A \[ ()RMAL 1{I 3 I{I,SI ~N 1AI I()N ()F Till'; 
1 \]ll,,MA I IC-I{I I~MAI \[C S I I/,U(, 1 U\]{I, ()1,' SI,,N I I~,N(,I,,S 
I~ASI~I) ()N A 'I'YI)I,\]I) A-_,AI~CUI~US 
Y()ichi \[3 Eq'A\](I'; 
Tokyo University of Mercantile Ms.fine * 
ABSTRACT: In this paper, we give a formal rep- 
resentation of the thematic-rhem~tic (T-II) structure 
of a n~tnr,'~l language discourse, b~tsed on ~ typed ~- 
calculus. 
I. INTII.ODUCTION 
In this paper, we give a formed representation of the 
thematie-rhematic ('P-ll) structure of a, nahum\] lsLn- 
triage discourse. Some pairs, triples, or in general 
n-tuples of sentences in a discourse may differ in the 
1)hLee of theh' information focus. The distribution of 
this inform~ttion focus is c~dled the thematic-thematic 
(T-R) structure, or dichotomy. \[n English, the use of 
paa'ticles tile and a (an) is deeply related to tile T-ll 
strnctu re. lit general, a noun with the particle a con- 
stitutes arheme part of tit(; sentence that appears at 
the \])cginiug of the discourse or text, while that noun 
with the p~Lrticle tl~e ~ppe~rs ht the second, third, etc. 
sentences ~LS themes. In Ja.p~Lnese, the %1{ dichotomy 
is we.ll represented by postpositlons wa and .qa. The 
Korean bLnguage h~s a similar system. Meanwhile, 
in Slavic langu~rges ~s Polish, Czech, and \]lnssian, 
the word order is free ~nd this degree of freedom is 
used h)r the represent~tion o\[ the T-R dichotomy. In 
Chinese, the word order is also used f(~r the T-R di~ 
chotomy. Besides theme and rheme, simibLr terms s.s 
old-information and new-itgormation, topic ~nd com- 
ment, topic ~tnd focus etc. ~re used in the literature 
concerning functionaL1 lingnistics (see, e.g., Va\]lduvi). 
In our ~tn~dysis, since we do not define these terms 
explicily, it is not essenti~d which terms are used. We. 
give implicit definition of these concepts a.z'iomati- 
cally. We eonside.r the problem mainly for J~p~Lnese. 
We propose to Its('. typed A-c~deulus to analyse the 
problent. A logie~d notation is seen as a typed X 
term. Batsi(: types sLre T and l{. t{.onghly speaking, 
7' and ll stand for a thenw, l)~rt ~Lnd a rhemc part 
of e~ sentence, respectively. Tile difference of T-It di- 
chotomy is given by different types. Thus tile same 
sentence *nay h;~ve different types depending on the 
*l';tchujim~ Kotc-ku Tokyo J~tpa, n 
sitnation. For utterstnces, type inference will be per- 
formed. The corre<:tness of ~t given discourse can bc 
proved by eheckh~g the correctness of the types of 
each utters, ace. \]in tills p~q)er, we elaborate on this 
ideaL. 
II. REPRESENTATION BASED ON A 
TYPED A-CALCULUS 
'I'll(; purpose of this psLper is to propose ~t formaJ 
model for nttersu~ce interprctsLtion of the them~ttic 
rhematic structure of a ,|~tp~tnesc scntel/ee using tt 
typed k-cedculus. In our sLn~dysls, a logical notsttion 
is seen as a typed A-term. B~tsic types stre. 51' a, nd 
/L Roughly spe~king, 7' and /{ sttuld R)r ~ theme 
ps~rt and ~ rheme part of a sentence, respeetlvely. 
