The default karaka chart for a verb or a class of verbs gives the mapping for the TAM la- bel tA_hE called basic.. It specifies the vibhakti per- mitted for the applicable karaka relatio
Trang 1P a r s i n g Free W o r d Order L a n g u a g e s in t h e
P a n i n i a n F r a m e w o r k
A k s h a r B h a r a t i
Rajeev Sangal
D e p a r t m e n t o f C o m p u t e r S c i e n c e a n d E n g i n e e r i n g
I n d i a n I n s t i t u t e o f T e c h n o l o g y K a n p u r
K a n p u r 2 0 8 0 1 6 I n d i a
I n t e r n e t : s a n g a l @ i i t k e r n e t i n
A b s t r a c t
T h e r e is a need to develop a s u i t a b l e c o m p u t a t i o n a l
g r a m m a r f o r m a l i s m for free word order languages
for two reasons: F i r s t , a s u i t a b l y designed formal-
ism is likely to be more efficient Second, such a
f o r m a l i s m is also likely to be linguistically more ele-
gant and satisfying In this p a p e r , we describe such
a formalism, called the P a n i n i a n framework, t h a t
has been successfully a p p l i e d to I n d i a n languages
T h i s p a p e r shows t h a t the P a n i n i a n f r a m e w o r k
a p p l i e d to m o d e r n I n d i a n languages gives an elegant
account of the relation between surface form (vib-
h a k t i ) and s e m a n t i c (karaka) roles T h e m a p p i n g
is elegant a n d c o m p a c t T h e s a m e basic account
also explains active-passives and complex sentences
T h i s suggests t h a t the solution is not j u s t adhoc b u t
has a deeper u n d e r l y i n g unity
A c o n s t r a i n t based p a r s e r is described for the
framework T h e c o n s t r a i n t s p r o b l e m reduces to bi-
p a r t i t e g r a p h m a t c h i n g p r o b l e m because of the na-
ture of constraints Efficient solutions are known
for these problems
It is interesting to observe t h a t such a parser (de-
signed for free word order languages) compares well
in a s y m p t o t i c t i m e c o m p l e x i t y with the parser for
context free g r a m m a r s ( C F G s ) which are basically
designed for p o s i t i o n a l languages
1 I n t r o d u c t i o n
A m a j o r i t y of h u m a n languages including I n d i a n
and other languages have relatively free word or-
der tn free word order languages, order of words
contains only s e c o n d a r y i n f o r m a t i o n such as em-
phasis etc P r i m a r y i n f o r m a t i o n relating to 'gross'
m e a n i n g (e.g., one t h a t includes s e m a n t i c relation-
ships) is contained elsewhere Most existing compu-
t a t i o n a l g r a m m a r s are based on context free g r a m -
m a r s which are basically p o s i t i o n a l g r a m m a r s It
g r a m m a r f o r m a l i s m for free word order languages for two reasons:
1 A s u i t a b l y designed f o r m a l i s m will be more ef- ficient because it will be able to m a k e use of
p r i m a r y sources of i n f o r m a t i o n directly
2 Such a f o r m a l i s m is also likely to be linguisti- cally more elegant and satisfying Since it will
be able to relate to p r i m a r y sources of informa- tion, the g r a m m a r is likely to be more econom- ical and easier to write
In this p a p e r , we describe such a formalism, called the P a n i n i a n framework, t h a t has been successfully
a p p l i e d to I n d i a n languages 1 It uses the notion
of k a r a k a relations between verbs and nouns in a sentence T h e n o t i o n of k a r a k a relations is cen- tral to the P a n i n i a n model T h e k a r a k a relations are s y n t a c t i c o - s e m a n t i c (or s e m a n t i c o - s y n t a c t i c ) re- lations between the verbals and o t h e r r e l a t e d con-
s t i t u e n t s in a sentence T h e y by themselves do not give the semantics I n s t e a d t h e y specify re- lations which m e d i a t e between v i b h a k t i of nom- inals and verb forms on one h a n d and s e m a n t i c relations on the o t h e r (Kiparsky, 1982) ( C a r d o n a (1976), (1988)) See Fig 1 Two of the i m p o r -
t a n t karakas are k a r t a k a r a k a a n d k a r m a karaka Frequently, the k a r t a k a r a k a m a p s to agent t h e t a role, and the k a r m a to t h e m e or goal t h e t a role Here we will not argue for the linguistic significance
