Christian Ebert
Lehrstuhl fur Computerlinguistik
Universitat Heidelberg
Karlstr. 2
D-69117Heidelberg
Marcus Kracht
II. Mathematisches Institut
Freie Universitat Berlin
Arnimallee3
D-14195Berlin
Abstract
In this paper the phenomenom of case
stack-ingis investigatedfrom aformal point of view.
We willdeneaformal languagewithidealized
casemarkingbehaviourand prove thatstacked
cases have the ability to encode structural
in-formationon thewordtherebyallowingfor
un-restricted word order. Furthermore, the case
stackshelptocomputethisstructurewithalow
complexitybound. Asasecondpartwepropose
a compositional semantics for languages with
stacked cases and show how this proposal may
work for our formal language as wellas for an
examplefrom Warlpiri.
1 Introduction
Case stacking is a phenomenomthat occurs in
many Australian languages (such as Warlpiri
and Kayardild) and e.g. Old Georgian. Case
stacking is known to pose problems for the
treatment of case in many formal frameworks
today 1
. In (Nordlinger, 1997) theproblem was
attacked by extending the framework of LFG.
Nordlinger claims that case morphology can
constructgrammaticalrelationsandlarger
syn-tactic contexts.
In Section 2 we will introduce an ideal
lan-guge,whichexhibitsperfectmarking. This
lan-guagecapturesNordlinger'sideaofcaseas
con-structors of grammatical relations, butis
inde-pendent of any syntactic framework. We will
prove ofthislanguagethat thecasestacks
pro-videalltheinformationneededfor
reconstruct-ingthefunctor-argumentrelationsandthe
syn-tactic context a wordappears in. Additionally,
sincestructureliesencodedinthesecase stacks
thereisnoneedto assumeanyphrasestructure
1
see(Nordlinger,1997)and(Malouf,1999)fora
dis-cussiononLFGandHPSGrespectively.
orrestriction onwordorder. At theendof this
sectionweconsiderthecomputational
complex-ity of ourlanguage and draw some conclusions
aboutgrammarformalismsthatareableto
gen-erate it.
In Section 3 we propose a compositional
se-mantics for our case stacking languages.
Un-like Montague semantics, where the
manage-ment of variables is not made explicit, we will
use referent systems to keep track of variables
and to make semantic composition eÆcently
computable 2
. The proposal will be applied to
our formal language and to an example from
Warlpiri.
2 Syntax
In this section a perfectly case marked formal
languagewillbe dened and investigated. The
denitionofthislanguageisbasedonterms
con-sistingoffunctorsandargumentsandthuscases
willbe taken to mark arguments.
In the following we let N denote the set of
non-negative integers and _
the concatenation
ofstrings, whichis oftenomitted. We shalluse
typewriter font to denote true characters in
print fora formallanguage.
2.1 Basic Denitions
Anabstract denitionof termsruns asfollows.
Let F be a set of symbols and : F ! N a
function. The pair hF;i is calleda signature.
We shall often write in place of hF;i . An
element f 2F iscalled a functorand (f) the
arity of f. We let ! := maxf(f) j f 2 Fg
denote the maximal arity. Terms are denoted
herebystringsinPolishNotation,forsimplicity.
Denition1. Let be a signature. A term
over is inductivelydened as follows.
2
2. If (f) > 0 and t
i
; 1 i (f), are
terms, sois ft
1 :::t
(f) .
The set C of case markers will be the set
f1;:::;!g,whichweassumetobedisjointfrom
F. Given a term, each functor will be case
marked according to the argument position it
occupies. This is achieved through the notion
of a unit, which consists of a functor and a
se-quenceof casemarkerscalled case stack.
Denition 2. Let tbea term over a signature
. The corresponding bag, (t), is
induc-tively dened as follows.
1. If t=f, then (t):=ftg.
2. If t = ft
1 :::t
n
, then (t) := ffg [
S
n
i=1 fz
_
ijz2(t
i )g.
