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\begin{document}
% \vspace*{1in}
\begin{center}
{\large \bf Blocking}\\%[1in]
Ted Briscoe \& Ann Copestake\\
Computer Laboratory, University of Cambridge, Pembroke Street,
Cambridge, CB2 3QG, UK\\
ejb / aac @cl.cam.ac.uk\\[2mm]
Alex Lascarides\\
Human Communication Research Centre, University of Edinburgh\\
2 Buccleuch Place, Edinburgh, EH8 9LW, UK\\
alex@cogsci.ed.ac.uk
\end{center}

\section{Introduction}

A major motivation for the introduction of default inheritance
mechanisms into theories of lexical organisation has been to account
for the prevalence of the family of phenomena variously described as
blocking (Aronoff, 1976:43), the elsewhere condition (Kiparsky, 1973),
or preemption by synonymy (Clark \& Clark, 1979:798).  In Copestake \&
Briscoe (1991) we argued that productive processes of sense extension
also undergo the same process, suggesting that an integrated account
of lexical semantic and morphological processes must allow for
blocking.  In this paper, we review extant accounts which follow from
theories of lexical organisation based on default inheritance, such as
Paradigmatic Morphology (Calder, 1989), {\sc datr} (Evans \& Gazdar,
1989), {\sc elu} (Russell et al., 1991, in press), Word Grammar
(Hudson, 1990; Fraser \& Hudson, 1992), or the {\sc lkb} (Copestake
1992; this volume; Copestake et al., in press).  We argue that these
theories fail to capture the full complexity of even the simplest
cases of blocking and sketch a more adequate framework, based on a
nonmonotonic logic that incorporates more powerful mechanisms for
resolving conflict among defeasible knowledge resources (Commonsense
Entailment, Asher \& Morreau, 1991).  Finally, we explore the
similarities and differences between various phenomena which have been
intuitively felt to be cases of blocking within this formal framework,
and discuss the manner in which such processes might interact with
more general interpretative strategies during language comprehension.
Our presentation is necessarily brief and rather informal; we are
primarily concerned to point out the potential advantages using a more
expressive default logic for remedying some of the inadequacies of
current theories of lexical description.

\section{Data}

The type of phenomenon which is typically discussed under the rubric
of blocking involves inflectional morphological irregularities; for
example, the past participle of {\it walk} is {\it walked} and we can
describe this regular morphological operation (roughly) in terms of a
rule of concatenation $\langle $stem$\rangle
${\it+ed}\footnote{Throughout this paper we will ignore complications
of morphographemics and morphophonemics.  For proposals on how such
complications can be dealt with within the accounts of the lexicon we
consider see e.g. Calder (1989), Cahill (1990), Russell et al. (1991)
or Bird (1992).}.  However, this rule should not be applied either to
non-verbal stems (e.g. {\it beast, *beasted}) or to the subset of
verbal stems which exhibit irregular behaviour (e.g.  {\it sink, sunk,
*sinked; bring, brought, *bringed; make, made, *maked}). These
observations are captured in default inheritance based approaches to
the lexicon by specifying that the rule in question applies to the
subclass of verb stems, but allowing subclasses of this class to
override its application (where in the limit a subclass may consist of
a specific word). Thus, the rule is interpreted as a default within a
class and more specific subclasses can prevent its application via
stipulative statements. In the case where a rule does not apply to a
subclass or single member of the class over which it is defined the
rule is said to be blocked. There are cases though, where both a
morphologically irregular and regular form coexist, so that blocking
cannot be treated as an absolute property of lexical organisation
(e.g. {\it dream, dreamt, dreamed; burn, burnt, burned}).  Proposals
for accounting for such exceptions to blocking have not treated
blocking itself as a default, but have either, in effect, denied that
any such general principle is at work in the lexicon by stipulating
its effect on a word-by-word basis (Fraser \& Hudson, 1992) or treated
those verbs which allow both irregular and irregular forms as a
further subclass of `dual-class' verbs which specify an irregular
variant but also inherit the regular pattern (Russell et al., 1991).
Even in the clear cases of blocking the productive forms surface as
common errors in the language of first and second language learners
and, though we may not be happy to accept {\it sleeped} as a word of
English, we are able to interpret utterances such as {\it I sleeped
well} with considerably more ease than many other forms of putative
ungrammaticality.

Blocking is not restricted to inflectional morphology, but appears to
be pervasive throughout the lexicon: we can see its effects in
derivational morphology, conversion and metonymic processes of sense
extension. It appears to account for some of the semi-productivity of
most synchronic, generative lexical rules (see Bauer, 1983:84f for a
discussion of other factors restricting productivity). However,
blocking rarely if ever appears to be an absolute constraint on word
(sense) formation.  For example, there is a derivational rule of noun
formation from adjectives by suffixation of {\it +ity} (e.g. {\it
curious, curiosity; grammatical, grammaticality}). This rule is
generally blocked by the existence of a non-derived synonymous noun
(e.g. {\it glorious, glory, *gloriousity; tropical, tropic(s),
*tropicality}). The similar rule of noun formation from adjectives by
suffixation of {\it +ness} is apparently not blocked or not blocked to
the same extent under these circumstances (e.g. {\it ?curiousness,
?grammaticalness, ?gloriousness, ?tropicalness}). Nevertheless, there
is a sense of markedness or awkwardness about such examples which is
not present with forms which do not undergo noun formation with {\it
+ity} or compete with a non-derived synonymous form (e.g. {\it
awkwardness, markedness, weakness, kindness}). Aronoff (1976) argues
that noun formation with {\it +ness}, in contrast to {\it +ity}, is
fully productive, hence the lack of blocking with {\it +ness} forms.
This account does not address the difference in acceptability between
those {\it +ness} forms which are preempted by a synonymous underived
form or one formed with {\it +ity}.

In general, even clearly productive derivational rules can be,
at least partially, blocked. Rappaport \& Levin (1990) argue
convincingly that noun formation via suffixation with {\it +er} on
verbs is highly productive and creates a meaning change where the
derived noun denotes the class of objects which can serve as the
verb's subject (or external argument in their terminology). Thus {\it
teacher} denotes the agent of {\it teach}, whilst {\it opener} can
denote the agent or more usually instrument of {\it open}. Bolinger
(1975:109) cites {\it stealer} as an example of a form blocked by {\it
thief}. In fact, {\it stealer} is fairly clearly blocked in contexts
where its meaning is synonymous with {\it thief}, but may be used in
contexts where its productively derived meaning has been specialised
or modified; for example, Bauer (1983:87-8) points out that {\it
stealer} can be used metonymically as in Shakespeare's {\it the ten
stealers} (fingers) or when the denotation is restricted via mention
of the object (e.g. {\it stealer of hearts / fast sports cars}). 

Nominalisation with {\it +er} also illustrates another apparent type
of blocking which cannot be described as preemption by synonymy but
which can be characterised as preemption by equality at some level of
representation. Rappaport \& Levin (1990) point out that the class of
words which allow middle formation (e.g. {\it John stuck the poster
(to the wall), The poster stuck easily (to the wall)}) tend to form
nominalisations with {\it +er} which denote stereotypical objects,
rather than subjects (e.g. {\it sticker, broiler,
bestseller})\footnote{Whether these cases can be subsumed under a
general rule based on subject / external argument nominalisation is
controversial, but it seems clear that this (sub-)pattern is less
productive in terms of the restrictions imposed on the denotation of
the derived nominal.}.  It is difficult for these nominalisations to
also denote subjects and, where this can be forced, the meaning is
inherently specialised; for example, {\it sticker} can be used to
refer to a person who is determined, as in {\it He is a sticker when
it comes to crosswords}, but is very odd in {\it ?He works as a
sticker of posters}.  Productive nominalisation with {\it +er} can
also be blocked by the existence of a non-productive sense of the
derived form; for example, {\it stationer} means a person who sells
stationary and not a person who stations, say, troops, and {\it banker
(at Barclays)} refers to a person who works in a (senior) position in
a bank, and not to one who banks his money there. In these cases, what
is blocked is generation of an orthographically or phonologically
identical word form with the productive meaning associated with the
relevant derivational rule.

In addition to rules of derivation there are similarly productive
processes of conversion and sense extension which also appear to be
subject to blocking.  In Briscoe \& Copestake (1991) and Copestake \&
Briscoe (1991) we argued in detail that derivation, coercion and sense
extension should be treated within the same general framework.
Conversion processes are often highly productive as the host of
noun-verb conversions such as {\it hoover} illustrate.  However,
derived forms such as {\it arrival} do not undergo this process
because of the existence of {\it arrive}. Briscoe \& Copestake (1991),
Copestake \& Briscoe (1991) and Ostler \& Atkins (1991) argue that
metonymic and metaphorical processes of sense extension must be
treated as lexical rules which can undergo blocking; for example, the
productive sense extension of `grinding' (see e.g.  Pelletier \&
Schubert, 1986), whereby a count noun denoting an individuated object
becomes a mass noun denoting a substance derived from that object,
when applied to animals typically denotes the meat or edible flesh of
that animal (e.g. {\it lamb, haddock, rabbit}).  However, this process
is blocked by the existence of a synonymous form (e.g. {\it pork, pig;
beef, cow}). In these cases, blocking does not prevent the more
general process of grinding but makes the typical interpretation
highly marked.  

It seems that blocking is pervasive, or at least that use of the term
in the literature is pervasive. However, all the cases we have
considered appear to share some properties: blocking itself is rarely
if ever absolute, and blocking is preemption of rule application by
equality at some level of lexical representation. In what follows, we
explore the hypothesis that a uniform account of blocking can be
provided which subsumes both the morphological and lexical semantic
cases.

\section{Inadequacies of Current Approaches}

Extant accounts of blocking in recent theories of lexical organisation
based on default inheritance have mostly treated blocking as an
absolute principle of lexical organisation.  That is, if some
derivational behaviour is given for a class, specifying some distinct
behaviour for a subclass {\em always} has the effect of preventing the
more general behaviour from applying.  The sole exception that we are
aware of is the theory of Word Grammar (Hudson, 1990; Fraser \&
Hudson, 1992) which treats blocking as the exception rather than the
rule.  Thus, in Word Grammar, by default, the lexical rules specified
for a class apply to all subclasses, even those that have been stated
to have exceptional behaviour --- the more specific information
augments, rather than overrides, the more general information.  This
approach would be natural if it were the case that the lexicon
appeared to tolerate the rule-governed production of synonymous or
homophonous words, but the cases considered in \S2 suggest that this
is not so.  As a result, it is necessary to stipulate that blocking
does occur with forms such as {\it sleeped}, whilst cases like {\it
dreamed} or {\it curiousness} are predicted to be the norm rather than
the exception.  At the very least, this obliges the lexicographer to
include many idiosyncratic exception statements in a lexical
description (e.g.  Cahill, 1992). In addition, there are theoretical
reasons for believing blocking to be a general principle of lexical
organisation: a lexicon which did not impose some such constraint
would be in danger of becoming dysfunctional with language change and
development, resulting in too many distinct word forms conveying the
same meaning or too many distinct meanings associated with homophonous
forms.