of k a r a k a relations and differences with t h e t a rela- tions, as t h a t has been done elsewhere ( B h a r a t i et
al (1990) and (1992)) In s u m m a r y , k a r t a k a r a k a
is t h a t p a r t i c i p a n t in the action t h a t is most inde- pendent A t times, it t u r n s o u t to be the agent But t h a t need not be so Thus, ' b o y ' and ' k e y ' are respectively the k a r t a karakas in the following sen- tences
1The Paninian framework was originally designed more than two millennia ago for writing a grammar of Sanskrit;
it has been adapted by us to deal with modern Indian languages
Trang 2- s e m a n t i c level (what the speaker
- k a r a k a level
I
- v i b h a k t i level
I
- s u r f a c e l e v e l ( u t t e r e d sentence)
Fig I: L e v e l s in the P a n i n i a n model
The boy opened the lock
The key opened the lock
Note that in the first sentence, the karta (boy) maps
to agent theta role, while in the second, karta (key)
maps to i n s t r u m e n t theta role
As part of this framework, a m a p p i n g is specified
between karaka relations and vibhakti (which covers A 2
collectively case endings, post-positional markers,
etc.) This m a p p i n g between karakas and vibhakti
depends on the verb and its tense aspect modality
(TAM) label The m a p p i n g is represented by two
structures: default karaka charts and karaka chart
transformations The default karaka chart for a verb
or a class of verbs gives the mapping for the TAM la-
bel tA_hE called basic It specifies the vibhakti per-
mitted for the applicable karaka relations for a verb
when the verb has the basic TAM label This basic
TAM label roughly corresponds to present indefinite
tense and is purely syntactic in nature For other B 1
TAM labels there are karaka chart transformation
rules Thus, for a given verb with some TAM la-
bel, appropriate karaka chart can be obtained using
its basic karaka chart and the transformation rule B.2
depending on its TAM label 2
In Hindi for instance, the basic TAM label is
tA_hE (which roughly stands for the present indef-
inite) T h e default karaka chart for three of the B.3
karakas is given in Fig 2 This explains the vibhak-
tis in sentences A.1 to A.2 In A.1 and A.2, ' R a m '
is karta and ' M o h a n ' is karma, because of their vib-
hakti markers ¢ and ko, respectively 3 (Note that B.4
' r A m a ' is followed by ¢ or empty postposition, and
' m o h a n a ' by 'ko' postposition.)
A.I rAma mohana ko pltatA hE
2The transformation rules are a device to represent the
karaka charts more compactly However, as is obvious, they
affect the karaka charts and not the parse structure There-
fore, they are different from transformational granmlars
Formally, these rules can be eliminated by having separate
karaka charts for each TAM label But one would miss the
liguistic generalization of relating the karaka charts based on
TAM labels in a systematic manner
3In the present examples karta and karma tm'n out to be
agent and theme, respectively
K A R A K A V I B H A K T I P R E S E N C E
K a r t a ¢ m a n d a t o r y
K a r m a ko or ¢ m a n d a t o r y
K a r a n a se or optional
dvArA Fig 2: A default karaka Chart TAM LABEL T R A N S F O R M E D
V I B H A K T I F O R KARTA
yA_gayA se or dvArA (and karta is
optional) Fig 3: Transformation rules
R a m M o h a n -ko beats is (Ram b e a t s Mohan.)
m o h a n a ko rAma p I t a t A hE
M o h a n -ko R a m b e a t s is (Ram b e a t s Mohan.)
Fig 3 gives some transformation rules for the default m a p p i n g for Hindi It explains the vibhakti
in sentences B.1 to B.4, where R a m is the karta b u t has different vibhaktis, ¢, he, ko, se, respectively
In each of the sentences, if we transform the karaka chart of Fig.2 by the transformation rules of Fig.3,
we get the desired vibhakti for the karta Ram
r A m a P a l a ko K A t A hE
R a m f r u i t -ko eats is (Ram eats the fruit.)
rAma ne P a l a KAyA
R a m -ne f r u i t ate
(Ram ate the fruit.)
rAma ko P a l a K A n A padA
Ram -ko f r u i t eat h a d to (Ram h a d to eat the fruit.)
r A m a se P a l a nahI K A y A g a y A Ram -se f r u i t not eat c o u l d (Ram c o u l d not eat the fruit.)
