ITERATIg-E OPERATIONS 
Sae Yamada 
Notre Dame Seishin University 
Ifuku-Ch5 2-16-9 
700 Okayama, Japan 
ABSTRACT 
We present in this article, as a part 
of aspectual operation system, a gene- 
ration system of iterative expressions 
using a set of operators called iterative 
operators. In order to execute the itera- 
tive operations efficiently, we have 
classified previously propositions 
denoting a single occurrence of a single 
event into three groupes. The definition 
of a single event is given recursively. 
The classification has been carried out 
especially in consideration of the dura- 
tire / non-durative character of the 
denoted events and also in consideration 
of existence / non-existence of a cul- 
mination point (or a boundary) in the 
events. The operations concerned with 
iteration have either the effect of giving 
a boundary to an event ( in the case of 
a non-bounded event) or of extending an 
event through repetitions. The operators 
concerned are: N,F .. direct iterative 
operators; I,G .. boundary giving opera- 
tors; I .. extending operator. There are 
direct and indirect operations: the direct 
ones change a non-repetitious proposition 
into a repetitious one directly, whereas 
the indirect ones change it indirectly. 
The indirect iteration is indicated with 
. The scope of each operator is not 
uniquely definable, though the mutual 
relation of the operators can be given 
more or less explicitly. 
I INTRODUCTION 
The system of the iterative opera- 
tions, which makes a part of aspectual 
operation system, is based on the assump- 
tion that the general mechanism of 
repetition is language independent and 
can be reduced to a small number of 
operations, though language expressions 
of repetition are different from language 
to language. It must be noticed that even 
in one language there are usually several 
means to express repetitious events. We 
know that "il lui cognait la t~te contre 
lemur" and "il lui a cogn~ deux ou trois 
fois la t~te contre lemur", the examples 
given by W. Pollak, express the same event. 
We have also linguistic means for 
iterative expressions on all lin- 
guistic levels: morphological, 
syntactical, semantic, pragmatic etc. 
As the general form of repetition 
we use ~ = (~i~ in which ~ is the 
whole event, ~ia single occurrence 
of a single event and* an iteration 
indicator. For example: 
: (a series of) explosions took 
place 
93: a single explosion took place 
: indefinit number of times 
~i denotes actually a proposition 
describing a single event S i. ~ sign will 
be replaced later by a singIe or complex 
operator or operators, which operate(s) 
on ~i- 
We hope also to be able to give various 
expressions to the same event and for 
that purpose we are planning to have 
a set of interpretation rules. 
The language mainly concerned is 
Japanese, but in this article examples 
are given in French, in English or in 
German. 
2 BASIC CONDITION OF THE ITERATION 
The iterative aspect is one of 
sentential aspect and denotes plural 
occurrence of an event or an action. The 
iterative aspect concerns therefore 
the property of countability. The itera- 
tire operations give the iterative aspect 
to a proposition and are concerned with 
the plurality of occurrences of the event. 
As we distinguish count nouns (count 
terms) from non-count nouns (mass terms), 
we distinguish countable events from non- 
countable events, or more precisely, 
the events of which the number of occur- 
rences is countable and those of which 
the number of occurrences is non-coun- 
table. 
14 
As a count noun has a clear boundary, 
a countable event also has to have a 
clear boundary. Countable events are 
for instance: he opens a window; he reads 
a book; he kicks a ball etc. Non-countable 
events are for instance: he swims; he 
sleeps deeply; he runs fast,etc. 
Only a countable event can be repeated: 
he opens three windows; he kicked the 
ball twice,etc. A n~n-countable event 
can't be repeated: ~he sleeps twice. 
The distinction of two kinds of events 
(and of two kinds of propositions), 
which also is called telic-atelic, cyclic- 
non-cyclic or bounded-non bounded dis- 
tinction" is therefore necessary for the 
execution of the iterative operations. 
It must be useful to give here some 
remarks on the terminology. 
The terms such as 'iterative', 'repeti- 
tive', 'frequentative' and 'multiplica- 
tire' are used very often as synonyms. 
However there are some works which 
distinguish them one from the other 
The term repetitive is used sometimes 
to indicate only one repetition and the 
term iterative to indicate more than two 
repetitions. And sometimes the term 
iterative is used for one repetition and 
the term frequentative is used for 
several repetitions. 
We use both of the terms 'iterative' and 
'repetitive'~ (hence 'iteration' and 
'repetition'~as synonyms. In this article 
'repetition' means, in most of cases, 
two or more occurrences of a same event. 
But in order to prevent a misunderstan- 
ding, we rather use the term 'iteration'. 