All other theories of lexical organisation which account for blocking
impose it as an absolute principle.  We will illustrate some
approaches to blocking within default inheritance based accounts by
discussing how inflectional morphology may be described within the
{\sc lkb} system and compare this to some other formalisms,
concentrating on those which are similar to the {\sc lkb}, in that
lexical information is represented as feature structures ({\sc fs}s).
The {\sc lkb}, in effect, incorporates two inheritance mechanisms ---
the type system, based on Carpenter (1992), enforces constraints on
{\sc fs}s which cannot be overridden, and psorts provide a default
inheritance mechanism.  Psorts are named {\sc fs}s which are used in
the {\sc lkb} description language in a manner similar to the use of
templates in {\sc patr}-II.  Psorts may be specifically defined, or be
ordinary lexical entries or rules, since these are also represented as
{\sc fs}s.  In the {\sc lkb}, psorts may be combined using either
ordinary unification or default unification, the latter giving default
inheritance (Copestake et al., in press).  In this case the {\sc fs}
description may specify information which conflicts with that of the
psort and which will override it.

For ease of comparison with other untyped systems, the examples which
follow are based on the use of inheritance via the psort mechanism,
rather than the type mechanism.  We will simply assume a most general
type {\bf lex-sign} which will constrain all the {\sc fs}s
representing lexical signs to have the three features {\sc orth}, {\sc
syn} and {\sc sem}.  Thus, if we specify the hierarchy of psorts
illustrated in Figure~\ref{psort-hier}, {\tt Past-Verb} and {\tt
Irreg-Past-Verb} introduce incompatible values for the feature {\sc
suffix} so that {\sc fs}s for verbs in the latter class will not be
subsumed by those for {\tt Past-Verb}.\footnote{For consistency we use
a notation based on Blackburn (1992a,b) to encode constraints on {\sc
fs}s.  We discuss this further in \S3.1}
\begin{figure}
\thicklines
\setlength{\unitlength}{1.5cm}
\begin{picture}(5,3)(0,1)
\put(1,4){\makebox(0,0){{\bf LEX\_SIGN}:}} 
\put(2,4){\makebox(0,0)[l]
{$\langle $ORTH$\rangle $ (orth) $\wedge$ $\langle $SYN$\rangle $(syn)
$\wedge$ $\langle $SEM$\rangle $(sem)}}
\put(1,3.8){\line(0, -1){0.6}}
\put(1,3){\makebox(0,0){Verb:}}
\put(2,3){\makebox(0,0)[l] 
{$\langle $SYN$\rangle $$\langle $CAT$\rangle $(v) $\wedge$ $\langle
$SYN$\rangle $$\langle $VFORM$\rangle $(base)}}
\put(1,2.8){\line(0, -1){0.6}}
\put(1,2){\makebox(0,0){Past-Verb:}} 
\put(2,2){\makebox(0,0)[l] 
{$\langle $SYN$\rangle $$\langle $VFORM$\rangle $(pastprt) $\wedge$
$\langle $ORTH$\rangle $$\langle $SUFFIX$\rangle $(+ed)}}
\put(1,1.8){\line(0, -1){0.6}}
\put(1,1){\makebox(0,0){Irreg-Past-Verb:}}
\put(2,1){\makebox(0,0)[l] 
{$\langle $ORTH$\rangle $$\langle $SUFFIX$\rangle $(+t)}}
\end{picture}\\[5mm]
\begin{tabular}{ll}
walk: & Verb\\
walked: & Past-Verb\\
slept: & Irreg-Past-Verb\\
dreamed: & Past-Verb\\
dreamt: & Irreg-Past-Verb\\ 
\end{tabular}
\caption{{\sc lkb} psort hierarchy and lexical entries}
\label{psort-hier}
\end{figure}

There are clearly some infelicities in this approach, since we are
forced to associate one and only one type of {\sc fs} with each type
or psort and therefore cannot capture the morphological relationships
between inflectional variants of a verb.  For this reason, the {\sc
lkb} includes a notion of lexical rule which can encode essentially
arbitrary mappings between {\sc fs}s, and it would be more natural to
treat past participle formation as a rule of this type.  One possible
treatment is shown in Figure~\ref{LKBrule}.  Here the inflectional
variants are generated by rule application but we need to introduce a
new feature {\sc morph} on lexical signs, in order to specify the
appropriate irregular forms.  We also must ensure that only the
correct rules apply and this is done by typing the stem values as
either {\bf regular} or {\bf irregular}.  {\tt Verb}, by default,
specifies that the value of $\langle${\sc orth}$\rangle \langle${\sc
stem}$\rangle$ has to be a subtype of {\bf regular}, but this is
overridden by the psort {\tt Irregbase}.  

A lexical rule of past participle formation, shown in
Figure~\ref{LKBrule}, maps regular base verbs to regular past
participle forms.  In the {\sc lkb}, lexical rules are themselves {\sc
fs}s, and thus can be treated as psorts and inherit information from
each other, thus the lexical rule for irregular suffixation default
inherits from the standard rule.  A lexical rule is applicable to a
{\sc fs} if its input half, that is the section of the lexical rule
found by following the path $\langle$1$\rangle$, is unifiable with
that {\sc fs}.  The result of lexical rule application is indicated by
the path $\langle$0$\rangle$.  In Figure~\ref{LKBrule}, we assume that
unspecified parts of the {\sc fs} are the same in the input and
output.  Thus the effect of the rule Past\_formation is to add the
suffix {\it ed} and to change the value of $\langle${\sc
syn}$\rangle\langle${\sc vform}$\rangle$.  The lexical entry for {\it
walk} simply inherits from {\tt Verb} and {\it sleep} inherits from
{\tt Irregbase}.  The regular lexical rule applies to {\it walk} but
the irregular one does not, because the types {\bf regular} and {\bf
irregular} are disjoint.  Note that we have to explicitly specify the
stem of {\it walk} as {\bf regular} to avoid the incorrect lexical
rule applying.  In order to allow the regular past participle form of
{\it dream} to be produced, we have to ensure that both lexical rules
can apply, and we do this by specifying that the value for
$\langle${\sc orth}$\rangle\langle${\sc stem}$\rangle$ for the psort
{\tt Dualpast} is the type {\bf reg/irreg} which subsumes both {\bf
regular} and {\bf irreg}.  Thus {\it dream} has both an irregular
suffix and a regular one.\footnote{The value of $\langle${\sc
orth}$\rangle \langle${\sc stem}$\rangle$ has actually to be
constrained to be equal to {\bf reg/irreg} rather than just subsumed
by it, in order that {\tt Dualpast} can inherit from {\tt Irregstem},
without $\langle${\sc orth}$\rangle \langle${\sc stem}$\rangle$ being
specified to be {\bf irreg}.  The {\sc lkb} incorporates a notion of
an equality or mutual subsumption specification as well as ordinary
value specification, which allows this to work.}
\begin{figure}
\setlength{\unitlength}{1.5cm}
\begin{picture}(5,3)(0,1)
\thicklines
\put(1,4){\makebox(0,0){{\bf LEX\_SIGN}:}} 
\put(2,4){\makebox(0,0)[l]
{$\langle $ORTH$\rangle $ (orth) $\wedge$ $\langle $SYN$\rangle $(syn)
$\wedge$ $\langle $SEM$\rangle $(sem)}}
\put(1,3.8){\line(0, -1){0.6}}
\put(1,3){\makebox(0,0){Verb:}}
\put(2,3){\makebox(0,0)[l] 
{$\langle $SYN$\rangle $$\langle $CAT$\rangle $(v) $\wedge$ $\langle
$SYN$\rangle $$\langle $VFORM$\rangle $(base) 
$\wedge$ $\langle $ORTH$\rangle $$\langle $STEM$\rangle $(regular)}}
\put(1,2.8){\line(0, -1){0.6}}
\put(1,2){\makebox(0,0){Irregbase:}} 
\put(2,2){\makebox(0,0)[l] 
{$\langle $MORPH$\rangle $$\langle $PASTPRT$\rangle $(+t) $\wedge$
$\langle $ORTH$\rangle $$\langle $STEM$\rangle $(irregular)}}
\put(1,1.8){\line(0, -1){0.6}}
\put(1,1){\makebox(0,0){Dualpast:}} 
\put(2,1){\makebox(0,0)[l] 
{$\langle $ORTH$\rangle $$\langle $STEM$\rangle $(reg/irreg)}}
\end{picture}\\[5mm]
\begin{picture}(5,2.4)(0,0.7)
\thicklines
\put(1,2.5){\makebox(0,0){Past\_formation:}}
\put(2.5,2.5){\makebox(0,0)[l]
{[ 1 : [ $\langle $SYN$\rangle $$\langle $VFORM$\rangle $(base) $\wedge$
$\langle $ORTH$\rangle $$\langle $STEM$\rangle $(regular)]}}
\put(2.5,2.2){\makebox(0,0)[l]
{\hspace*{2.5mm}0 : [ $\langle $SYN$\rangle $$\langle $VFORM$\rangle
$(pastprt) $\wedge$
$\langle $ORTH$\rangle $$\langle $SUFFIX$\rangle $(+ed)]]}}
\put(1,2.3){\line(0, -1){1.1}}
\put(1,1){\makebox(0,0){Irreg\_suffixation:}}
\put(2.5,1){\makebox(0,0)[l]
{[ 1 : [ $\langle $MORPH$\rangle $$\langle $PASTPRT$\rangle $($\alpha$)
$\wedge$ $\langle $ORTH$\rangle $$\langle $STEM$\rangle $(irregular)]}}
\put(2.5,0.7){\makebox(0,0)[l]
{\hspace*{2.5mm}0 : [ $\langle $ORTH$\rangle $$\langle $SUFFIX$\rangle
$($\alpha$)]]}}
\end{picture}\\[5mm]
\begin{tabular}{ll}
walk: & Verb\\
sleep: & Irregbase\\
dream: & Dualpast 
\end{tabular}
\caption{{\sc lkb} lexical rules ($\alpha$ is a variable
indicating reentrancy)}
\label{LKBrule}
\end{figure}

Although this approach is better than the first, it is still far from
perfect.  The lexical rule is not directly blocked by the existence of
a `competing' form, instead we have associated the existence of the
irregular form with a class which also specifies that the stem is
irregular, and it is this specification that blocks the application of
the lexical rule.  This, in turn, is overridden for dual class verbs
such as {\it dream}.  Thus we have made blocking non-absolute at the
cost of increasing the complexity of the representation.  A third
option is to give the suffixes themselves lexical entries which are
combined with the base forms in more or less the same way as word
forms are combined in the grammar.  This approach however suffers from
much the same problem with respect to blocking.