Although we analyse ma.hdy Japanese sentences, the 
resnlts can be tLpp\]ied to other langn,Lges. The T-\[~ 
dichotomy of ~t Japanese sentence is represented by 
the postpositions wa a.nd ga. For extLmple, the folo 
lowing two sentences ~re different in T-I{ dichotomy, 
~nd used in dit\[rent situtttions: (a) Taroo wa Gakusei 
,lea,,. (Speaking of Ta,'oo, he is a student) (b) 'l'aroo 
ga (?akusei desu. ((Of all the people we are talkino 
abo,a) "*~,'oo (and o,~l,a 7a,'oo) i.~ ~, .,t,,,~e,~t.) The 
mo~ning or both (,,) a~,(l 0') i,~ ~'~,,.0o i., a .~,,t~,,~, 
~nd thus ma.y be wrlten ~s student(Tn.roo). I\[ow~ 
ever this representation is obviously not sufficient for 
an ~tcc'ount of the utter~Lnce interpretation of (~) and 
(b). The NI' (noun phrase) of (a) marked with wa 
functions as ~ theme, i.e., it should h~Lve ~dready ,~p- 
petered in the preceding discourse and thns can be 
considered ~m ~n old information. Therefore, in the 
discourse, sentence (~) should t)e preceded by ~t sen- 
tenee that contains "l'a~'oo sm a rhe.me (new informa- 
tion), l,'or example, Taroo in the. fol\]owing sentence 
can be considered as ~L new information: (c) 7a,'0o ga 
ima.su. (llc,'e is 7h,'oo.) The pair (e), (a)in this 
order is ~L correct discourse utterance. On the other 
h,~nd, the p~dr (c), (b)cannot be considered correct 
since student functions ,as ~ theme in (b) while it h~m 
not appeared in the preceding context. As is seen 
1105 
from (b) and (c), an NP marked with postposltion 
.qa fnnctions as a theme (i.e., information focus). To 
explain the difference between (a) and (b) in the ut- 
terance level, we annotate Ax.student(x) of (a) and 
(b) by different typed A-terms. Roughly speaking we 
assign T ---* It and R to each Ax.studeM(x) of (a) 
and (b), respectively. Based on this, if we can show 
student(Taroo): 12 then we say sentence (a) (or (b)) 
of the discourse is correct. For example, if Taroo of 
(@ has a type T then by the fl-reduction of typed 
A-calculus, we have student(Taroo) : 12. For Taroo 
to have a type T, we impose a constraint that Taroo 
must have appeared in a preceeding sentence. Other 
cases can be treated similarly. See the following de- 
scriptions for details. Thus the correctness of the 
discourse CaLL be proved by checking the correctness 
of the types of eavh formula. In general, given a dis- 
course so,sa,"',s,~ in logical forms, what we have 
to show is that (k so : 12), (so : 12 k st : R), ..., 
(so : R,..., s,~-L : R ~" s,~ : R), succesively. 
First consider the following discourse consisting of a 
single sentence. 
Taroo ga imasu. (Here is Taroo.) (1) 
The meaning of this sentence is: 
so = here_is(Taroo) (2) 
We define this discourse to be correct if so : R. This 
is done in the following way: Translate Taroo ga into 
Af.f(Taroo). We let this formula have either type of 
T --+ 12 or 12 --~ 12 when the proper noun Taroo is 
marked with the postposition ga. Thus we have the 
following translation rules: 
Taroo ga ~ Af.f(Taroo) E so : T ---, R (3.1) 
Taroo ga ~ Af.f(Taroo) ~ s0: R ~ R (3.2) 
This can be writen for short as 
Taroo ga ~ A f.f(Taroo) ff so: (T or 12) -+ tt (4) 
In the above, t ~_ s0 means that t is a typed A-term 
component of the logieM formula so. That is 
t gso if\] (?tl,t2)tLtt2=so (5) 
A sentence of neatral description in the Jap~nese lan- 
guage was first found and named by Kuroda (1965). 
This kind of sentence has no theme part. For this 
kind of verb, we assign a type R and write ms follows: 
k Ax.here_is(:~) E_ so: 12 (8) 
Now by (6) and (8) we can deduce the following judge- 
ment. 
eo : Ao, ea : A1 F 
( A f .f( Taroo) )( Acv.here_is(x) ) 
= (Ax.here_is(*))(Taroo) 
= hereSs(Taroo) = ,so: R (9) 
where e0 : A0 and el : A1 stand for (6) and (8), 
respectively. Thus ,so : 12 has been proved and the 
correctness of the discourse (1) has been established. 
To deduce (9), we have of course used the inference 
rule of the typed A-caJculus given by 
co : a --+ fl, el : cY P eoel : fl (10) 
Note that the type used for (Af.\](Taroo)) in deduc- 
tion (9) is R -+ R. In general, for a neutral descrip- 
tion, fl-reduction for 12 + R ~nd R occur. Next we 
consider the discourse consisting of the following two 
sentences. 
Taroo 9a imasu. (Here is Taroo.) (11.1) 
Taroo wa 9akusei desu.(Taroo is a student.) 
(11.'2) 
The T-R dichotomies of tile above sentences are as 
follows: 
Taroo ga imasu. 
Rheme Rheme (12.1) 
Taroo wa gakusei desu. 