In general, the transformations affect not only the vibhakti of karta b u t also that of other karakas They also 'delete' karaka roles at times, that is, the 'deleted' karaka roles must not occur in the sen- tence
The P a n i n i a n framework is similar to the broad class of case based grammars W h a t distinguishes the P a n i n i a n framework is the use of karaka re- lations rather than theta roles, and the neat de- pendence of the karaka vibhakti m a p p i n g on TAMs
Trang 3and the transformation rules, in case of Indian lan-
guages The same principle also solves the problem
of karaka assignment for complex sentences (Dis-
cussed later in Sec 3.)
2 C o n s t r a i n t B a s e d P a r s i n g
The P a n i n i a n theory outlined above can be used
for building a parser First stage of the parser takes
care of morphology For each word in the input
sentence, a dictionary or a lexicon is looked up, and
associated grammatical information is retrieved In
the next stage local word grouping takes place, in
which based on local information certain words are
grouped together yielding noun groups and verb
groups These are the word groups at the vibhakti
level (i.e., typically each word group is a noun or
verb with its vibhakti, TAM label, etc.) These in-
volve grouping post-positional markers with nouns,
auxiliaries with main verbs etc Rules for local word
grouping are given by finite state machines Finally,
the karaka relations among the elements are identi-
fied in the last stage called the core parser
Morphological analyzer and local word grouper
have been described elsewhere (Bharati et al., 1991)
Here we discuss the core parser Given the local
word groups in a sentence, the task of the core
parser is two-fold:
1 To identify karaka relations among word
groups, and
2 To identify senses of words
The first task requires karaka charts and transfor-
mation rules The second task requires lakshan
charts for nouns and verbs (explained at the end
of the section)
A data structure corresponding to karaka chart
stores information about karaka-vibhakti mapping
including optionality of karakas Initially, the de-
fault karaka chart is loaded into it for a given verb
group in the sentence Transformations are per-
formed based on the TAM label There is a sep-
arate data structure for the karaka chart for each
verb group in the sentence being processed Each
row is called a karaka r e s t r i c l i o n in a karaka chart
For a given sentence after the word groups have
been formed, karaka charts for the verb groups
are created and each of the noun groups is tested
against the karaka restrictions in each karaka chart
When testing a noun group against a karaka re-
striction of a verb group, vibhakti information is
checked, and if found satisfactory, the noun group
becomes a candidate for the karaka of the verb
group
The above can be shown in the form of a con-
straint graph Nodes of the graph are the word
b a c c A h A T a se k e l A K A t A h E
Fig 4: C o n s t r a i n t g r a p h
groups and there is an arc labeled by a karaka from
a verb group to a noun group, if the noun group satisfies the karaka restriction in the karaka chart
of the verb group (There is an arc from one verb group to another, if the karaka chart of the former shows that it takes a sentential or verbal karaka.) The verb groups are called demand groups as they make demands about their karakas, and the noun groups are called source groups because they sat- isfy demands
As an example, consider a sentence containing the verb KA (eat):
b a c c A h A T a se k e l A K A t A hE
c h i l d h a n d -se b a n a n a eats (The c h i l d eats the b a n a n a w i t h his hand.)
Its word groups are marked and KA (eat) has the same karaka chart as in Fig 2 Its constraint graph
is shown in Fig 4
A parse is a sub-graph of the constraint graph satisfying the following conditions:
1 For each of the m a n d a t o r y karakas in a karaka chart for each demand group, there should be exactly one out-going edge from the demand group labeled by the karaka
2 For each of the optional karakas in a karaka chart for each demand group, there should be
at most one outgoing edge from the demand group labeled by the karaka
3 There should be exactly one incoming arc into each source group
If several sub-graphs of a constraint graph satisfy the above conditions, it means that there are multi- ple parses and the sentence is ambiguous If no sub- graph satisfies the above constraints, the sentence does not have a parse, and is probably ill-formed There are similarities with dependency grammars here because such constraint graphs are also pro- duced by dependency grammars (Covington, 1990) (Kashket, 1986)
Trang 4It differs from t h e m in two ways First, the
Paninian framework uses the linguistic insight re-
garding karaka relations to identify relations be-
tween constituents in a sentence Second, the con-
straints are sufficiently restricted t h a t they reduce
to well known b i p a r t i t e graph matching problems
for which efficient solutions are known We discuss
the l a t t e r aspect next
If karaka charts contain only m a n d a t o r y karakas,
the constraint solver can be reduced to finding a
matching in a b i p a r t i t e graph 4 Here is what
needs to be done for a given sentence (Perraju,
1992) For every source word group create a node
belonging to a set U; for every karaka in the karaka
chart of every verb group, create a node belonging
to set V; and for every edge in the constraint graph,
create an edge in E from a node in V to a node in
U as follows: if there is an edge labeled in karaka
k in the constraint graph from a d e m a n d node d
to a source node s, create an edge in E in the bi-
p a r t i t e graph from the node corresponding to (d,
k) in V to the node corresponding to s in U The
original problem of finding a solution parse in the
constraint graph now reduces to finding a complete
matching in the b i p a r t i t e graph {U,V,E} t h a t covers
all the nodes in U and V 5 It has several known effi-
cient algorithms The time complexity of augment-
ing p a t h a l g o r i t h m is O (rain (IV], [U]) ]ED which
in the worst case is O ( n 3) where n is the number
of word groups in the sentence being parsed (See
P a p a d i m i t r o u et al (1982), i h u j a et al (1993).)