In contrast to the {\sc lkb}, {\sc elu} and {\sc datr} represent
inflectional paradigms by inheritance, without introducing the concept
of a lexical rule.  {\sc elu} makes use of a variant set mechanism
which causes a verbal class or paradigm to generate separate {\sc fs}s
for each inflectional variant.  Constraints on {\sc fs}s are specified
as path equations and hierarchical class membership is used to enforce
default inheritance so that blocking is absolute whenever a subclass
introduces a constraint which clashes with that introduced by a
superclass.  Therefore, Russell et al. (1991) introduce a separate
class of dual past verbs and treat {\it dream} as a member of both
this and the regular class.  A slight variant of their analysis is
reproduced in Figure~\ref{ELU}.\footnote{We have modified their
notation and analysis to make it more consistent with Blackburn (1992a,b)
and the {\sc lkb} descriptions above.} Vertical bars separating
constraints indicate {\sc elu} variant sets, that is, disjunctive
constraints on (distinct) {\sc fs}s.
%
\begin{figure}
\begin{tabular}{ll}
Verb:  &  $\langle $SYN$\rangle $$\langle $CAT$\rangle $(v) $\wedge$
$\langle $MORPH$\rangle $$\langle $PASTPRT$\rangle $(+ed) \\
       & $|$  \\                  
       & $\langle $VFORM$\rangle $(base)\\ 
       & $|$  \\                  
       & $\langle $VFORM$\rangle $(pastprt) $\wedge$ $\langle
$ORTH$\rangle $$\langle $SUFFIX$\rangle $($\alpha$) $\wedge$ 
$\langle $MORPH$\rangle $$\langle $PASTPRT$\rangle $($\alpha$)\\[0.2in]

Dual-past: & $|$ \\
           & $\langle $VFORM$\rangle $(pastprt) $\wedge$ $\langle
$ORTH$\rangle $$\langle $SUFFIX$\rangle $(+t) \\
           & $|$ \\[0.2in]                    

walk: & Verb\\[0.2in]

sleep: & Verb\\
       & $\langle $MORPH$\rangle $$\langle $PASTPRT$\rangle $(+t)\\[0.2in]  

dream: & Dual-past Verb
\end{tabular}
\caption{{\sc elu}-like fragment}
\label{ELU}
\end{figure}
%
The empty variant set in {\tt Dual-past} allows {\tt dream} to inherit
all the information from {\tt Verb} as well as the variant specified
by {\tt Dual-past}.  The order of inheritance in this example is
dependent on textual order --- that is {\tt dream} inherits first from
{\tt Dual-past} and then from {\tt Verb}.  An equivalent approach
would be to make {\tt Dual-past} inherit from {\tt Verb}.  This would
be slightly more elegant, because it would remove the need to specify
that the lexical entries inherited from two classes, and avoid the
dependency on textual order.  However, in both cases the same form of
redundancy is present in the analysis as in the {\sc lkb} description
considered above: the irregular past specification has been made in
several parts of the lexical description in order to circumvent the
absolute effects of blocking.

{\sc datr} (Evans \& Gazdar, 1989) also employs a notion of
hierarchical class membership to specify lexical entries and to
control default inheritance.  However, unlike the {\sc lkb} and {\sc
elu}, {\sc datr} is a language which is intended to be specific to
lexical representation and to be usable by any grammar which can be
encoded in terms of attributes and values.  {\sc datr} does not
specify or manipulate {\sc fs}s directly --- {\sc datr} queries result
in the values of paths at particular nodes being returned.  There are
essentially two components --- a monotonic component which explicitly
specified values for paths and a nonmonotonic operation of path
closure which can intuitively be thought of as `filling in the gaps'
to give values for all node/path pairs not defined in the theory.
However, despite the differences between {\sc datr} and the other
theories, similar redundancy arises as in the analyses sketched above
when we attempt to account in a straightforward way for alternative
forms such as {\it dreamed} and {\it dreamt}, since the specification
of a value for a node blocks any alternative inherited value.

To summarise, the inelegancy and redundancy in these descriptions
arises from the fact that each is forced to treat blocking as an
absolute constraint enforced via default inheritance.  The undesirable
effects of this are circumvented by introducing additional classes
which are otherwise unmotivated.  Less restricted approaches such as
those of Daelemans (1987) or de Smedt (1984) might, in principle, be
able to use additional object-oriented techniques to achieve a less
redundant lexical description, by allowing the option of accumulating
rather than overriding inherited information, for example. However,
this is difficult to evaluate in view of the open-ended nature of
these techniques.  Furthermore, it is far from clear that any
inheritance based treatment of blocking can allow for the examples
discussed in the previous section where the lexical item responsible
for the blocking of the productive process is morphologically
unrelated (e.g.  {\it stealer}/{\it thief}, {\it pig}/{\it pork}).
This would imply that we would have to structure the lexicon so that,
for example, {\it thief} was related to {\it steal} by some form of
the process of {\it +er} nominalisation.  Clearly there is some
semantic relationship, but requiring that a relationship between the
lexical items as a whole be specified directly as though it were a
variant of the productive process is very unappealing.

\section{A New Framework}

In this section, we outline a novel approach to lexical description in
which we utilise a default logic as a constraint language on possible
lexical entries specified as {\sc fs}s.  (For a detailed description
of constraint-based languages in linguistic description, see Shieber
(1992).)  Within this framework we believe that it is possible to
characterise blocking more satisfactorily.  The default logic that we
utilise is the propositional variant of Commonsense Entailment ({\sc
ce}, Asher \& Morreau, 1991). This logic is appropriate because we can
encode {\sc fs}s using nominal modal propositional logic (Blackburn,
1992a,b); {\sc ce} supports the patterns of default reasoning
required to characterise blocking; and the propositional variant is
decidable which is one crucial (and minimal) requirement for a theory
of lexical description.

\subsection{A Language for describing FSs}


Blackburn (1992a,b) demonstrates that attribute value structures
expressed as directed graphs are, in effect, languages of
propositional modal logic in which attributes are modal operators and
values are propositional variables.  Furthermore, he proves that the
subset of attribute value structures that can be expressed as directed
acyclic graphs ({\sc dag}s) incorporating re-entrancy are
characterised by the nominal propositional modal logic ({\sc npl}).
This is a modal logic augmented with an atomic propositional variable
sort called {\it nominals}, where a nominal is constrained to be true
on a unique node.  Blackburn (1992b) discusses why modal logics are
particularly natural ways of describing FSs.  In effect, the standard
notations for describing FSs, such as AVMs, assume an internal view:
than describing what happens at arbitrary nodes, the graph structure
is described in terms of transitions from some particular node.  Thus
these notations are equivalent to a language where all quantifiers are
bounded and the richness of the variable binding machinery of the
first order languages is unnecessary.  Modal operators and AVMs both
precisely capture this restriction.  Blackburn's results extend those
of Kaspar \& Rounds (1990) and others concerning the logical
characterisation of unification-based formalisms utilising {\sc dag}s
(e.g.  Shieber, 1986).

In what follows, we develop a theory in which lexical signs are
encoded as {\sc fs}s; that is {\sc dag}s incorporating re-entrancy.
And we utilise a notational variant of {\sc npl} to describe {\sc
fs}s. An example of a lexical entry represented as a {\sc fs} in {\sc
avm} notation is given in Figure \ref{FSsleep}. We will mostly refer
to such {\sc fs}s as lexical signs and assume that all such signs are
minimally specified for {\sc orth}, {\sc syn} and {\sc sem} (e.g.
Pollard \& Sag, 1987).

\begin{figure}
\begin{center}
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
                         \attvaltyp{STEM}{sleep}}}\\
         \attval{SYN}{\avmplus{
                  \attvaltyp{CAT}{verb}\\
                  \attval{SUBJ}{\avmplus{
                           \attval{SYN}{\avmplus{
                                         \attvaltyp{CAT}{np}}}\\
                           \attval{SEM}{\ind{i}}\ \ \myvaluebold{obj}}}\\
                  \attvaltyp{VFORM}{base}}}\\
         \attval{SEM}{\avmplus{
                  \attvaltyp{PRED}{sleep}\\
                  \attval{ARG1}{\ind{i}}}}}$
}\\[1mm]
\caption{{\sc fs} for {\it sleep}}
\label{FSsleep}
\end{center}
\end{figure}

Descriptions in {\sc npl} can be partial in the sense that more than
one {\sc fs} may satisfy a set of {\sc npl} formulas; we will use a
sorted version of {\sc npl}, in order to achieve the required
fine-grainedness of descriptions, in which a partial specificity
ordering induces a lattice over propositional variables from which we
can derive a subsumption relation (Mellish, 1988). The minimal model
which satisfies a lexical description will correspond to the set of
well-formed {\sc fs}s or lexical signs according to that description.
The sortal axioms in \exnum{+1}{} define some useful sorts and values for
verbal features.
\begin{ex}
\begin{subex}
$vform \equiv base \sqcup finite \sqcup infinite \sqcup pastprt \sqcup pass
\sqcup presprt$
\end{subex}
\begin{subex}
$suffix \equiv +ed \sqcup +t \sqcup ...$
\end{subex}
\begin{subex}
$stem \equiv vstem \sqcup nstem \sqcup ...$
\end{subex}
\begin{subex}
$vstem \equiv walk \sqcup dream \sqcup sleep \sqcup ...$
\end{subex}
\begin{subex}
$irregvstem \equiv dream \sqcup sleep \sqcup ...$
\end{subex}
\begin{subex}
$dualvstem \equiv dream \sqcup ...$
\end{subex}
\begin{subex}
$cat \equiv verb \sqcup noun \sqcup ...$
\end{subex}
\end{ex}
A partial description of the class of {\sc fs}s that we have been assuming
for verbs is given in \exnum{+1}{}.
\begin{ex}
$\langle ${\sc syn}$\rangle (\langle ${\sc cat}$\rangle $(verb)
$\wedge$ 
$\langle ${\sc vform}$\rangle $(vform)) $\wedge$ $\langle ${\sc
orth}$\rangle \langle ${\sc stem}$\rangle $(vstem)
\end{ex}
Modal operators representing attributes are written in upper case
between angle brackets, propositional variables are lower case, and
round brackets, delimiting the scope of operators, are mostly
suppressed where this is unambiguous.  This description is satisfied
by any of the {\sc fs}s containing values for {\sc cat}, {\sc vform} and
{\sc stem} subsumed by the propositional variables `verb', `vform' and
`vstem', such as that in Figure \ref{FSsleep}.  We can define the closure of a
lexical description as the set of {\sc fs}s which satisfy all theorems of
that description. An algorithm for computing (membership of) this set
can be defined in terms of substitutions of atomic propositional
variables (such as `verb' in the sort system defined in \exnum{-1}{})
guided by the subsumption relation.  Whether the closure of a lexical
description remains finite will depend, firstly, on the finiteness of
the sort system and, secondly, on the manner in which we restrict the
inference mechanisms available via the constraint language.