Theme Rheme (12.2) 
The NP (noun phrase) of (12.2) marked with wa func- 
tions as a theme. It should have already appeared in 
the preceding discourse as a rheme. The discourse 
(12) satisfies this constraint since Taroo appears 
a rhemc in (12.1) since it is marked with the post- 
position ya. The discourse (12) is ax:tnally correct. 
We now formally state the correctness of (12). The 
tlere t~ and/or t2 may be empty. Thus so _. so. From logical forms of (:12.1) and (12.2) are given as 
(3), we have 
~- A/./(T~roo) C .90 : (T or R) --, n (6) 
The verb imasu allows a neutrM description. A neu- 
tral description has the following T-R dichotomy: 
Taroo ga imasu. 
Rheme Rheme (7) 
.~o = ~ere_i4 T~roo) (13.1) 
sL = ~t~de,~t(Taroo) (13.2) 
First we must show so : 12, however we have M- 
ready seen this. Thus we show sL : R. Note that 
so = (Ax.student(x))(Taroo). It is natural to ~ssigu 
Ax.student(x) a type T ~ R since (12.2) contains 
1106 
the i)ostposition wa. This postposition is catlled the 
themattic wa. We write this ~s follows. 
wa flakusci desu ==~ 
Ax.studt:nt(a:) ~ s, : 7' ~ R 
(l'h¢~s we ht~ve 
(14) 
2'hcrefore if Taroo h~us at type T, we hKve st : 1~ by 
fl-reduction. The NP ca,n be ,~ theme if it has MreMy 
atppeared in the preceding discourse ms t~ rheme. This 
rule e~tn be written as follows: 
&f./(To.roo) ~_ so : ('1' or ~) --~ ~ ~ ~r,,.o,, g s, : "r 
(~6) 
Now .st: \]/, e~n be show,, ats follows. By (6) amd (H0, 
~- T,.roo ~ st : 7' (17) 
Applying the fl-reduction rule to (15) atnd (17), we 
hatve st : R. Thus the discourse (11) is correct. 
in Japatnese, the following sentence art the beginning 
of the discourse is not n~turM. 
~r,,.oo ,,,,,. ,,,a:.,,,s~; des,,.(:r~,.o~ is ~,..~,,.d~,,~.) (~s) 
This is bec~Luse Taroo atppe~trs ats t~ thente but it is not 
proceeded by ~ sentence in which ~lhroo atppeatrs aa ~ 
rheme, in our formM description, the incorrectness 
of the discourse (18) is described ~us at fatihlre of type 
cheekhlg. We define the discourse to be incorrect if 
either so : I?. or Sl : R is not proved. Indeed, so : I~, 
where so = st~Ment(Taro,o) is not proved since we do 
not have Ta.roo E .so : T. 
We now consider the following discourse consisting 
of two sentences. 
Ga.k'l~.s~i ga imasu. (19.1 ) 
7'o.roo (.la flokusci desu. (19.2) 
The logleM forms re," (~9.1) atnd (19.2) ~re given ,'us 
fol low s. 
.so = (9:c)studcnl(x) A here_is(x) (20.1) 
S 1 :: s~,l(tg'l\],l(T(troo) (20.~) 
Since flakusci (student) is m~rked with the postposi- 
tlou Oa, atnd the verb imasu ~fllows ~ nentratl descrlp- 
tiolG we h~tve 
l"rom this we hatve, 
(3,)st~,~,,t(~,) ~ .%: ~ (uu) 
In genera\] we hnpose the following postub~te. 
AABEsI: I?,F AEs~: 1~ (23) 
Furthermore we atdd the following postula, t,e. 
(&;y(x) E so : I~ ~- ~x'/(:'0 E s, : "r (24) 
where Q stands for a qn~ud;ifier V or 3. This postul~d;e 
means thatt at predlc~tte thatt ~ppeatred as ~t rheme catn 
be treatted sts at theme in the succeeding sentences. 
From this ~uM (22) we c~n deduce 
a:,,.s,,,ee,,,,~(.~) C s,: T (~s) 
We. now show s, : R. I"irst l)y (4) we h~ve (6). Ap- 
plying the fl- reduction rule (10) to (6) ~t,~<l (25) wc 
hatve sl = sbu.denl(Ta.roo) : R. Therefore, the dis- 
course (19) is correct. Note that the type used for 
A/.f(Taroo) is 7'-~ ll. Compatre this with (9). 
We now consider the following discourse consisting 
of at single sentence~ 
Taroo 9a gakusei desu. (26) 
\[n the atbove sentence type checking fMls ~,s follows. 