The fastest known algorithm has a s y m p t o t i c corn-
of O (IV[ 1/2 [E[) and is based on max flow
plexity
] problem (Hopcroft and S a r p (1973))
If we p e r m i t optional karakas, the problem still
has an efficient solution It now reduces to finding
a matching which has the m a x i m a l weight in the
weighted m a t c h i n g problem To perform the reduc-
tion, we need to form a weighted b i p a r t i t e graph
We first form a b i p a r t i t e graph exactly as before
Next the edges are weighted by assigning a weight
of 1 if the edge is from a node in V representing
a m a n d a t o r y karaka and 0 if optional karaka The
problem now is to find the largest m a x i m a l match-
ing (or assignment) t h a t has the m a x i m u m weight
(called the maximum bipartite matching problem or
assignment problem) The resulting matching rep-
resents a valid parse if the matching covers all nodes
in U and covers those nodes in V t h a t are for manda-
tory karakas (The m a x i m a l weight condition en-
4 We are indebted to Sonmath Biswas for suggesting the
connection
5A matching in a bipartite graph {U,V,E)is a subgraph
with a subset of E such that no two edges are adjacent A
complete matching is also a largest maximal matching (Deo,
197"4)
sures t h a t all edges from nodes in V representing
m a n d a t o r y karakas are selected first, if possible.) This problem has a known solution by the Hun- garian method of time complexity O(n 3) arithmetic operations (Kuhn, 1955)
Note t h a t in the above theory we have m a d e the following assumptions: (a) Each word group
is uniquely identifiable before the core parser ex- ecutes, (b) Each d e m a n d word has only one karaka chart, and (c) There are no ambiguities between source word and d e m a n d word Empirical d a t a for Indian languages shows that, conditions (a) and (b) hold Condition (c), however, does not always hold for certain Indian languages, as shown by a cor- pus Even though there are m a n y exceptions for this condition, they still produce only a small num- ber of such ambiguities or clashes Therefore, for each possible d e m a n d group and source group clash,
a new constraint graph can be produced and solved, leaving the polynomial time complexity unchanged The core parser also disambiguates word senses This requires the p r e p a r a t i o n of lakshan charts (or discrimination nets) for nouns and verbs A lak- shan chart for a verb allows us to identify the sense
of the verb in a sentence given its parse Lakshan charts make use of the karakas of the verb in the sentence, for determining the verb sense Similarly for the nouns It should be noted (without discus- sion) t h a t (a) disambiguation of senses is done only after karaka assignment is over, and (b) only those senses are disambiguated which are necessary for translation
The key point here is t h a t since sense disambigua- tion is done separately after the karaka assignment
is over it leads to an efficient system If this were not done the parsing problem would be NP-complete (as shown by Barton et al (1987) if agreement and sense a m b i g u i t y interact, they make the problem NP-complete)
3 A c t i v e - P a s s i v e s a n d C o m -
p l e x S e n t e n c e s
This theory captures the linguistic intuition t h a t in free word order languages, v i b h a k t i (case endings or post-positions etc.) plays a key role in determining karaka roles To show t h a t the above, though neat,
is not j u s t an adhoc mechanism t h a t explains the isolated phenomena of semantic roles m a p p i n g to vibhaktis, we discuss two other phenomena: active- passive and control
No separate theory is needed to explain active- passives Active and passive turn out to be special cases of certain TAM labels, namely those used to mark active and passive Again consider for exam- ple in Hindi
Trang 5Ram Mohan -ko beat pres
(Ram beats Mohan.)
F.2 r A m a dvArA m o h a n a ko pItA gayA (passv.)