In general, to properly constrain the set of lexical signs compatible
with a description, it will be necessary to specify re-entrancy or
structure sharing within and between lexical signs. Blackburn (1992a)
uses nominals, denoted by alphabetical indices, to describe re-entrant
{\sc fs}s.  Thus, the {\sc fs} in Figure \ref{FSsleep} specifies that
{\sc arg1} of the predicate `sleep' is the same as the value of {\sc
sem} under {\sc subj}. A constraint enforcing this equivalence is
given in \exnum{+1}{}.
\begin{ex}
$\langle ${\sc sem}$\rangle \langle ${\sc arg1}$\rangle $(i) $\wedge$
$\langle\mbox{{\sc syn}}\rangle\langle ${\sc subj}$\rangle \langle${\sc
sem}$\rangle $(i)
\end{ex}

We will use {\sc npl} to model the lexicon as a whole, including the
relationships between individual lexical signs.  We will employ a
notion of lexical rule similar to that presented in Pollard \& Sag
(1987:209f) in which such rules are treated as conditional assertions
relating one (basic) lexical sign to another (derived) lexical sign.
(This notion of lexical rule is essentially equivalent in expressive
power to that utilised in the {\sc lkb}, allowing arbitrary
relationships between {\sc fs}s to be specified.)  We represent this
in terms of a structure where derived signs `fan out' from the basic
signs, using a conventionally-named set of modal operators, LR1, LR2
\ldots LRn, in order to denote the transitions described by the
different lexical rules.  In {\sc npl}, we can express the fact that
conjunctive constraints must hold for a single lexical sign by naming
the superordinate node in the {\sc dag} corresponding to the
appropriate {\sc fs}.  For example, \exnum{+1}{} illustrates a simple
lexical rule for past participle formation expressed in {\sc npl}:
here $i$, $j$ and $k$ are metavariables ranging over nominals.  Models
corresponding to descriptions which include lexical rules may include
more than one {\sc fs} to which the rule has been applied.
\begin{ex}
$(i \implies (\langle \mbox{{\sc sem}}\rangle(\mbox{j}) \wedge
\langle\mbox{{\sc orth}}\rangle \langle\mbox{{\sc
stem}}\rangle(\mbox{k}) 
\wedge
\langle\mbox{{\sc syn}}\rangle \langle\mbox{{\sc
vform}}\rangle(\mbox{base})))$\\[1mm]
$\implies$\\[1mm]
$(i \implies (\langle \mbox{{\sc LR1}}\rangle(\langle\mbox{{\sc
syn}}\rangle \langle\mbox{{\sc vform}}\rangle(\mbox{pastprt})\wedge
\langle\mbox{{\sc sem}}\rangle(\mbox{j})\wedge
\langle\mbox{{\sc orth}}\rangle(\langle\mbox{{\sc
stem}}\rangle(\mbox{k})\wedge\langle\mbox{{\sc suffix}}\rangle(\mbox{+ed})))))$
\end{ex}
In this framework, we are treating lexical rules as constraints on
connected FSs containing one basic and further derived lexical entries
connected by a distinguished attribute naming the relevant rule.  This
notation is verbose, so we will represent lexical rules using more
familiar {\sc avm} notation in which boxed values translate as
nominals and standard {\sc avm} formatting conventions are used to
indicate the scope of modal operators, conjunction, and so forth
(Blackburn, 1992a).  Rather than explicitly indicating the conventional
modal operators, LR1 etc, we will use the syntax $\implies_{LR1}$ etc
to indicate that the implied feature structure is connected by an LR
operator.  We will also omit the explicit specification of the
outermost nominal ($i$) in the AVM diagrams; thus
\exnum{0}{} will become \exnum{+1}{}.
\begin{ex}
\begin{center}
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
                  \attval {STEM} {\ind{k}}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{base}}}\\
          \attval{SEM}{\ind{j}}}$                  
}
$\implies_{LR1}$
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
          \attval {STEM} {\ind{k}}\\
          \attval {SUFFIX} {+ed}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{pastprt}}}\\
          \attval{SEM}{\ind{j}}}$
}
\end{center}
\end{ex}

\subsection{Default Logic as a Constraint Language}

The lexical rule of past participle formation in \exnum{0}{} is far
too strong since it does not hold of all (English) verbs and would
therefore produce many linguistically unmotivated {\sc fs}s. The
approach taken in theories of lexical description based on default
inheritance is, firstly, to restrict the domain of application of such
a constraint to a subclass of verbs and, secondly, to allow the
constraint to be overridden on stipulated further sub-classes or
cases. We can restrict the application of a rule such as
\exnum{0}{} to an appropriate subclass via the sort system and we can
express its default nature by relaxing $\implies$ to default
implication ($>$), as in \exnum{+1}{} ($\phi > \psi$ is to be read as
{\it If $\phi$ then by default, $\psi$}).
\begin{ex}
\begin{center}
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
                  \attval {STEM} {\ind{k}}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{base}}}\\
          \attval{SEM}{\ind{j}}}$                  
}
$>_{LR1}$
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
          \attval {STEM} {\ind{k}}\\
          \attval {SUFFIX} {+ed}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{pastprt}}}\\
          \attval{SEM}{\ind{j}}}$
}
\end{center}
\end{ex}
Default logics incorporate a rule of defeasible modus ponens (DMP:
$\phi > \psi, \phi\csmodels \psi$, where $\csmodel$ stands for
nonmonotonic validity) which ensures that within the class of verbs
{\it vstem}, \exnum{0}{} will apply unless the result is inconsistent
with the premises. We cannot, though, prevent the application of
\exnum{0}{} to irregular verbs merely by adding the more specific rule
in \exnum{+1}{}.
\begin{ex}
\begin{center}
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
                  \attval{STEM}{irregvstem \wedge \ind{k}}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{base}}}\\
          \attval{SEM}{\ind{j}}}$                  
}
$>_{LR2}$
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
          \attval{STEM}{\ind{k}}\\
          \attval{SUFFIX}{+t}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{VFORM}{pastprt}}}\\
          \attval{SEM}{\ind{j}}}$
}
\end{center}
\end{ex}
This is because as things stand, there is no statement in the premises
asserting that the consequents of \exnum{-1}{} and \exnum{0}{} don't
usually both hold in any given knowledge base ({\sc kb}).  So no
inconsistency arises through the application of DMP to both
\exnum{-1}{} and \exnum{0}{} (which we can represent schematically as
$\phi > \psi, \chi > \zeta, \phi, \chi \csmodels \psi, \zeta$).
Therefore, as the system stands we do not incorporate blocking. And,
in fact, this is the result that we want because, although we
intuitively understand these values for {\sc suffix} to be in
conflict, the existence of forms such as {\it dreamed} and {\it
dreamt} tells us that we do not want this conflict to be as strong as
logical inconsistency. In order to create a conflict, we can include a
further constraint \exnum{+1}{}, which represents the effect of
blocking in irregular past participle formation.
\begin{ex}
$\langle${\sc LR2}$\rangle $($ \langle ${\sc orth}$\rangle $($\langle
${\sc stem}$\rangle$(irregvstem) $\wedge$ $\langle ${\sc suffix}$\rangle $(+t))) $>$
$\neg \langle${\sc LR1}$\rangle $($ \langle${\sc orth}$\rangle 
$($\langle ${\sc stem}$\rangle
$(irregvstem) $\wedge$ $\langle ${\sc suffix}$\rangle $(+ed)))
\end{ex}
\exnum{0}{} captures the intuition that the {\sc suffix}s {\em +t} and
{\em +ed} are usually, but not always, in conflict. The motivation
for including \exnum{0}{} is to override the default application of
\exnum{-2}{} in appropriate circumstances. In fact, this is an
inference that will not follow in all theories of default logic.
However, we postpone detailed discussion of the precise theory of
nonmonotonic reasoning required to validate the desired patterns of
reasoning until \S4.4.  In addition, we want irregular past participle
formation to continue to apply. The (minimal) pattern of reasoning
required is thus schematised in
\exnum{+1}{}.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{ss} \wedge \psi_o$
\end{subex}
\begin{subex}
$\chi_{ls} > \zeta_{ss} \wedge \zeta_o$
\end{subex}
\begin{subex}
$\zeta_o > \neg\psi_o$
\end{subex}
\begin{subex}
$\phi_{ls} \sqsupset \chi_{ls}$
\end{subex}
\begin{subex}
$\chi_{ls}$
\end{subex}
\begin{subex}
$\csmodels \zeta_{ls}$\\
$\notcsmodels \psi_{ls}$
\end{subex}
\end{ex}
Lower case Greek symbols range over lexical signs and subscripts are
used to refer to the values of paths, {\sc syn} and {\sc sem}
($\phi_{ss}$) and {\sc orth} ($\phi_o$) within a derived lexical sign
($\phi_{ls}$).  In this case, \exnum{0}{a,b} represent the rules of
regular and irregular past participle formation respectively.
\exnum{0}{d,e} follow from the sort system, and \exnum{0}{c}
represents the blocking constraint for past participle formation
(i.e., the default rule \exnum{-1}{} above).  However, note that
$>_{LR1}$ is a special case of $>$ so that the schema can be thought
of as generalising over all lexical rules (and other defeasible
conditionals) in the description language.  When we have an irregular
verb ($\chi$) we wish to conclude that we have an irregular derived
past participle ($\zeta$) and to prevent the derivation of a regular
past participle ($\psi$); that is, not infer $\psi$ and optionally
infer $\neg\psi$. \exnum{0}{c} is a specific default rule relevant to
the blocking of regular past formation by irregular past participle
formation. In \S5 we argue that this and many other blocking rules can
be derived from an indefeasible principle of blocking, so it is not
necessary to stipulate all the specific rules required to implement
every individual case of blocking.

\subsection{Unblocking}

The fact that specific blocking rules, such as \exnum{-1}{}, are
defaults can be exploited to develop a more elegant and conceptually
clearer account of `unblocking', where `competing' rules do both fire.
In the case of dual past verbs, such as {\it dream}, given the sort
system defined in \exnum{-4}{} the dual past class of verbs are
subsumed by the class of irregular verbs. Therefore, these verbs will
be treated like {\it sleep} with respect to past participle formation.
In order to unblock regular past participle formation for these verbs
we can override the blocking default by explicitly adding the
assertion that dual past verbs are exceptional in that they tolerate
both regular and irregular past participle forms.  This information is
represented in \exnum{+1}{}.
\begin{ex}
$\langle${\sc LR2}$\rangle $($ \langle ${\sc orth}$\rangle 
$($\langle ${\sc stem}$\rangle
$(dualvstem) $\wedge$ $\langle ${\sc suffix}$\rangle $(+t))) $\implies$
$\langle${\sc LR1}$\rangle $($ \langle${\sc orth}$\rangle 
$($\langle ${\sc suffix}$\rangle $(+ed))))
\end{ex}
By adding \exnum{0}{}, we introduce an indefeasible assertion
concerning dual class verbs' orthographic forms which will override
the blocking default rule of \exnum{-2}{}, since indefeasible
information always has priority over defeasible information (this is
supported in all nonmonotonic logics).  The schematic pattern of
inference involves the addition of the subsumption relation and
\exnum{0}{} to the pattern given in \exnum{-1}{} above, as
\exnum{+1}{} illustrates.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{ss} \wedge \psi_o$
\end{subex}
\begin{subex}
$\chi_{ls} > \zeta_{ss} \wedge \zeta_o$
\end{subex}
\begin{subex}
$\zeta_o > \neg\psi_o$
\end{subex}
\begin{subex}
$\delta_o \wedge \psi_o$
\end{subex}
\begin{subex}
$\phi_{ls} \sqsupset \chi_{ls}$
\end{subex}
\begin{subex}
$\chi_{ls} \sqsupset \delta_{ls}$
\end{subex}
\begin{subex}
$\delta_{ls}$
\end{subex}
\begin{subex}
$\csmodels \zeta_{ls}, \psi_{ls}$
\end{subex}
\end{ex}
The unblocking constraint is represented in \exnum{0}{d} and will
prevent application of the blocking default in \exnum{0}{c}, thus both
\exnum{0}{a} and \exnum{0}{b} will apply to dual class verbs.
So far, we have developed an account in which dual past verbs can be
represented as a subclass of irregular verbs and the regular and
irregular rules of past participle formation need only be stated once.