An elementf 2(t) is called a unit and 2
C
its case stack.
For example, if f;g, and x are functors of
arity 2;1 and 0, respectively, the bag (fxgx)
is ff;x1;g2;x12g. The meaning of a unit x12
could be described by `x is the functor of the
rst argument of the second argument of the
term'.
Denition 3. Let tbea term over a signature
, (t) the corresponding bag and (t)=fÆ
i j
i ng an arbitrary enumeration of its units.
ThenthestringÆ
1 _
Æ
2 _
::: _
Æ
n 1 _
Æ
n
issaid to
bea (t)-string.
Some of the (fxgx)-strings are e.g.
fx1g2x12andg2x1fx12. We arenowprepared
to dene a formal language over the alphabet
F [C bycollecting all (t)-strings fora given
signature:
Denition 4. Let bea signature. Theideal
case marking language ICML
over this
signatureconsists ofall(t)-stringssuch that
t isa term over .
2.2 Trees and Unique Readability
Thereis a strongcorrespondencebetween bags
andlabelledtreessincecasestackscanbe
iden-tiedwithtree addresses:
Denition 5. A nonempty nite set D N
+
2. If d
1 d
2
2Dthen d
1 2D.
3. If di2D; i2N then dj2D for all j<i.
The elements of a tree domain are called tree
addresses. A -labelled treeis a pair hD;i
such that D is a tree domain and : D ! F
a labelling function such that the number of
daughtersof d2D isexactly ((d)).
To formalize the correspondence we dene a
function T that assigns every bag (t) a
-labelledtree:
T(ff
1
1 ;:::;f
n
n
g):=hf R
i
j1ing;
: R
i 7!f
i
(1in)i
ThefunctionT reversesthecasestacksof all
units to get a set of tree addresses. Then the
functor of the unit is assigned to the tree
ad-dress. E.g. ifthe bag contains a unit g321the
resultingtreedomainwillcontainatreeaddress
123 and the labellingfunction will assign g to
it.
Similarly one can dene an inverse function
assigning a bag to each -labelled tree. Thus
thereisabijectionbetween-labelledtreesand
bags. Therefore dierent bags correspond to
dierentorderedlabelledtrees. Thisshowsthat
we have unique readability for bags and since
every ICML
string can be uniquely
decom-posed into its unitswe maystate thefollowing
proposition.
Proposition 6. Letbeasignature. Then
ev-ery ICML
string isuniquely readable.
2.3 Pumpability and Semilinearity
We will rst consider the property of being
nitelypumpableasdenedin(Groenink,1997).
Denition7. A language L is nitely
pumpable if there is a constant c such that
for any w 2 L with jwj > c, there are a nite
number k and strings u
0 ;::: ;u
k andv
1 ;:::;v
k
such that w=u
0 v
1 u
1 v
2 u
2 v
k u
k
and for each
i;1 jv
i
j < c and for any p 0 the string
u
0 v
p
1 u
1 v
p
2 u
2 v
p
k u
k
belongs to L.
Proposition 8. Let be a signature. Then
ICML
partscannotcontain afunctorsincethatwould
lead to pumped strings containing the same
units more than once. Hence the number of
units cannot be increased by pumping and all
pumpable parts must consist of case markers
solely. ButsincethelengthofanICML
string
consistingofaxednumberofunitsisbounded
eachpumpablestringcouldbepumpedupsuch
thatitexceedsthisbound. ThusICML
isnot
nitelypumpable atall.
Now we areconcernedwith semilinearity.
Denition 9. Let M N n
. Then M is a
1. linear set, if for some k 2 N there are
u
0 ;:::;u
k 2 N
n
, such that M = fu
0 +
P
k
i=1 n
i u
i jn
i 2Ng,
2. semilinear set, if for some k 2 N there
arelinearsetsM
1
;:::;M
k N
n
,suchthat
M = S
k
i=1 M
i .
