}, \avmbox{2}) holds exactly when
\avmbox{1} and \avmbox{2} are of types whose \textsc{case} dimensions
are the same. Similar relations could be defined as needed for
\textsc{number}, \textsc{gender}, and \textsc{person}. This may have
an intuitive feel of ``one step forward, two steps back'' (as
relations are now needed to provide what once came ``for free'') but
it does account for a wider range of the data than any standard theory
of agreement. At any rate, more work is certainly needed in this area.

\section{Levy 2001}

Having developed a theory of feature neutrality and unlike
coordination for HPSG, it is now possible to compare it to the other
extant HPSG account of these phenomena, namely \citep{levy01}.

As mentioned above, Levy uses sets of sets to model neutrality and
coordination; the internal sets represent coordination and the
external set represents neutrality. In particular, a neutral form like
\emph{Frauen} is assigned \savm{case}{\{\,\{acc\}\,,\,\{dat\}\,\}\,},
while the coordination of an accusative NP and a dative NP would be
given \savm{case}{\{\,\{acc, dat\}\,\}}.

As in \citep{dalrymple00}, verbs impose a lower-bound constraint of
the form $\hbox{\savm{case}{\avmbox1}} \land \hbox{\avmbox1}
\trianglerighteq \{\{\dat\}\}$ (where the $\mathord\trianglerighteq$
symbol can be thought of as ``at least'').

The central difference, then, is that Levy does not modify the type
hierarchy in any way other than to change the appropriateness
conditions on \textsc{case}: instead of taking values of type
\emph{case}, it now takes values of type \emph{set(set(B))}, where $B$
is the set of basic cases for the language. Levy thus avoids having
both pure and non-pure types. Then, since \textsc{case} values are no
longer ordered with respect to the type hierarchy, the bounded nature
of subcategorization must be made explicit with the additional
$\mathord\trianglerighteq$ constraint. (In the proposal given in
section \ref{sec4}, subcategorization is effected directly through
supertypes, which implicitly represent the disjunction of all
maximally-specific types they dominate.)

When given a fully-neutralizing language, Levy's double-set lattice
and the type hierarchy proposed in section \ref{sec4d} are isomorphic;
this is shown in \citep{levy-pollard02}. For a partially-neutralizing
language, however, the analysis in section \ref{sec4} can model this
partiality directly, as exemplified by (\ref{napnphier}) as compared
to (\ref{natnphier}); Levy's analysis must treat all non-neutralizing
combinations of values as accidental gaps.

\section{Conclusion}

The constraints that relate a mother's feature values to those of its
daughters are quite complex, and this paper only begins to explore the
nature of these relationships. By examining \textsc{case} as both a
valence feature (as is relevant to the analysis of feature neutrality)
and as an inherent morphosyntactic property (as relevant to the
analysis of unlike coordination), I hope to have made some progress in
this area.

In general, Ingria's assessment of unification-based theories of
grammar does not apply to constraint-based theories like HPSG: the
type hierarchy can directly represent indeterminate feature values as
well as coordinations of unlike categories, resulting in a theory of
feature neutralization and unlike coordination that does not need to
add anything new to the HPSG framework.

\bibliography{hpsg}
\bibliographystyle{forum}

\end{document}