\subsection{Choosing a Suitable Logic}

We have proposed that to account for blocking, the statements about
conflict among feature values must be defeasible.  This had
ramifications on the inferences we need to validate, and hence on the
kind of logic that will be suitable.  In this section, we introduce a
logic for defeasible reasoning called commonsense entailment (or {\sc
ce}) (Asher and Morreau, 1991), and argue that for our purposes it has
certain features that make it more attractive than other candidate
logics.  In particular, we'll suggest that {\sc ce}'s logical
consequence relation supports the patterns of inference \exnum{-2}{}
and \exnum{0}{} in a straightforward and more elegant way than the
other candidate logics.

The default constraint language that we use is a sorted version of
{\sc npl}, augmented with a nonmonotonic conditional operator $>$;
{\sc ce} supplies a modal semantics for such default formulas.
Intuitively, $\phi > \psi$ is read as {\em If $\phi$ then normally
$\psi$}. The constraint language incorporates two logical consequence
relations.  The first relation is represented as $\models$; it is
monotonic and supra-classical. It applies to the truth conditional
component of the logic, and supports intuitively compelling monotonic
patterns of inference involving defeasible statements, such as
Facticity.
\begin{itemize}
\item	{\tt Facticity}: $\models \phi > \phi$
\end{itemize}
In words, Facticity validates
{\em If $\phi$ then normally $\phi$}.

The second logical consequence relation is nonmonotonic and written as
$\csmodel$.  It underlies a partial dynamic theory of belief, which is
defined on top of the truth conditional semantics for defeasible
statements.  It supports intuitively compelling patterns of defeasible
inference, such as those given below:
\begin{itemize}
\item	{\tt Defeasible Modus Ponens:} $\phi > \psi, \phi \csmodels
\psi$\\
(e.g., birds normally fly, Tweety is a bird $\csmodels$ Tweety flies)
\item	{\tt The Penguin Principle:} $\phi > \psi, \chi > \neg \psi,
\chi \rightarrow \phi, \chi \csmodels \neg\psi$\\
(e.g., birds fly, penguins don't fly, penguins are birds, 
Tweety is a penguin $\csmodels$
Tweety doesn't fly.)
\item	{\tt The Nixon Diamond:}
$\phi > \psi, \chi > \neg \psi,
\phi,\chi \notcsmodels \psi$ (and $\notcsmodels \neg\psi$)\\
(e.g., Quakers are pacificists, Republicans are non-pacifists, Nixon
is a Quaker and Republican $\notcsmodels$ Nixon is a pacifist, and
$\notcsmodels$ Nixon is a non-pacifist.)
\end{itemize}
Defeasible Modus Ponens captures the intuition that one can infer the
consequent of a default rule if it is consistent with what is already
known.  The Penguin Principle contains two conflicting default rules
in the premises, in that the consequents of both laws cannot hold in a
consistent knowledge base ({\sc kb}).  The antecedents of these rules
are logically related because $\chi$ entails $\phi$; in this sense
$\chi > \neg\psi$ is more specific than $\phi > \psi$.  The Penguin
Principle therefore captures the intuition that one prefers specific
defaults to general ones when they conflict.  The Nixon Diamond is
like the Penguin Principle in that two conflicting default laws apply.
However it differs in that these default laws do not stand in a
relation of specificity to each other.  {\sc ce} is a logic where no
conclusion is drawn under these circumstances without further
information.  In other words, it supports the intuition that conflict
among default laws is irresolvable when they're unrelated.

The way {\sc ce} is set up gives at least two advantages over other
logics for defeasible reasoning.  First, it splits the monotonic
component of the theory from the nonmonotonic component, and this can,
on occasion, prove conceptually useful when the premises form complex
logical structures.  Second, unlike Hierarchical Autoepistemic Logic
(Konolige, 1988) and Prioritised Circumscription (McCarthy, 1980;
Lifschitz, 1984), it captures the Penguin Principle without the
intervention of a device which is extraneous to the semantic
machinery.  We find this latter property of {\sc ce} attractive, since
intuitively, whatever kind of reasoning default reasoning is, that
more specific information takes precedence is intrinsic to it and so
should be captured in the semantics of defaults.

We will shortly give a brief description of the way the dynamic
partial theory of belief supports nonmonotonic patterns of inference,
and in particular how it supports the inferences \exnum{-2}{} and
\exnum{0}{}.  But first, we consider the truth conditional component
of the theory.  The semantics of defeasible statements are defined in
terms of a function $\ast$ from worlds and propositions to
propositions.  This function forms part of the model.  And
intuitively, $\ast(w,p)$ is the set of worlds where the proposition
$p$ holds together with everything else which, in world $w$, is
normally the case when $p$ holds.  So $\ast$ encodes assumptions about
normality. The truth conditions of defeasible statements are defined
as follows:
\begin{itemize}
\item	$M,w\models \phi > \psi$ if and only if $\ast(w,\sem{\phi})
\subseteq \sem{\psi}$
\end{itemize}
In words, this says that {\em If $\phi$ then normally $\psi$} is
true with respect to a  model $M$ at a possible world $w$ if the set
of worlds that defines what is normally the case when $\phi$ is true
in $w$ all contain the information that $\psi$ is also true.  Thus
{\sc ce} is a conditional logic in the Acqvist (1972) tradition; it
differs from previous conditional logics in the constraints it puts on
$\ast$.  

There are two constraints: the first is Facticity, which
intuitively states that however few propositions normally hold when
$p$ is the case, one of them is $p$.  
\begin{itemize}
\item	{\tt Facticity:} $\ast(w,p) \subseteq p$
\end{itemize}
This constraint enables the monotonic inference Facticity stated above
to be verified.  The second constraint is Specificity: it is essential
for capturing the Penguin Principle as we will see shortly.  It
expresses the intuition that penguins aren't normal birds, because
penguins don't have the flying property associated with normal birds.
To see this, think of $p$ as {\em Tweety is a penguin}, and $q$ as
{\em Tweety is a bird}; so $*(w,p)$ is {\em Tweety is a normal
penguin} and $*(w,q)$ is {\em Tweety is a normal bird}.
\begin{itemize}
\item	{\tt Specificity:} If $p \subseteq q$ and
$\ast(w,p)\cap\ast(w,q) = \emptyset$, then $\ast(w,q)\cap p =
\emptyset$
\end{itemize}

The dynamic semantics of information states is set up so that
intuitively, Defeasible Modus Ponens goes as follows: first, one
assumes the premises $\phi>\psi$ and $\phi$ and no more than this.
Second, one assumes that $\phi$ is as normal as is consistent with
these premises.  Finally, from these two assumptions, one concludes
that $\psi$.  The theory makes precise the notion of assuming no more
than the premises of an argument (in terms of an update function), and
assuming everything is as normal as is consistent with those premises
(in terms of a normalisation function).  Information states are
considered to be sets of possible worlds.  The update function $+$ is
defined on an information state $s$ and a set of {\sc wffs} $\Gamma$,
and it outputs a new information state that supports everything in the
old information state as well as supporting $\Gamma$.
\begin{itemize}
\item	{\tt Update Function:} $s+\Gamma = \{w\in s: w\models
\Gamma\}$
\end{itemize}
The normalisation function $N$ uses a definition of $\ast$ on information
states (as opposed to possible worlds):
\begin{itemize}
\item {\tt Definition of $\ast$ on information states:} $\ast(s,p) =
\cup_{w\in s}\ast(w,p)$
\end{itemize}
$N$ takes an information state $s$ and a {\sc wff} $\phi$ as input,
and it outputs a new information state $s'$.  Intuitively, $s'$
isolates those worlds in $s$ where either $\phi$ isn't true at all, or
its true but normal, if there are any worlds where $\phi$ is normal;
otherwise it `gives up' and simply returns the original state $s$.
\begin{itemize}
\item	{\tt The Normalisation function:} \\
$
\begin{array}{lll}
N(s,\phi) & = & \{w \in s: w \not\in \sem{\phi}\setminus
\ast(s,\sem{\phi})\}  
\mbox{ if }\ast(s,\sem{\phi}) \cap s \not= \emptyset\\
& = & s\mbox{ otherwise}
\end{array}
$
\end{itemize}
Or to put it another way, if it is consistent in $s$ to assume that
$\phi$ is normal (and therefore by Facticity it is also consistent to
assume $\phi$ is true), then $N(s,\phi)$ eliminates all those worlds
from $s$ where $\phi$ is true but abnormal.  So normalisation is the
process where one revises the information state to include the
assumption that $\phi$ is as normal as possible, given what is already
known.  Pictorially, we can represent the effect of normalisation as
in Figure~5, 
%\ref{N(s,phi)}, 
where the shaded areas define the value of $N(s,\phi)$.
%
%\begin{figure}
%
%{\em Insert Diagram 1 in here}
%
%\caption{The definition of Normalisation}
%\label{N(s,phi)}
%\end{figure}

Suppose one knows the premises $\Gamma$.  Then from the normalisation
function, one can build a normalisation {\em chain} for $\Gamma$.
First, one orders the antecedents of the default rules in $\Gamma$
into a sequence $\mu$.  One then builds the normalisation chain by
repeated applications of $N$: one applies $N$ to the current
information state and the first element in $\mu$; then one applies $N$
to the result of this first application of $N$, and the {\em second}
element in $\mu$; and so on until one reaches a fixed point (such a
point will be reached since $N$ is non-increasing).  In order to see
what nonmonotonically follows from the premises $\Gamma$, one starts
with the information state $\happyface + \Gamma$, where $\happyface$
is the information state containing nothing but the laws of logic, and
one sees what holds in all the fixed points of all the normalisation
chains for all possible enumerations $\mu$.\footnote{
\happyface\ is defined to be the set of worlds in the canonical model,
that's used to prove completeness of $\models$.  Since this
corresponds to knowing nothing but the laws of logic,
\happyface$+\Gamma$ corresponds to knowing nothing but $\Gamma$ and
the laws of logic.}
In propositional {\sc ce}, $\models$ and $\csmodel$ are both sound,
complete and decidable (see Lascarides and Asher, in press).