A language L over an alphabet = fw
i j
0 i < ng is called a semilinear language
if its image under the Parikh mapping is a
semilinear set, where the Parikh mapping :
!N n
isdened as follows:
7! h0;:::;0i
w
i
7! e (i)
for 0i<n
7! ()+ () for all ; 2
where e (i)
is the i+ 1-th unit vector, which
consists of zeros except for the i-th component,
which is 1.
Notethat{givenatermt{theParikhimage
of all (t)-strings is the same since these are
justconcatenationsof dierent permutationsof
theunitsin(t).
In thefollowingwemake useofa proof
tech-nique used in (Michaelis and Kracht, 1997) to
showthatOldGeorgianisnotasemilinear
lan-guage. We cite a specialinstance of a
proposi-tion giventherein:
Proposition 10. Let P(k)= !
2 k
2
+ 2 !
2
k and
M be a subset of N n
, where n 2, which has
the properties
1. For any k 2 N
+
there are some numbers
l (k)
2
;:::;l (k)
n 1
2 N for which the n-tuple
hk;P(k);l (k)
;:::;l (k)
i belongs toM.
+
upper bound for the second component l
1
of any n-tuple hk;l
1 ;:::;l
n 1
i 2 M (that
means l
1
P(k) for any such n-tuple).
ThenM isnot semilinear.
In order to investigate the semilinearity of
ICML
we choose distinct symbols f;x 2 F,
such that (f) = ! and (x) = 0. We shall
construct terms s
i
by the following inductive
denition:
1. s
0 :=x
2. s
n
:=f(s
n 1
;x;::: ;x) for n>0
Itiseasy to observethat byvirtue of
construc-tions
n
consistsofnleadingfunctorsf andthat
in each iteration the number of x increases by
(! 1).
Lemma11. Let F [ C = ff;1;x;2;::: ;!,
f
1 ;:::;f
jFj 2
gbean enumeration of the
alpha-bet underlying ICML
, where f
1 ;:::;f
jFj 2
are the remaining functors in F ff;xg Then
the Parikh imageof some (s
n
)-string Æ
n is
(Æ
n )=hn;
!
2 n
2
+ 2 !
2 n;
(! 1)n+1;n;:::;n
| {z }
! 1
;0;:::;0
| {z }
jFj 2 i
Furthermore, !
2 n
2
+ 2 !
2
n imposes an upper
bound on the second component of (Æ
n ).
Proof. The rst part of the lemma can be
proved in a straightforward way by induction
on n. The claim on the upper bound follows
fromtheobservationthatthenumberof
occur-rences of case marker 1 can be maximized by
repeated embedding of terms in the rst
argu-ment position.
Proposition 12. Let be a signature. Then
ICML
isnot semilinear.
Proof. Letn=!+jFjand considerthelinear,
andhence semilinear,setR:=
f(n
2
(! 1)+1)e (2)
+ !+2
X
i=0;i6=2 n
i e
(i)
jn
R
Parikh map consists of all strings which
con-tain n
2
((! 1)+1) occurrences of the symbol
x(where n
2
isanynumber) andanynumberof
occurrences of the symbols f;1;:::;!, and no
other symbols. We dene the language L
M as
the set of all strings belonging to L
R
and the
ideal case marking languages. Then L
M
con-tainsall (s
n
)-strings.
Considering the Parikh image M of L
M we
get
M = [L
M
]= [L
R
]\ [ICML
]
=R\ [ICML
]
because of the denitionof L
R
as the full
pre-image of R . But then the set M fullls the
conditionsof Proposition10 dueto Lemma 11.
HenceMisnotsemilinear. SinceRissemilinear
by denition and semilinearity is closed under
intersection ICML
isnotsemilinear.
2.4 Computational Complexity
In this subsection the computational
complex-ity of ICML
is considered. The results are
achieved by dening a 3-tape-Turing machine
acceptor (dependingon agiven signature)that
recognizesICML
.