A 'proposition' denotes an event and it 
is a neutral expression in the sense that 
the tense, aspect and mode operators 
operate on it. 
3 SOME PREVIOUS REMARKS ON ITERATION 
3.1 Regular and irregular iteration 
Two kinds of iterations are distin- 
guished: regular and irregular iterations, 
i.e. the iterations which correspond to 
cardinal count adverbials and the itera- 
tions which correspond to frequency 
adverbials. 
A regular iteration is defined either by 
a regular frequency of the occurrence of 
the event, (called 'fixed frequency' by 
Stump), or by a constant length of 
intervals between occurrences. 
(I) We ate supper at six o'clock every 
night last week. (Frequency) 
The busses started at five-minute 
intervals. (Interval) 
I These termes are used by Garey, Bull and 
Allen respectively. 
The extreme case of the regular itera- 
tion is called 'habitude'. 
(2) En ~t~, elle se levait ~ quatre 
heure s. 
A regular frequency or a constant inter- 
val is indicated by the operator F. 
An irregular iteration is indicated 
either with a number of occurrences of an 
event or with irregular lengths of 
intervals between occurrences. 
(3) Linda called you several times last 
night. (Frequency) 
Nous avons entendu le m~me bruit par 
intervalles. (Interval) 
Both the numerical indications and the 
indications of irregular intervals are 
given with the operator N. 
3.2 Repeated constituent of the event 
Considering the structure of a 
repeated event, we can distinguish 
several forms of repetitions, according 
as which constituent is affected. If we 
say,"She changes her dress several 
times a day", it is the object which is 
affected by the repetition. 
Using grammatical category-names we can 
indicate the repeated constituent as the 
following. 
Simple repetition 
(4) Subj (Pred)~: Mr. Wells is publishing 
a novel year by year; L'une apr~s 
l'autre le pilote v~rifia des chiffrea 
(Subj Pred~ : People walked across 
the lawn; Each boy in the room stood 
up and gave his name. 
Complex repetition 
(5)(Subj(Pred)~)*: Lorsqu'elle venait 
avec sa m~re, souvent celle-ci cares- 
salt ce vieux pilier central... 
((Subj Pred) ~ : Les habitants de ce 
quartier r~p~tent toujours:~Si nous 
avions un arr~t d'autobus pr%s d'ici.~ 
On the actual stage we have no such a 
detailed mechanism to be able to diffe- 
rentiate the repeated constituent. Nor 
do we consider the differentiation neces- 
sary. We treat all these repetitions as 
having the type (Subj Pred)~,(in a more 
general form ~), and we find no incon- 
venience doing so. 
3.3 Repeated phase of the event 
An event consists of several phases: 
the beginning, the middle, the end and 
eventually the result and the imminent 
phase, i.e. the phase directly preceding 
the beginning point. 
15 
As for the repetition is concerned only 
a phaseincluding a culmination point is 
capable of repetition, because the repe- 
tition presuppos~ that the event has a 
(real or hypothetical) boundary. 
(6) (Inchoative)~: Lorsqu'il arrivait .., 
M~re et Mme van Daan se mettaient 
pleurer ~ chaque fois. ~ 
Terminative~ : Une ~ une les villes 
talent englouties. 
(Imminent Phase)*: Trois fois ou 
quatre fois au cours de l'entretien 
le commissaire avait ~t~ sur le 
point de lui appliquer sa main sur 
la figure. (Hypothetical culmination 
point) 
(Resultative~ : Chaque fois que je 
vais chez elle, je trouve toute la 
maison bien nettoy~e. 
Like the distinction of the repeated 
constituent, the distinction of the 
repeated phase is not especially signifi- 
cative in the iterative operations. 
Besides, if necessary, we can treat each 
phase as an independent event: the begin- 
ning part ~' of the event ~ can be 
considered as an event. Thus, for the 
time being, the distinction of phases is 
also neglected in the iterative opera- 
tions. 
3.4 Homogeneous iteration and hetero- 
geneous iteration 
A homogeneous iteration is an ordinary 
iteration of the type(~)~ and a hetero- 
geneous iteration is what is called by 
Imbs 'la r~p~tition d'alternance'. It is 
not the iteration of a simple event but 
the iteration of two or more mutually 
related events. It has the form: (~'÷~' '...)~ 
(7) J'allume et j'~teins une fois par 
minute. 
The most frequent case is the combina- 
tion of two events, but the combination 
of three events is still possible: 
(8) Depuis une heure il va ~ la fen~tre 
tousles trois minutes, s'arr~te un 
moment et revient encore. 