To clarify this, let's see informally how $\ast$, $+$ and $N$ interact
to ensure that $\csmodel$ supports The Penguin Principle.  Let $\Gamma
= \{\phi>\psi,\chi >
\neg \psi,\chi \rightarrow \phi, \chi\}$ (i.e., the premises of a
Penguin Principle).  Then there are essentially two possible
enumerations $\mu_1$ and $\mu_2$, where in the former we normalise on
$\chi$ first and then on $\phi$, and in the latter we normalise on $\phi$
first and then $\chi$.  Consider the normalisation chain for $\mu_1$.  
We must first calculate the value of the function
$N($\happyface$+\Gamma,\chi)$.  Note that
\happyface$+\Gamma$ contains worlds where $\chi$ is normal, and so the
first clause in the definition of $N$ applies.  Thus, given the truth
definition of $\chi > \neg\psi$, 
$N($\happyface$+\Gamma,\chi) = s$ for some information state
$s$ that is contained in $\sem{\neg\psi}$.  Pictorially, $s$
corresponds to the shaded areas in the diagram in Figure~6. 
%\ref{Penguin1}.
%\begin{figure}
%
%{\em Insert Diagram 2 in here}
%
%\caption{The Penguin Principle: application of $N$ to $\chi$ with
%respect to sequence $\mu_1$}
%\label{Penguin1}
%\end{figure}

Now we must work out the
value $s'$ of $N(s,\phi)$.  Since
$s\subseteq\sem{\neg\psi}$, and by the definition of $\phi>\psi$
$\ast(w,\sem{\phi})\subseteq\sem{\psi}$, there
are no worlds in $s$ where $\phi$ is normal.  This situation is
depicted in Figure~7:
%\ref{Penguin2}:
the second clause in
the definition of $N$ applies.  Therefore $s'=s$, and so it also
supports $\neg\psi$.  
%
%\begin{figure}
%
%{\em Insert Diagram 3 in here}
%
%\caption{The Penguin Principle: application of $N$ to $\phi$ with
%respect to sequence $\mu_1$}
%\label{Penguin2}
%\end{figure}
%
Since $N$ is non-increasing, it should be clear that $\neg\psi$ will
be in the fixed point of this normalisation chain.

Now consider the enumeration
$\mu_2$.  We must first work out the value of
$N($\happyface$+\Gamma,\phi)$.  By the truth conditions of $\phi
>\psi$ and $\chi > \neg\psi$,
$\ast(w,\sem{\phi})\cap\ast(w,\sem{\chi})=\emptyset$.  And
$\chi\rightarrow\phi$ means that $\sem{\chi}\subseteq\sem{\phi}$.  So
the Specificity Constraint on $\ast$ applies here; for all worlds $w$
in $\mbox{\happyface}+\Gamma$, $\ast(w,\sem{\phi})\cap\sem{\chi}=\emptyset$.
But \happyface$+\Gamma\subseteq \sem{\chi}$.  So there are no worlds
in $\mbox{\happyface}+\Gamma$ where $\phi$ is normal.  Therefore
$N(\mbox{\happyface}+\Gamma,\phi) = $\happyface$+\Gamma$.  Now we
calculate the value of $N(\mbox{\happyface}+\Gamma,\chi)$, and as
before, $N($\happyface$+\Gamma,\chi)\subseteq \sem{\neg\psi}$.
So this normalisation chain also supports $\neg\psi$ in its fixed
point.  From this brief sketch, one can intuitively see that
$\csmodel$ supports the Penguin Principle, since $\neg\psi$ holds in
all fixed points of all normalisation chains.  What's more, it has done
so without having to assume a particular order of application of the
default rules.  This is in sharp contrast to
{\sc hael} and Prioritised Circumscription.  In these logics, one must
constrain the reasoning process so that one applies the more specific
default rule first, which then blocks applying the second more general
rule.  

The advantages of having specificity baked into the semantics of
default statements really come into their own when one considers the
pattern of inference in \exnum{+1}{}, which we claim underlies blocking.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{ss} \wedge \psi_o$
\end{subex}
\begin{subex}
$\chi_{ls} > \zeta_{ss} \wedge \zeta_o$
\end{subex}
\begin{subex}
$\zeta_o > \neg\psi_o$
\end{subex}
\begin{subex}
$\phi_{ls} \sqsupset \chi_{ls}$
\end{subex}
\begin{subex}
$\chi_{ls}$
\end{subex}
\begin{subex}
$\csmodels \zeta_{ls}$\\
$\notcsmodels \psi_{ls}$
\end{subex}
\end{ex}
To see this, we will indicate how {\sc ce} supports \exnum{0}{}, and
then briefly consider how {\sc hael} and prioritised circumscription
would deal with it.  Let the premises in \exnum{0}{} be $\Gamma$.
There are three default rules in $\Gamma$, making six possible
enumerations.  First consider the enumeration $\mu_1$ where we
normalise in the order: $\phi_{ls}$, $\chi_{ls}$, $\zeta_o$.  The
Specificity Constraint doesn't apply here, since even though the
premises entail $\sem{\chi_{ls}}\subseteq\sem{\phi_{ls}}$, they don't
allow us to assume
$\ast(w,\sem{\phi_{ls}})\cap\ast(w,\sem{\chi_{ls}})=\emptyset$.  So we
cannot assume $\ast(w,\sem{\phi_{ls}})\cap\sem{\chi_{ls}}=\emptyset$.
Therefore, the first clause in the definition of $N$ applies when
calculating $N(\mbox{\happyface}+\Gamma,\phi_{ls})$, for there are
worlds in
\happyface$+\Gamma$ where $\phi_{ls}$ is normal.  So
$N($\happyface$+\Gamma,\phi_{ls}) = s$ where
$s\subseteq \sem{\psi_{ss}\wedge\psi_o}$.  This normalisation is
depicted in figure~8, 
%\ref{blocking1}, 
where $s$ is characterised by the
shaded areas.

%\begin{figure}
%
%{\em Insert Diagram 4 in here}
%
%\caption{Blocking: Normalising on the general default first}
%\label{blocking1}
%\end{figure}

Now $s$ must contain worlds
where $\chi_{ls}$ is normal, since this is consistent with
what is known, and therefore $N(s,\chi_{ls}) = s'$,
where $s'\subseteq\sem{\zeta_{ss} \wedge \zeta_{o}}$.  Normalisation
on the specific default is given in figure~9: 
%\ref{blocking2}: 
$s'$ is the shaded area.

%\begin{figure}
%
%{\em Insert Diagram 5 in here}
%
%\caption{Blocking: Normalising on the specific default second}
%\label{blocking2}
%\end{figure}

Furthermore $N$ is non-increasing, and so $s'\subseteq s\subseteq\sem{\psi_o}$.
By
the truth conditions of \exnum{0}{c},
$\ast(w,\sem{\zeta_o})\cap\sem{\psi_o} = \emptyset$.  
So there are no
worlds in $s'$ where $\zeta_o$ is normal.  Therefore
$N(s',\zeta_o) = s'$.  Thus the final link in the normalisation chain
is depicted in figure~10. 
%\ref{blocking3}.
%
%\begin{figure}
%
%{\em Insert Diagram 6 in here}
%
%\caption{Blocking: Normalising on the blocking default last}
%\label{blocking3}
%\end{figure}
%
In fact, the fixed point of this
normalisation chain $\mu_1$ contains $\sem{\zeta_{ls}}$ and
$\sem{\psi_{ls}}$. 

Now consider the enumeration $\mu_2$, where we normalise in the order:
$\chi_{ls}$, $\zeta_o$, $\phi_{ls}$.
$N(\mbox{\happyface}+\Gamma,\chi_{ls}) = s$, where
$s\subseteq\sem{\zeta_{ss} \wedge \zeta_{o}}$.  And
$N(s,\zeta_o) = s'$, where $s'\subseteq\sem{\neg\psi_o}$.
By the truth definition of \exnum{0}{a},
$\ast(w,\phi_{ls})\cap\sem{\neg\psi_{ls}}=\emptyset$.  So 
$s'$ doesn't
contain any worlds where $\phi_{ls}$ is normal (because
$s'\subseteq\sem{\neg \psi_o}\subseteq\sem{\neg\psi_{ls}}$).  So
$N(s',\phi_{ls}) = s'$.  In fact, the fixed point of this
normalisation chain $\mu_2$ contains $\sem{\zeta_{ls}}$ and
$\sem{\neg\psi_{ls}}$.  

These normalisation chains show that the following holds, as we
required in \exnum{0}{}:
\begin{itemize}
\item	$\Gamma \notcsmodels \psi_{ls}$\\
$\Gamma \notcsmodels \neg\psi_{ls}$
\end{itemize}
In fact, using a similar line of reasoning, 
the other four normalisation chains lead to fixed points that all 
support $\zeta_{ls}$ (the details are omitted for reasons of
space).  So the following also holds, as we require:
\begin{itemize}
\item	$\Gamma \csmodels \zeta_{ls}$
\end{itemize}
Hence the pattern of inference \exnum{0}{}, which underlies lexical
blocking, is supported in {\sc ce}.  So {\sc ce} provides a suitable
default logic with which to formalise the constraint language.

There are other candidate logics for defeasible reasoning, so why
choose {\sc ce}?  Let's assess how the other logics deal with
\exnum{0}{}.  A few comments about the
logical structure of \exnum{0}{} will help clarify the discussion.
The default rule \exnum{0}{a} is more specific than \exnum{0}{b}
because of \exnum{0}{d}.  Furthermore, because of \exnum{0}{c}, the
consequents of these rules {\em conflict by default}.  The inference
in \exnum{0}{} resolves this conflict in favour of the more specific
rule.  So a logic will be adequate for our purposes only if it has the
means to resolve conflict.  Not all logics for defeasible reasoning
have this facility; for example, Reiter's (1980) default logic,
autoepistemic logic (Moore 1984) and circumscription (McCarthy 1980).
There are at least two logics apart from {\sc ce} that have the
facility to resolve conflict among defeasible rules: {\sc hael}, and
prioritised circumscription.  As we've mentioned, these logics resolve
the conflict by constraining the reasoning processes with machinery
that is extraneous to the semantics of default statements.

{\sc hael} is an extension of autoepistemic logic motivated in part by
the need to validate the Penguin Principle.  An autoepistemic logic is
a theory of {\sc kb} extension: if the consequent of a default law is
consistent with the rest of the contents of the {\sc kb}, then it is
added to that {\sc kb}.  In {\sc hael}, certain {\sc kb} expansions
are preferred on the basis that the defaults used in them have a
higher priority in some well-defined sense than the ones used in the
alternative expansions.  Informally, the contents of a {\sc kb} are
structured into a hierarchy.  The information at each level in the
hierarchy represents different sources of information available to an
agent, while the hierarchy expresses the way in which this information
is combined.  Facts and indefeasible laws are placed at the first
level, and more specific default laws are placed at a lower level in
the hierarchy than less specific default laws; the way the logic works
ensures that information at level $n$ has priority over information at
level $m$ where $n<m$; in other words the effects of default laws at
lower levels in the hierarchy can block the action of more general
defaults, placed higher in the hierarchy.  The only constraint on the
hierarchy is that more specific defaults must be at a lower level.
Since
\exnum{0}{b} and \exnum{0}{c} have unrelated antecedents, they can
appear in any relative order in the hierarchy, even though they
conflict.  This proves problematic, if we wish to support \exnum{0}{}
in {\sc hael}.  For suppose we put
\exnum{0}{b} at a lower level in the hierarchy than \exnum{0}{c}.
This means that in the {\sc kb} extension, \exnum{0}{b} will have
priority over
\exnum{0}{c}.  Hence
$\psi_{ls}$ will follow from the premises \exnum{0}{a-e}, contrary to
our requirements.  To avoid this undesirable result, we would have to
constrain the way we construct the hierarchy, in order to ensure that
\exnum{0}{c} is always placed at a lower level than \exnum{0}{b},
(if they're at the same level, we would in fact still infer
$\psi_{ls}$).  But this has serious ramifications on the translation process of
default statements into the object language.  We would be committed to
constraining the hierarchy in this case so that two defaults with
logically unrelated antecedents appear in a particular order.  But it
is not always the case that we wish to constrain the order of two such
defaults (cf. the premises of a Nixon Diamond).  So we have sacrificed
the means to translate default statements into the logic in a uniform
way.