Proposition 13. Let be a signature. Then
ICML
2DTIME(n p
nlogn)
ICML
2DSPACE(n):
Proof. In the following we let n denote the
lengthoftheinputstring. The Turingmachine
algorithm can be subdivided into three main
parts:
1. Theinputstringissegmentedintoitsunits:
Thealgorithmstepsthroughtheinputand
adds separation markers in between two
units. This canbe doneinO(n)time.
2. The units are sorted according to their
casestacks: Moreformallya2-way straight
mergesortisperformed. Thissorting
algo-rithm is known for its worst case optimal
complexity: it performs the sort of k keys
in O(klogk) steps. In our case the keys
are units and thus their number is clearly
bounded by n. The additionalsquare root
One can show that the maximal length of
a case stackoccuringinan ICML
string
of length n is bounded above by O( p
n).
Hence a comparison of two units takes at
most O( p
n) steps. Thusthe overall
com-plexityofthesortingpartisO(n p
nlogn).
3. The sorted sequence of units is checked:
The algorithm successively generates case
stacksaccordingtothefunctorsithasread.
Each case stack iscompared to theunit of
the input. If they coincide the algorithm
advancesto thenext unitontheinputand
generatesthenextcasestack. Afterallcase
stackshavebeengeneratedthewholeinput
string must have beenworked through. In
this case the algorithm accepts. This can
be doneinO(n)time.
Summing up the complexities of these three
parts shows that the time complexity is as
claimed in the proposition. Furthermore, the
algorithm uses onlythe cells needed bythe
in-put plusat most k 1 cells foradditional
sep-aration markers (due to the rst part), where
k is the number of units the input string
con-sists of. This shows that the space complexity
is linear.
2.5 Discussion
A rst conclusion we may draw is that cases
have the ability to construct the context they
appear in. ICML
strings encode the same
structural informationasorderedlabelledtrees
do thereby allowing unconstrained order of
units. Additionallyeachsuchstringcanberead
unambigously. This was shown by means of
a bijection between bags and ordered labelled
trees.
The fact that ideal case marking languages
are neither nitely pumpable nor semilinear
meansthattheyfalloutofalotofhierarchiesof
formallanguages. As(Weir,1988)shows,
multi-component tree adjoining grammars 3
generate
only semilinear languages. Consequently, ideal
casemarkinglanguagesarenotMCTALs.
How-ever, (Groenink, 1997) denes a class of
gram-mars,calledsimpleliteral movementgrammars,
3
andhencelinearcontext-freerewrite systems,which
areshowntobeweaklyequivalenttoMCTAGsin(Weir,
which generate all and onlythe PTIME
recog-nizablelanguages. Idealcasemarkinglanguages
should therefore be generated by some simple
literalmovement grammar.
We note furthermore that the (theoretical)
timecomplexityis signicantlybetter thanthe
best known for recognizing context-free
gram-mars. Infact,weimplementedapractically
ap-plicable algorithm which constructs the
corre-sponding tree outof a given ICML
string in
linear time(inaverage).
3 Semantics
We are now going to propose a semantics for
languages with stacked cases. The basic
prin-ciple is rather easy: we are going to identify
variables by case stacks thereby making use of
referent systems.
3.1 Referent Systems
The semantics uses two levels: a DRS-level,
which contains DRSs, and a referent level,
which talks about the names of the referents
used by theDRS. Referent systems were
intro-duced in (Vermeulen, 1995). We keep theidea
of areferent system asadevice which
adminis-trates thevariables(or referents) undermerge.
Thetechnicalapparatusishoweverquite
dier-ent. In particular, the referent systems we use
dene explicit global string substitutions over
thereferentnames.
ThereisoneadditionalsymbolÆ. Itisa
vari-ableovernamesof referents. Ifwe assumethat
afunctorghasmeaninggasimplelexicalentry
forg looks like this:
/g/
Æ:Æ
?