The combination of more than three 
events is not natural. 
4 APPLICATION ORDER OF TENCE AND 
ASPECT OPERATOR 
In the present article we are exclu- 
sively concerned with aspect operators and 
tense operators are not treated, though 
past tense sentenses are used as examples. 
We will be contented just to say that 
tense operators come after aspect opera- 
tors in the operation order. 
(9) I1 travaille. --- I1 se met enfin 
travailler. (Inchoative) --- I1 s'est 
nis enfin ~ travailler. (Inchoative + 
Past) 
CLASSIFICATION OF BASIC PROPOSI- 
TIONS 
A sentential aspect is the sythesis 
of the aspectual meanings of all consti- 
tuents of the sentence. 
For the efficient execution of iterative 
operations as well as all aspectual 
operations we have to classify previously 
propositions ~i denoting events S i. For 
this classification we take accoufit of 
durative/non-durative and bounded/non- 
bounded characters of events. 
The distinguished propositions are: 
~ = durative proposition; ~2 = accom- 
plishment proposition; ~ = momentaneous 
(or non-durative) proposltion. This clas- 
sication is basically identical with 
Verkuyl's. The criteria we have used and 
examples of propositions of each groupe 
are as the following. (For pragmatic 
reason, sentences are given instead of 
propositions.) 
Criteria 
~I: the event is represented with an open 
interval; satisfies the additivity (or 
partitivity) condition; co-occurrence 
with durative adverbials such as a yea~ 
an hour .. Ok; co-occurrence with 
momentaneous adverbials such as in five 
minutes, at that moment .. No 
~2: the event is represented with a 
closed interval; a culmination point 
(or a boundary) is included; if the 
culmination point is excluded, it 
satisfies the additivity condition, 
otherwise .. ~o 
~: the event can be considered as a 
~momentaneous one; co-occurrence with 
durative adverbials .. No; co-occur- 
rence with momentaneous adverbials ..Ok 
I Cf. Verkuyl (80) p145. Verkuyl distin- 
guishes durative VP, terminative VP and 
momentaneous VP. 
16 
Examples of expressions 
~I: he sleeps, he sings, he walks 
~2: he swims across the river, he 
reaches the top of the hill, he builds 
a sandcastle 
@3: he hits the ball, a bombe explodes, 
-he kicks at a ball 
This classification is necessary also 
for other aspectual operations. In order 
to show the varidity of the classifi- 
cation, we give an example of other 
aspectual operations: the inchoative 
operation. Inch is a boundary giving 
operator and gives the initial border 
to any proposition, but the meaning of 
Inch(@ i) is different according to @i- 
With ~\[, which doesn't imply any boundarz 
Inch functions to give the initial boun- 
dary. 
ex. ~I it rains; Inch(~l) .. It 
begins to rain 
With @o, which implies an end point, 
inch fiEes the initial boundary. 
ex. @2 "" Bob builds a sandcastle; 
Inch(@2) .. Bob began to build a sand- 
castle. 
The length of the event is the time 
stretch, at the end of which Bob is 
supposed to complete the sandcastle. 
With @3 the condition is quite different. 
~3, momentaneous proposition, implies no 
length (or no meaningful length) and the 
beginning point and the end point overlap 
each other. Inch(~3) gives automatical\]y 
the iteration of the event and the 
initial boundary becoms the initial 
boundary of the prolonged event. 
ex. @3 "" he knocks (one time) on the 
door; Inch(@3) .. He began knocking 
(repeatedly) on the door. 
The function of the Inch is the same for 
all of three examples, but the meaning 
of the beginning is different one from 
another. The third case (that of ~3) is 
an example of the fact that a non-repe- 
titious operator can produce certain 
repetitions. This is the repetitious 
effect of a non-repetitious operator, to 
which we will return later. 
6 BASIC OPERATORS 
An iterative operation is noted as 
Rj(~i), of which Rj is either a single 
operator or operators. As it was already 
said t a necessary condition of the itera- 
tion is that the event in question has 
a clear boundary. Thus the operators 
concerned with the iterative operations 
have either the effect of giving a certain 
boundary, (in the case of non-bounded 
event): B@i , or the effect of repetition. 
The following operators indicated with 
capital letters are not individual opera- 
tors,but group names. An individual 
operator has for instance a form like N 2 
or F1/w(eek). 