A similar problem arises in prioritised circumscription, where the
abnormality predicates are ordered into a hierarchy, and one prefers
models that satisfy the premises and which minimise on abnormality
predicates higher in the hierarchy than those lower down.  Again, we
would have to be careful about the order in which we place the
abnormality predicates associated with the default rules
\exnum{0}{b} and \exnum{0}{c}, if the logic is to capture the pattern
of inference we need to do blocking.  Again, we would fail to supply a
uniform translation of default statements into the formal language.
By contrast, our formalisation of blocking in {\sc ce} captures the
pattern of inference we need, while at the same time maintaining a
uniform translation of default statements into the object language.
This is desirable because it allows for a more direct, perspicuous and
elegant encoding of lexical rules and other constraints without the
need to impose any external, linguistically-unmotivated ordering.

\section{Blocking -- A Unified Phenomenon?}

The account we have developed would be unsatisfactory if we were
unable to derive any general principle of blocking, since intuitively
the same principle is involved in many other examples including those
cited in \S2. We argue that specific blocking defaults, such as that
for irregular past verbs, follow from an indefeasible rule which can
express a version of `preemption by synonymy' restricted via the
subsumption relation. This rule enforces the principle that, if there
are two lexical rules which validate derived signs with equivalent
{\sc syn} and {\sc sem} components but distinct {\sc orth} components,
and the antecedents of these rules stand in a subsumption relation,
then a blocking default is validated negating the derived orthography
of the more general rule.  We express the first version of the
blocking principle as a deductive propositional schema; that is, as a
`metalevel' indefeasible implication over lexical descriptions, in
\exnum{+1}{}.\footnote{It's important to stress that \exnum{+1}{} is a
metalevel rule only in the sense that it is a schema; its desired
effects are captured in the logic without have to assume metalevels of
reasoning.}
\begin{ex}
a) $(\phi_{ls} > \psi_{ls}) \wedge (\chi_{ls} > \zeta_{ls}) \wedge$\\
b) $(\chi_{ls} \implies \phi_{ls}) \wedge \neg(\phi_{ls} \implies
\chi_{ls}) \wedge$\\
c) $\psi_{ss} \leftrightarrow \zeta_{ss} \wedge \neg(\psi_{o}
\leftrightarrow \zeta_{o}))\\
\implies (\zeta_o > \neg\psi_o)$
\end{ex}
The schema in \exnum{0}{a} represents any two lexical rules, and
\exnum{0}{b} states that there must be an (indirect) unidirectional 
subsumption relation between the antecedents of these rules
(inferrable from the sort system). \exnum{0}{c} states that these
rules must result in derived lexical signs with equivalent values for
{\sc syn} and {\sc sem} and distinct values for {\sc orth}. If the
conditions expressed in the antecedent of \exnum{0}{} are met the
principle validates the deduction of a specific blocking default
negating the orthography of the more general derived rule.  This
deduction takes place in the monotonic component of the constraint
language (since it is an indefeasible implication).  Hence if
\exnum{-1}{a,b,d} are verified, the default statement \exnum{-1}{c} of
conflict is inferred {\it before} normalisation starts.  And as we saw
in \S4.4, the nonmonotonic component can use this default statement of
conflict to achieve the desired effect of blocking.  Moreover, it's
important to stress that the statement of conflict is a default; so
even if its antecedent is verified by the {\sc kb}, its effects can be
overridden by indefeasible unblocking assertions of the form $\psi_o
\wedge \zeta_o$ (as in the case of the dual class verbs in \S4.3).  

The blocking principle in \exnum{0}{} is adequate to generate the
blocking default required to prevent {\it *sleeped} discussed in \S4.2
and many others. The case of {\it +ness} and {\it +ity}, introduced in
\S2, can be treated in a manner analogous to past participle
formation, since we have two lexical rules which clearly stand in a
subsumption relation and generate differing forms with the same syntax
(and meaning). However, an account of the {\it curious, curiosity;
glorious, glory, *gloriousity} case cannot be obtained for the
blocking principle as stated in \exnum{0}{} because we require the
blocking of a (derivational) morphological rule in the presence of an
underived synonymous but orthographically distinct lexical sign. A
similar situation obtains with the rule of `animal grinding' sense
extension or conversion which produces the meat reading of {\it lamb},
and so forth, but is blocked in the case of {\it pig} by {\it pork}.
In these cases, it appears that we must state that `preemption by
synonymy' also occurs when a non rule-generated synonymous word
exists. We will concentrate on developing an account of the blocking
of animal grinding, since this putative lexical rule has motivated
much of our interest in blocking (Copestake \& Briscoe, 1991). We can
recast the rules of grinding and animal grinding as lexical rules, as
in \exnum{+1}{a,b}.
\begin{ex}
\begin{subex}
\begin{center}
{\tiny $\avmplus{\attval{ORTH} {\ind{j}}\\
\attval{SYN}{\avmplus{ \attval{CAT}{noun}\\ \attval{CNT}{+}}}\\
\attval{SEM}{\avmplus{ \attval{REF}{ind\_obj \wedge \ind{k}}}}}$ }
$\implies_{LR3}$
{\tiny 
$\avmplus{
\attval{ORTH} {\ind{j}}\\
\attval{SYN}{\avmplus{ \attval{CAT}{noun}\\ \attval{CNT}{-}}}\\
\attval{SEM}{\avmplus{ \attval{REF}{substance}\\
\attval{ORIGIN}{\ind{k}}}}}$ }
\end{center}
\end{subex}
\begin{subex}
\begin{center}
{\tiny $\avmplus{\attval{ORTH} {\ind{j}}\\
\attval{SYN}{\avmplus{ \attval{CAT}{noun}\\ \attval{CNT}{+}}}\\
\attval{SEM}{\avmplus{ \attval{REF}{animal \wedge \ind{k}}}}}$ } 
$>_{LR4}$
{\tiny 
$\avmplus{
\attval{ORTH} {\ind{j}}\\
\attval{SYN}{\avmplus{ \attval{CAT}{noun}\\ \attval{CNT}{-}}}\\
\attval{SEM}{\avmplus{ \attval{REF}{nat\_food\_subst}\\
\attval{ORIGIN}{\ind{k}}}}}$ }
\end{center}
\end{subex}
\end{ex}
We assume that the class of `animals' is subsumed by the class of
`ind(ividuated)\_obj(ects)', that \exnum{0}{a} is an indefeasible
rule, and that \exnum{0}{b} applies by default to further specify the
interpretation of grinding applied to animals (see Copestake \&
Briscoe, 1991) for a more detailed justification of this analysis,
explication of {\sc origin}, and so forth). In order to not generate this
more specified meaning for {\it pig}, by default, we must express the
manner in which {\it pork} takes precedence over {\it pig}. As it
stands the blocking principle apparently states that blocking applies
when two lexical rules are involved, but in this case blocking is
caused by the presence of an underived lexical sign (standing in the
same general relationship to the derived sign). But the underived sign
does not stand in a subsumption relation to either sign in the lexical
rule. Intuitively, though the underived sign is more `reliable' than
the derived one because it represents a fact, rather than a
(nonmonotonic) inference. We can incorporate this intuition into the
blocking principle without modification, given that we use {\sc ce},
because the axioms of {\sc ce} in \exnum{+1}{a,b} allow us to make
this pattern of inference parallel to the earlier case.
\begin{ex}
\begin{subex}
$\models \bot \implies \psi$
\end{subex}
\begin{subex}
$\models (\psi \implies \phi) \implies (\psi > \phi)$
\end{subex}
\end{ex}
In \exnum{+1}{} $\chi_{ls}$ represents a basic lexical sign such as
{\it pork} and we use the axioms in \exnum{0}{a,b} to obtain the
formula in \exnum{+1}{b} and \exnum{0}{a} to obtain \exnum{+1}{c}.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{ls}$
\end{subex}
\begin{subex}
$\bot > \chi_{ls}$
\end{subex}
\begin{subex}
$\bot \implies \phi_{ls} \wedge \neg(\phi_{ls} \implies \bot)$
\end{subex}
\begin{subex}
$\psi_{ss} \leftrightarrow \chi_{ss} \wedge \neg(\psi_{o}
\leftrightarrow \chi_{o}))$
\end{subex}
\end{ex}
If \exnum{0}{a} represents animal grinding then application of this
rule to {\it pig} will produce a lexical sign which stands in the
relation to {\it pork} (i.e., $\chi_{ls}$) defined by \exnum{0}{d},
and the formulas of \exnum{0}{} together will instantiate the blocking
principle given in \exnum{-3}{}. Therefore, we will derive the
specific blocking default that we require to prevent animal grinding
validating {\it pig}, meaning meat; schematically, $\chi_o >\neg
\psi_o$; and the application of animal grinding in this case will be
blocked because the premises will be analogous to the case of
irregular past participle formation in \S4.2.  Thus, these axioms
allow us to formalise the similarity between these two examples of
blocking without stating that {\it pork} is subsumed by {\it pig}
(meat), as we would have been forced to do in a purely inheritance
based account.