Ram by M o h a n -ko b e a t e n was
(Mohan was b e a t e n by Ram )
Verb in F.2 has TAM label as yA_gayA Conse-
quently, the vibhakti ' d v A r A ' for karta (Ram) fol-
lows from the transformation already given earlier
in Fig 3
A major support for the theory comes from com-
plex sentences, that is, sentences containing more
than one verb group We first introduce the prob-
lem and then describe how the theory provides an
answer Consider the ttindi sentences G.1, G.2 and
G.3
In G.1, R a m is the karta of both the verbs: KA
(eat) and bulA (call) However, it occurs only once
The problem is to identify which verb will control
its vibhakti In G.2, karta R a m and the karma
Pala (fruit) both are shared by the two verbs kAta
(cut) and KA (eat) In G.3, the karta 'usa' (he) is
shared between the two verbs, and 'cAkU' (knife)
the karma karaka of 'le' (take) is the karana (instru-
mental) karaka of ' k A t a ' (cut)
G.I rAma Pala K A k a r a m o h a n a ko bulAtA hE
Ram fruit h a v i n g - e a t e n Mohan -ko calls
(Having eaten fruit, Ram calls Mohan )
G.2 rAma ne Pala k A t a k a r a KAyA
Ram ne fruit having-cut ate
(Ram ate having cut the fruit.)
G.3 Pala kAtane ke liye usane cAkU liyA
fruit to-cut for he-ne knife took
(To cut fruit, he took a knife.)
The observation that the matrix verb, i.e., main
verb rather than the intermediate verb controls the
vibhakti of the shared nominal is true in the above
sentences, as explained below The theory we will
outline to elaborate on this theme will have two
parts The first part gives the karaka to vibhakti
mapping as usual, the second part identifies shared
karakas
The first part is in terms of the karaka vibhakti
mapping described earlier Because the interme-
diate verbs have their own TAM labels, they are
handled by exactly the same mechanism For ex-
ample, kara is the TAM label 6 of the intermedi-
ate verb groups in G.1 and G.2 (KA (eat) in G.1
and kAta (cut) in G.2), and nA 7 is the TAM label
6,kara, TAM label roughly means 'having completed the
activity' But note that TAM labels are purely syntactic,
hence the meaning is not required by the system
ZThis is the verbal noun
TAM LABEL T R A N S F O R M A T I O N
not be present K a r m a is optional
nA K a r t a and karma are op-
tional
not be present K a r m a is optional
Fig 5: More transformation rules
of the intermediate verb (kAta (cut)) in G.3 As usual, these TAM labels have transformation rules that operate and modify the default karaka chart
In particular, the transformation rules for the two TAM labels (kara and nA) are given in Fig 5 The transformation rule with kara in Fig 5 says that karta of the verb with TAM label kara must not be present in the sentence and the karma is optionally present
By these rules, the intermediate verb KA (eat) in G.1 and kAta (cut) in G.2 do not have (indepen- dent) karta karaka present in the sentence R a m is the karta of the main verb Pala (fruit) is the karma
of the intermediate verb (KA) in G.1 b u t not in G.2 (kAta) In the latter, Pala is the karma of the main verb All these are accommodated by the above transformation rule for 'kara' The tree structures produced are shown in Fig 6 (ignore dotted lines for now) where a child node of a parent expresses a karaka relation or a verb-verb relation
In the second part, there are rules for obtaining the shared karakas K a r t a of the intermediate verb
KA in G.1 can be obtained by a sharing rule of the kind given by S1
R u l e S I : K a r t a of a verb with TAM label 'kara' is the same as the karta of the verb it modifies s The sharing rule(s) are applied after the tentative karaka assignment (using karaka to vibhakti map- ping) is over The shared karakas are shown by dotted lines in Fig 6
4 C o n c l u s i o n a n d f u t u r e w o r k
In summary, this paper makes several contributions:
• It shows that the P a n i n i a n framework applied
to modern I n d i a n languages gives an elegant account of the relation between vibhakti and karaka roles The mapping is elegant and com- pact
8The modified verb in the present sentences is the main verb
Trang 6bulA (call)
karta.~5 ~ a r m a
(~ruit)
KA ( e a t )
( f r u i t )
karta karma
le (take)
k a r t / ~ a r m a ~ ~ u r p o s e
U-
(he) (fruit)
• karana
( k n i f e )
Fig 6: Modifier-modified relations for sentences
G.1, G.2 and G.3,respectively (Shared karakas
shown by dotted lines.)