Operators 
N: operators indicating directly the num- 
ber of repetitions 
F: operators indicating a frequency or 
regular intervals between occurrences 
I: operators indicating a temporal 
length; effect of prolonging and 
bordering 
B: boundary giving operators 
G: prolonging operators 
Examples of expressions 
N: two times, three times, several times 
F: every day, three times a week, 
several times a day 
I: for an hour, from one to three 
B: begin to, finish -ing, (teshimau..J) 
G: continue to, used tO, (te iru.. J) 
7 OPERATIONS 
7.1 Single operators N~F~I 
7.1.1 Direct operations 
The operation of N, F, repetitious 
operators, on ~2, ~3 give as the output 
N~2, N~ 3, F~2, F~. These are direct (ex- 
plicit) repetitiofis operations, namely 
those which change a non-repetitious 
proposition into a repetitious one. The 
result of the operations is exactly what 
the operators indicate. 
(lO) N~: He crossed the road twice. 
N~: He knocked on the door twice. 
F~2: He goes to Tokyo Station once a 
week. 
F~3: It s~arkles every two minutes. 
7.1.2 Indirect Operations 
The operator I gives a temporal 
limit to a proposition. Usually it ope- 
rates on ~I" 
ex. ~I .. he walks; I~I .. he walks 
for two hours 
t7 
It is not a proper repetitious operator. 
However, if the operator I operates on 92 
or on 9x, a bounded proposition, it turns 
the proposition into that of repeated 
event. In this case, the iterative opera- 
tion is effectuated indirectly. We call 
this iteration 'implicative iteration'. 
ex. 92 -- John walks to the door; 
I .. for hours; I92 .. John walked to 
the door for hours. 
In order to differentiate this I92 from 
I91, we use the symbolXfor an implicative 
iteration: I(~92). (exactly~is~1 oral2) 
~appears not only with the operator I, 
but also with N and F. 
(11) N(~ 93): The top spun three times 
(= several times on three occasionsl). 
F(~93): The bell rings three times 
a day. 
As we have already seen, other aspectual 
operators can also have the effet of 
repetition. 
(12) Inch 93 = Inch(~ 93): It began to 
spin. 
Term 93 = Term(~93): It stopped to 
beat. 
As for the strings N91 and F91, they 
don't satisfy the basic condition of the 
iteration, i.e. 91 has no boundary. With 
some special interpretation rules, how- 
ever, we can interprete them as N92 and 
F92 respectively. 
ex. F91: ?He walks three times a week. 
--@ He walks from the house to the 
station three times every week (F92). 
7.2 Complex operators of N,F,I 
7.2.1 Direct Operations 
The above operators N,F,I can be 
applied successively one after the other, 
but not every combination nor every 
application order is acceptable. F.I, 
I.F, F-N and N-I are acceptable, but N.F 
is not natural. 
(13) F(I91): Ii y alla souvent pendant 
une quinzaine de jours; I .. 15 jours, 
F .. souvent, 91 .. il y alla (pour y 
rester) 
N(I91): J'~tais ~ Tokyo en tout 
trois fols, chaque lois pendant quel- 
ques semaines;N .. trois fois; I .. 
I The distinction of the situation and 
the occasion is clear in Mourelatos. 
quelques semaines; 91 .. J'gtais 
Tokyo 
I(F93): Ii prend le medicament 
trois lois par Sour pendant une 
semaine; I .. une semaine; F .. trois 
fois par jour; 93 .- il prend le medi- 
cament 
I~N gives in a certain operational 
order the same effect as a single opera- 
tor F, but in other orde~ other effects. 
Using complex operators, we get the out- 
put I(F92), I(F93!, F(N92), F!N93), 
N(I91), F(I91), I(N92), I(N93). 
Combination of more than two operators 
are also possible. 
(14) II(F(I291)): Es hat heute ab und zu 
eine Stunde lang geregnet; II heute 
F .. ab und zu; 12 .. eine Stunde; 
91 .. es regnete 
Cf. Es hat heute eine Stunde lang ab 
und zu geregnet. 
II(F(I291)!: Toutes les fins de 
semaine en gte, on gtait toujours 
parti; II .. en gt~; F .. chaque 
semaine; !o -- pendant le week-end 
91 -. on @~ait parti I 
II(F(I291)): Ein Jahr fang hat 
Peter t~glich 3 Stunden lang trainiert; 
I1 .. ein Jahr; F .. t~glich; I2 .. 
d~ei Stunden; 91 .. Peter trainierte 
7.3 Operators B and G 
7.3.1 Direct Operations 
Adding B, boundary giving operators, 
and G, prolonging operators, to the above 
operators, we can further extend the 
iterative operations. B is by it-self no 
repetitious operator. Its proper function 
is to give a boundary to a non-bounded 
proposition. One of the B-operators is 
Inch: Inch 91 .. he begins to write. 