The type of blocking which is occurring in the case of words such as
{\it sticker} or {\it banker} is apparently rather different to
`preemption by synonymy'. In these cases, a different meaning for an
orthographically identical form is blocked. Ostler \& Atkins (1991)
refer to this as `lexical preemption'. Nevertheless, the pattern of
inference required to block the derived, productive sign resulting
from {\it +er} nominalisation is very similar to the cases of semantic
preemption we have considered. We assume that the non-productive
meaning of forms such as {\it sticker} means that underived lexical
signs are present expressing such senses.  Application of {\it +er}
nominalisation will be blocked by a specific blocking default which
asserts that the `more reliable' monotonically derivable semantics of
the basic sign for {\it sticker} negates the productive default
semantics produced by the rule.  In \exnum{+1}{} we give a
(simplified) rule of {\it +er} nominalisation.
\begin{ex}
\begin{center}
{\tiny
$\avmplus{\attval{ORTH}{\avmplus{
                 \attval{STEM}{\ind{j} \wedge vstem}}}\\
         \attval{SYN}{\avmplus{
                  \attval{CAT}{verb}\\
                  \attval{SUBJ}{\avmplus{
                           \attval{SYN}{\avmplus{
                                         \attval{CAT}{np}}}\\
                           \attval{SEM}{\avmplus{
                                        \attval{REF}{\ind{k}}}}}}\\
                  \attval{VFORM}{base}}}\\
         \attval{SEM}{\avmplus{
                  \attval{PRED}{pred}\\
                  \attval{ARG1}{\ind{k} \wedge anim\_obj}}}}$
}
$>_{LR5}$
{\tiny
$\avmplus{
          \attval{ORTH}{\avmplus{
                  \attval{STEM}{\ind{j}}\\
                  \attval{SUFFIX}{+er}}}\\
          \attval{SYN}{\avmplus{
                  \attval{CAT}{noun}}}\\                  
          \attval{SEM}{\avmplus{
                  \attval{REF}{\ind{k}}}}}$

}
\end{center}
\end{ex}
This rule will produce a meaning for {\it sticker} which can be
glossed as `person who sticks (things)'. We represent the pattern of
inference required to block derivation of this sign in \exnum{+1}{}.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{se} \wedge \psi_{sy} \wedge \psi_o$
\end{subex}
\begin{subex}
$\chi_{se} > \neg\psi_{se}$
\end{subex}
\begin{subex}
$\phi_{ls}$
\end{subex}
\begin{subex}
$\chi_{ls}$
\end{subex}
\begin{subex}
$\notcsmodels \psi_{ls}$
\end{subex}
\end{ex}
This pattern results in $\psi_{ls}$ being blocked, because there is
irresolvable conflict between the default rules \exnum{0}{a} and
\exnum{0}{b}.  In order to derive the blocking default in \exnum{0}{b}
we need to introduce a second related blocking condition which
introduces semantic blocking defaults when the conditions for `lexical
preemption' are satisfied. We give this version of the blocking
principle in \exnum{+1}{}.
\begin{ex}
a) $(\phi_{ls} > \psi_{ls}) \wedge (\chi_{ls} > \zeta_{ls}) \wedge$\\
b) $(\chi_{ls} \implies \phi_{ls}) \wedge \neg(\phi_{ls} \implies
\chi_{ls}) \wedge$\\
c) $\psi_{os} \leftrightarrow \zeta_{os} \wedge \neg(\psi_{se}
\leftrightarrow \zeta_{se}))\\
\implies (\zeta_{se} > \neg\psi_{se})$
\end{ex}
This second principle is symmetric with the first in \exnum{-6}{}
above, but reverses the conditions for preemption. $\psi_{os}$
abbreviates values for {\sc orth} and {\sc syn} on a lexical sign and
$\psi_{se}$ those for {\sc sem}.  As before, \exnum{0}{} will generate
the required blocking default when an underived lexical sign bears the
relation defined in \exnum{0}{c} to a lexical sign derived by default.
So in particular, \exnum{0}{} will generate \exnum{-1}{b}.

It might be thought that the availability of the axioms in
\exnum{-2}{} makes the blocking principles too unconstrained. For
example, what is there to stop us blocking one sense of a homonymous
form such as {\it bank} by applying the axioms of \exnum{-2}{} to two
basic signs which satisfy the conditions of the second principle in
\exnum{0}{}? In fact, condition b) in the antecedent of the
blocking principles prevents two underived forms satisfying them,
despite the availability of axioms which allow any underived sign to
be expressed as the consequent of a default implication.  Condition b)
fails to be satisfied in such a situation because $\neg(\bot \implies
\bot)$ can never be true in any model.

The symmetry between the two versions of the blocking principle,
encoding the conditions of semantic and lexical pre-emption mean that
it is possible to merge the two principles into one schematic blocking
principle, which illustrates more clearly the similarities between the
two subcases. \exnum{+1}{} is the final version of the blocking
principle and replaces both previous versions.
\begin{ex}
a) $(\psi_{ls} > \phi_{ls}) \wedge (\chi_{ls} > \zeta_{ls}) \wedge$\\
b) $(\chi_{ls} \implies \psi_{ls}) \wedge \neg(\psi_{ls} \implies
\chi_{ls}) \wedge$\\
c) $\phi_{syn} \leftrightarrow \zeta_{syn} \wedge$\\
d) $\phi_{x} \leftrightarrow \zeta_{x} \wedge$\\
e) $(\neg(\phi_{y} \leftrightarrow \zeta_{y}))\\
\implies (\zeta_y > \neg\phi_y)$
\end{ex}
Clauses c) -- e) express two subcases: for preemption, syntactic
identity is always required, {\it x} and {\it y} are variables ranging
over {\sc sem} and {\sc orth}. If {\it x} is instantiated to {\sc sem}
and {\it y} to {\sc orth}, we have semantic preemption, otherwise we
have lexical preemption.

We've seen how a general blocking principle can be encoded in our
constraint language.  It is interesting to note that such a principle
wouldn't have the desired effect in {\sc hael}.  This is because in
{\sc hael}, the default information is `drip-fed' into the reasoning
process as one climbs the hierarchy.
\exnum{0}{} will be known from the start (i.e., at level 0)
since it is indefeasible knowledge, but its antecedent won't be
verified until the reasoning process reaches the level where the
general default $\psi_{ls}>\phi_{ls}$ is introduced.  Therefore, given
the way {\sc hael} works, the consequents of both \exnum{0}{} and this
general default will be inferred at the next level in the hierarchy.
Consequently, the blocking default is eventually inferred, but it is
too late, because $\phi_{ls}$ has also been inferred: the blocking
principle has failed to block it.  Here, we see another advantage in
the way {\sc ce} is set up.  In contrast to {\sc hael}, {\em all} the
defeasible information is known from the start of the reasoning
process, and all monotonic inferences are carried out before the
nonmonotonic ones.

\section{(Un)Blocking and Interactions with Interpretation}

A major motivation for developing a default account of blocking was to
provide an account of interactions during interpretation, which would
allow, say, {\it pig} to have the meaning of meat in an appropriate
context; for example, {\it I ate pig last night in this awful
restaurant}.  In the extant theories of lexical organisation which we
have considered in \S3, there is no way of blocking the application of
animal grinding to a form such as {\it pig} deriving a meaning
synonymous with {\it pork}.  In addition, in those theories, if it
were possible to block {\it pig} meaning meat via the existence of the
underived form {\it pork}, then there would be no way of unblocking
it; just as in those theories there is no way of unblocking {\it
dreamed} in a manner which also captures the intuition that {\it
dreamed} is a subclass of the irregular past participle formation
verbs.

A full account of the unblocking of animal grinding is beyond the
scope of this paper.  However, we can illustrate the approach that
might be taken within this framework with respect to the blocked form
{\it *sleeped}.  In a context where a (first or second) language
learner utters, say, {\it I sleeped well}, it seems plausible that
listeners infer an unblocking default `on-line'.  The problem is: why
do we interpret {\it sleeped} as the past participle of {\it sleep} in
this case?  Intuitively, one recognises that the more general default
for constructing past participle verb forms applies.  One can express
this intuition declaratively in {\sc ce} plus {\sc npl}.  Let $\Gamma$
be \exnum{+1}{a-e}; i.e., the premises of the inference that enables
us to infer that the past participle form of {\it sleep} is {\it
slept}.
\begin{ex}
\begin{subex}
$\phi_{ls} > \psi_{ss} \wedge \psi_o$ where 
$\psi_o = i \implies LRn \langle \mbox{{\sc orth}} \rangle (\alpha_o)$
\end{subex}
\begin{subex}
$\chi_{ls} > \zeta_{ss} \wedge \zeta_o$
\end{subex}
\begin{subex}
$\zeta_o > \neg\psi_o$
\end{subex}
\begin{subex}
$\phi_{ls} \sqsupset \chi_{ls}$
\end{subex}
\begin{subex}
$\chi_{ls}$
\end{subex}
\begin{subex}
$\csmodels \zeta_{ls}$\\
$\notcsmodels \psi_{ls}$
\end{subex}
\end{ex}
We formalise the effect of the utterance of {\it sleeped} as adding
$\Diamond\alpha_o$ to these premises.  The possibility operator has
the effect that we are asserting that $\alpha_o$ (i.e., the
orthography ``sleeped'') is true at some node in the feature
structure, since $\Diamond\alpha_o$ means that $\alpha_o$ is true at
some node, but not necessarily the current one.  Then the embedded
default in \exnum{+1}{} captures the intuition that, upon learning
that $\alpha_o$ must hold at some node (through the utterance of {\it
sleeped}), one assumes that by default, the consequent of the more
general default rule holds and thus $\alpha_o$ is true as the value of
the orthography for $\psi_{ls}$.
\begin{ex}
$(\Gamma \wedge \Diamond\alpha_o) > \psi_{ls}$
\end{ex}
So, suppose the {\sc kb} verifies $\Gamma$ and $\Diamond\alpha_o$.  Then
\exnum{0}{} is the most specific default that applies.  So
by the Penguin Principle its consequent is inferred.  This means that
in the case of hearing {\it sleeped}, one infers that it is to be
interpreted as the past participle form of {\it sleep}.  

Our use of a default logic incorporating a dynamic theory of belief
guarantees us intertranslatibility between the constraint language we
use to describe the lexicon and the logic in which we characterise
processes of interpretation (e.g. Lascarides \& Asher, in press). This
enables us to elegantly capture such interactions by seemlessly
extending the lexical {\sc kb} into the more general modular, but
interacting {\sc kb} underlying language interpretation.  In addition,
the fact that we model the inference underlying the interpretation of
{\it sleeped} using an embedded default has ramifications on the kind
of logic that can account for these phenomena.  Embedded defaults are
syntactically ill-formed in {\sc hael} and default logic, but
well-formed in circumscription and {\sc ce}.

\section{Conclusion}

In this paper, we have shown that it is possible to derive a default
account of blocking.  This account is more compatible with linguistic
facts.  By representing conflict among feature values as default
statements, we were able to support inferences that underly
blocking without making blocking an absolute principle of lexical
organisation.   This paves the way for more insightful accounts of 
the status of words, such as {\it pig} meaning meat, {\it
?curiousness},  and so forth.  In this paper, we have sketched
preliminary accounts of a few of these cases in the logic {\sc ce}, 
and argued that this is a more appropriate for this application.  In
addition we have argued that our approach provides a basis for
characterising interactions between lexical and wider interpretative
principles.  

Much work remains to be done: we think that the approach could be
improved by being recast within a typed framework (Carpenter 1992)
which would disallow arbitrary extensions of feature structures.
Within such a treatment the notion of lexical rule could be
constrained: here we have assumed an arbitrary mapping between feature
structures is possible, but we would like to limit this to monotonic
operations plus restricted mappings between particular types in the
hierarchy.  In general the computational complexity and consequent
tractability of this approach needs to be investigated.  We intend to
explore the applicability of this approach to other examples of
blocking; and to examine interactions between lexical blocking and
other knowledge resources used in interpretation.

\section*{References}
\newcommand{\book}[4]{\item #1 (#4) {\it #2,} #3.}
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\newcommand{\forthart}[6]{\item #1 (#6, forthcoming) `#2' in #4 (eds.),
{\it #3,} #5.}
\newcommand{\journart}[6]{\item #1 (#6) `#2', {\it #3,} {\it vol.#4,} #5.}
\newcommand{\procart}[6]{\item #1 (#6) `#2', {\it Proceedings of the
#3,} #4, pp.~#5.}
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to appear in {\it Proceedings of the #3,} #4}
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#3,} #4.}

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    \parsep 0pt plus 1pt}

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\end{list}

\end{document}