• The same basic account also explains active- passives and complex sentences in these lan- guages This suggest that the solution is not just adhoc but has a deeper underlying unity
• It shows how a constraint based parser can be built using the framework The constraints problem reduces to bipartite graph matching problem because of the nature of constraints Efficient solutions are known for these prob- lems
It is interesting to observe that such a parser (designed for free word order languages) com- pares well in asymptotic time complexity with the parser for context free grammars (CFGs) which are basically designed for positional lan- guages
A parser for Indian languages based on the Paninian theory is operational as part of a machine translation system
As part of our future work, we plan to apply this framework to other free word order languages (i.e., other than the Indian languages) This theory can also be attempted on positional languages such as English What is needed is the concept of general- ized vibhakti in which position of a word gets inco- porated in vibhakti Thus, for a pure free word or- der language, the generalized vibhakti contains pre-
or post-positional markers, whereas for a pure posi- tional language it contains position information of a word (group) Clearly, for most natural languages, generalized vibhakti would contain information per- taining to both markers and position
A c k n o w l e d g e m e n t
Vineet Chaitanya is the principal source of ideas
in this paper, who really should be a co-author
We gratefully acknowledge the help received from K.V Ramakrishnamacharyulu of Rashtriya San- skrit Sansthan, Tirupati in development of the the- ory For complexity results, we acknowledge the contributions of B Perraju, Somnath Biswas and Ravindra K Ahuja
Support for this and related work comes from the following agencies of Government of India: Ministry
of Human Resource Development, Department of Electronics, and Department of Science and Tech- nology
References
Ahuja, R.K., Thomas L Magnanti, and James
B Orlin, Network Flows: Theory, Algorithms
Trang 7and Applications, Prentice-Hall, 1993 (forth-
coming)
Barton, G Edward, Robert C Berwick, and Eric
S Ristad, Computational Complexity and Nat-
ural Language, MIT Press, Cambridge, MA,
1987
Bharati, Akshar, Vineet Chaitanya, and Rajeev
Sangal, A Computational Grammar for Indian
Languages Processing, Journal of Indian Lin-
guistics, IL-51 (Also available as TRCS-90-96,
Dept of CSE, IIT Kanpur, 1990.)
Bharati, Akshar, Vineet Chaitanya, and Rajeev
Sangal, Local Word Grouping and Its Rel-
evance to Indian Languages, in Frontiers in
Knowledge Based Computing (KBCS90), V.P
Bhatkar and K.M Rege (eds.), Narosa Publish-
ing House, New Delhi, 1991, pp 277-296
Bharati, Akshar, Vineet Chaitanya, and Rajeev
Sangal, LFG, GB, and Paninian Frameworks:
An NLP Viewpoint, Part of NLP tutorial for
CPAL-2: UNESCO 2nd Regional Workshop on
Computer Processing of Asian Languages, 12-
16 March 1992, I.I.T Kanpur (Also available
as TRCS-92-140, Dept of CSE, IIT Kanpur.)
Cardona, George, Panini: A Survey of Research,
Mouton, Hague-Paris, 1976
Cardona, George, Panini: His Work and Its Tra-
dition (Vol 1: Background and Introduction),
Motilal Banarsidas, Delhi, 1988
Covington, Michael A., Parsing Discontinuous
Constituents in Dependency Grammar (Tech-
nical Correspondence), Computational Lin-
guistics, 16,4 (Dec 1990), p.234
Deo, Narsingh, Graph Theory, Prentice-Hall, 1974
Hopcroft, J.E and R.M Karp, "A n 5/2 Algorithm
for Maximum Matching in Bipartite Graphs,"
J SIAM Comp 2 (1973), pp.225-231
Kashket, Michael B., Parsing a free-word-order
language: Warlpiri, Proc of 24th Annual
Meeting of ACL, pp 60-66
Kiparsky, P., Some Theoretical Problems in
Panini's Grammar, Bhandarkar Oriental Re-
search Institute, Poona, India, 1982
Kuhn, H.W "The Hungarian Method for the As-
signment Problem", Naval Research Logistics
Quarterly, 2 (1955), pp.83-97
Papadimitrou, Christos H., and K Steiglitz, Com-
binatorial Optimization, Prentice-Hall, Engle-
wood Cliffs, 1982
Perraju, Bendapudi V.S., Algorithmic Aspects
of Natural Language Parsing using Paninian Framework, M.Tech thesis, Dept of Com- puter Science and Engineering, I.I.T Kanpur, Dec 1992