Æ :
=g (1 _
Æ;2 _
Æ;:::;(g) _
Æ)
Here,theupperpartisthereferentsystem,and
the lower part an ordinary DRS, with a head
section,containingasetofreferents,andabody
section,containingaset ofclauses. Thismeans
that the semantics of a functor g is given by
theapplicationof gto itsarguments. However,
insteadofvariablesx;y,etc. wend1 _
Æ;2 _
Æ,
etc. The semantics of a 0-ary functor x and a
case marker, say2, are:
Æ:Æ
?
Æ :
=x
Æ:2 _
Æ
?
?
Whentwosuchstructurescometogetherthey
will be merged. The merge operation takes
twostructuresandresultsinanewonethereby
using the referent systems to substitute the
names of referents if necessary and then
tak-ing the union of the sets of clauses. E.g. the
resultof themerge/g/ /2/is
/g2/
Æ:Æ
?
2 _
Æ :
=g (12 _
Æ;22 _
Æ;::: ;(g)2 _
Æ)
The meaning of Æ :2 _
Æ is as follows. If some
structureAismergedwithonebearingthat
ref-erent system, then all occurrences of the v
ari-able Æ in A are replaced by 2 _
Æ. As the
re-sulting referent system we get Æ : Æ. This is
exactlywhatisdoneinthemergeshownabove.
We shall call a structure with referent system
Æ:Æ plain. Merge is onlydened ifat least on
structure isplain.
3.2 Semantics for ICML
Tosee how thesemanticsworkswe shall
repro-duce an earlier example and take the ICML
stringg2x1fx12. Motivatedbythedenitionof
the ideal case marking languagewe shall agree
to the conventionsthat
1. Casemarkersmayonlybe suÆxes
2. Case markers may only be attached to
functors orcasemarkedfunctors
By these conventions the string under
consid-eration must be parsed as (g2)(x1)(f)((x1 )2).
They force us to combine the functors with
their case stacks rst and afterwards combine
the units. We shall understand that this is a
syntacticrestrictionand notduetoany
seman-tics.
The composition of g and 2 was already
shown above and is repeated on the left hand
side, using that (g) = 1. The result of
Æ:Æ
?
2 _
Æ :
=g (12 _
Æ)
Æ:Æ
?
1 _
Æ :
=x
Mergingthese twostructures we get
/g2x1/
Æ:Æ
?
2 _
Æ :
=g (12 _
Æ)
1 _
Æ :
=x
Together withfthisgives
/g2x1f/
Æ:Æ
?
2 _
Æ :
=g (12 _
Æ)
1 _
Æ :
=x
Æ :
=f(1 _
Æ;2 _
Æ)
Bycomposingthestructuresforxand1we get
thestructure/x1/shownabove. We mergethis
one withthat for2and get
/x12/
Æ:Æ
?
12 _
Æ :
=x
The merge of the two structures above nally
gives
/g2x1fx12/
Æ:Æ
?
12 _
Æ :
=x
2 _
Æ :
=g (12 _
Æ)
1 _
Æ :
=x
Æ :
=f(1 _
Æ;2 _
Æ)
We shall verify that the value of Æ is actually
the same as the value of f(x;g (x)). Notice rst
thatinthebodyoftheDRSwendthat12 _
Æ
and 1 _
Æ have the same value as x. We may
therefore reduce thebodyof thisstructureto
2 _
Æ :
=g (x)
Æ :
=f(x;2 _
Æ)
Finallywe mayreplace 2 _
Æ by g (x) inthe
sec-ond line. We get then
:
whichis theintended result.
Afterthesemanticsoftheunitshasbeen
com-putedtheorder ofmerge isunimportant. Ifwe
chooseto mergethesesemanticsinanorder
dif-ferent from the one above, we get the same
re-sult.