Since the postposition 9a is ,~ttavhed to Taroo, we. 
hatve (6). Therefore, Ax.student(x) E .so must hatve 
~t type of either T or 1L Ilowever this is impossible. 
Since flakusei desu c~tn not be used in at sentence 
of neutrM description, Am.student(x) F si never has 
at type /L The sentence x ga gakusei dcsu Mwatys 
meatns thatt it is x who is a student ~nd is used only 
in the situattkm where gakusci is a theme. Accord- 
ing to Kuno (1973), this use of predicate is cMled the 
exhgustlve-listing. On the other h~nd, Ax.student(x) 
catn have ~ type T only when student ha.q atppeatred a.s 
in (21) in the preceeding context atnd the postulatte 
(24) catn be used. Since (26) does not h~ve ~ pre- 
ceeding text, it never hatppens. Thus it fMls to prove 
so : II ~md it h~ts been estM)lished thatt (26) is not n 
correct disconrse. 
So fitr we h~ve considered discourses consisting of two 
sentences, fIowever the atbove method ctLn be easily 
extended to a discourse that is consisting of more 
th~n three sentences. In this case, the inference rules 
used over severM sentences atre modified. For exatm- 
pie, (16) can be modified ats follows: 
Af.f(Taroo) E sl,i < j : (T or R) -+ R 
H Taroo E ,sj : T (16') 
where si denotes the logleM h)rm correspond ing to the 
i-th sentence of ~ discourse. Furthermore, 7'aroo c~n 
of course be atrbitratry term, a~nd thus we (:atn estM)\]ish 
the following more genera\] rule: 
(1c,") 
1107 
III. CONCLUSIONS 
In this paper, we have given a formM representation 
of the T-R structure of a natural language discourse. 
We have proposed using a notion of typed /k-cMculus. 
A logical notation has been seen as a typed ),-term. 
The correctness of a given discourse can be proved by 
checking the correctness of the types of each utter- 
ante. Although we have analysed mainly Japanese 
sentences, the results can be applied to other lnn- 
guages by considering adaquate translation rules to 
encode a given sentence to formal representations. 
in Uetake (1993, 1994), the author has proposed an- 
other tool for the analysis of the T-R structure. The 
tool nsed there is a logical notation called ontologi- 
cal promiscuity of Ilobbs (1985), which is first-order 
and nonintensionM. Using this description, a proof 
process of utterance interpretation of a discourse is 
obtained. It is interesting that two concepts sim- 
ilar to these (i.e., typed A-c~dculus and ontological 
promiscuity) used in the analysis of the T-R struc- 
ture of a discourse are used in the theory of con- 
structive mathematics (r-realizability and construc- 
tive type theory). The concept of ontological promis- 
cuity in Uetake(1993, 1994) corresponds to the r- 
reMizability and the typed ),-c~lculns of this paper 
to the constructive type theory. See Uetake (1994) 
for more detailed discussion. 
One of the reviewers noted that Barbara Partee is re- 
cently working on logically reconstructing the Prague 
school's notion of topic-focus articulation. The au- 
thor would llke to thank him/her for this informa- 
tion. 
to Combinators and A-Calculus, Cambridge Univer- 
sity Press. 
Ilobbs, J. If. (1985). OntologieM Promiscuity, Proc. 
o,f the 23rd Annual Meeting, Association .for Compu- 
tational Linguistics, pp. 611-69. 
Knno, S. (1973). The Structure o,f the Japanese Lan- 
guage, Cambridge, Mass., MIT Press. 
Kuroda, S.-Y. (1965). Generative Grammatical Stud- 
ies in the Japanese Language, Ph.D Dissertation, 
MIT. 
Uetake, Y. (1991-1992). Analysis of the theme and 
rheme structure of a Japanese sentence, Lingua Pos- 
naniensis, vol. XXXIV, pp.125-134. 
Uetake, Y. (I993). Two formal representntions of 
the thematic-rhematic structure of sentences, Proc. 
o,f Pacific and Asian Conference on .formal and com- 
putational Linguistics, pp. 256-264. 
Uetake, Y. (1994). The thematic-rhematlc struc- 
ture of natural languages meets constructive math- 
ematies, preprint presented at the 6th International 
Workshop Open Systems and In,formation Dynamics, 
Tarufi, Poland, April 6-8. 
VMlduvi, E. h~,formation packaging:A survey, Univer- 
sity of Edinburgh, IIuman Communication Research 
Center, Research Paper tICRC/RP-44. 
ACKNOWLEDGEMENTS 
The author would like to thank Professor Akira 
Ishikawa for valuable discussions and comments. 
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1108 