3.3 An example from Warlpiri
Toshowhowthisproposalmayworkfornatural
languages we give an example from Warlpiri 4
in which case stacking occurs. We have to
dealwiththefourcasemarkersergative (erg),
pasttense(pst),absolutive(abs), andlocative
(loc).
Japanangka-rlu
Japanangka-erg
luwa-rnu
shoot-pst
marlu
kangaroo-abs
pirli-ngka-rlu
rock-loc-erg
Japanangkashot thekangaroo (while) onthe
rock
We extend the proposal by taking into
ac-countthatcasesmaynotonlyfunctionas
argu-ment markers but have a semantics, too. This
actually does not make much of a dierence
forthis calculus. We propose the following
se-manticsforthelocativeandthepasttense case
marker
/loc/ /pst/
Æ:loc _
Æ
?
located'(Æ;loc _
Æ)
Æ:Æ
?
past'(Æ)
So, when the locative is attached, it says that
thething to which it attaches is located
some-where. Here, Æ represents the thing that is
located, while loc _
Æ is the location. The
pasttensesemanticssimplysays thatthething
whichit attaches to happenedinthepast.
We construe the meaning of the ergative as
being theactor andthe meaningof the
absolu-tiveasbeingthe theme 5
.
4
Thisexampleistakenfrom(Nordlinger,1997),p.171
5
In fact, ergative and absolutive should mark for
grammatical functions, but since linking of
grammati-calfunctionsand actantsis quiteacomplicatedmatter
/erg/ /abs/
Æ:erg _
Æ
?
actor'(Æ) :
=erg _
Æ
Æ:abs _
Æ
?
theme'(Æ) :
=abs _
Æ
Again, we shall agree in using the conventions
statedinsubsection3.2. Firstwehavetoattach
thecasemarkerstocomposetheresulting
struc-tures afterwards. The semantics of the proper
nounJapanangkaistakentobeaplainstructure
withbodyÆ :
=japanangka 0
. Thecompositionof
thisstructureandtheergative semanticsyields
/Japanangka-erg/
Æ:Æ
?
erg _
Æ :
=japanangka 0
actor'(Æ) :
=erg _
Æ
We assume that the nominals /pirli/ and
/marlu/have thefollowinglexical entries:
/pirli/ /marlu/
Æ:Æ
fÆg
rock'(Æ)
Æ:Æ
fÆg
kangaroo'(Æ)
Thus the semantic composition of pirli, loc,
and erggives:
/pirli-loc-erg/
Æ:Æ
floc _
erg _
Æg
rock'(loc _
erg _
Æ)
actor'(Æ) :
=erg _
Æ
located'(erg _
Æ, loc _
erg _
Æ)
Similarlywecan composemarlu withthe
abso-lutive case:
/marlu-abs/
Æ:Æ
fabs _
Æg
kangaroo'(abs _
Æ)
theme'(Æ) :
=abs _
Æ
The semantics of the verb is shown on the left
hand side and its composition with /pst/ on
theright handside.
/luwa/ /luwa-pst/
Æ:Æ
?
shoot'(Æ)
Æ:Æ
?
shoot'(Æ)
erg/, /luwa-pst/, /marlu-abs/, and
/pirli-loc-erg/ in any order, we get the following
result.
Æ:Æ
fabs _
Æ,loc _
erg _
Æg
erg _
Æ :
=japanangka 0
shoot'(Æ)
past'(Æ)
actor'(Æ) :
= erg _
Æ
theme'(Æ) :
=abs _
Æ
kangaroo'(abs _
Æ)
rock'(loc _
erg _
Æ)
located'(erg _
Æ,loc _
erg _
Æ)
It says that there was an event of shooting in
thepast,whose actor is Japanangka and whose
themeissomethingthatisakangaroo,andthat
thereis arock,suchthat Japanangkais located
on it. Note that the only syntactic restriction
were the conventions stated in subsection 3.2
and thatwedid notmake anyfurther
assump-tionsonsyntacticstructure orwordorder.
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