Once a event gains a boundary, it can be 
repeated. 
(15) N(B91): He began to write three 
times. 
Another application order of N and B 
gives another kind of output. 
(16) B(N92): Bob began to build three 
sandcastles; N .. 3; B .. Inch; 92 -. 
Bob built a sandcastle 
I Example borrowed from Sankoff/Thibault. 
'en ~te' can be also interpreted as F. 
In this case, we have two F-operators F I 
and F2: FI (F2(I~I)); FI .. en ~t~ = 
chaque ~t~; F2 .. chaque semaine. 
18 
The prolonging operators G is not a 
repetitious operator either. If G performs 
on ~I, it has only the effect of prolon- 
ging orextending the event• 
(17) G~I: He is working; G .. ING; ~I -- 
he works 
7.3.2 Indirect Operations 
In some cases, the operation of B 
brings about repetitions, as we have seen 
with the operator Inch. It is done in the 
combination of B and ~3" 
(18) B~ = B(~3): She began to cough; 
it began to sparkle; I stopped his 
calling you. 
B(I~ I) = B(F(I~I)): He began jog- 
ging of half an hour (= half an hour 
each day). 
G gives the effect of iteration too, if 
G is associated with a bounded propositio~ 
such as ~2, ~3' I~I" 
(19) G~ 2 =~2: He continues going to 
Tokyo Station; G .. Cont; ~2 .- he 
goes to Tokyo Station 
Combination of the operators F,G with 
other operators can also give similar 
effects• 
(20) I(G~) = I(~ ~3): It was sparkling 
for an hour. 
G(F(X ~)) = F(~ ~3): It continued 
to spark~ ~very two mlnutes. 
7.4 Multiple Structure of Iteration 
A repeated event, (which in fact has 
durative character like ~I), can again 
be given a boundary. And this renewed bou~ 
ded event can again be repeated• This 
makes a multiple iteration• The iteration 
can be explicit or implicative. 
(21) G~2: Elle prend des legons de piano. 
B(Z ~2): Elle a commenc~ ~ prendre 
des le$ons de piano. 
N(B(X ~2)): A trois reprises elle a 
commenc~ ~ prendre des legons de piano. 
The following examples given by Freed 
have also a multiple iterative structure, 
'a series of series' according to her ter- 
minology. 
(22) N(~ ~3): She sneezes a lot. 
B(G(~3) : She began to cough 
(after years of smoking)• 
7.5 Order of Operations 
The scope of each operator is not 
unambiguously definable. However their 
mutual relation can be indicated more or 
less like the following• 
f • N~ 
• F~ 
i 
• • B~ 
. G-- 
Figure I 
The direction of an arrow in the figure 
indicates the written order of two ooera- 
tors in a form. The order of application 
in the operation is therefore inverse. 
8 EVENT AND BACKGROUND 
It is often proposedto distinguish 
an event from its background (or its 
occasion)• The background is a time 
stretch in which the event takes place• 
From a pure theoretical viewpoint, the 
idea of the double structure of event- 
background is very helpful for analysis 
of ambiguous structures• 
I ex. La toupie a tourn~ trois fois. 
In this expression, 'trois fois' can be 
either the number of occurrences of 
the event (i.e. number of spins of the 
top) or the number of occasions on which 
the top spun. With the iterative operators 
the difference can be given clearly: N~3 
and N(~3)• In the former case, the top 
spun three times on one occasion and in 
the latter case, the top spun several 
times on three occasions. 
The operators N,F,I are related with 
both the event and the background. 
Graphically the difference can be indi- 
cated as the figure 2. 2 
I This example is borrowed from Rohrer. 
2 The first graph (hT~3) is also borrowed 
from Rohrer. 
t9 
La toupie a tourn4 trois fois. 
La toupie a tourn~ trois foiso 
(= ~ trois occasions) 
La toupie a tourn~ pendant 
une minute. 
N(Z ~3) 
N=3 
I(x ~3) 
~~' I& I minute 
Figure 2 
Operationally, if we differentiate the 
background from the event on the level of 
iterative operations, the rules must be 
too complicated. For the time being 
the operators N,F, I are used regardless 
whether they operate on the event or on 
the occasion. 
9 NAGATION OF THE ITERATIVE PROPOSITIONS 
As for the negative cases of itera- 
tire operations, there are several 
possibilities. Either a negeted iterative 
proposition remains still iterative or it 
becomes a non-iterative proposition. In 
other words, the negation affects 
the whole proposition in the case of total 
negation, and affects just the number of 
repetitions or the frequency in the case 
of partial negation. In the former case 
the scope of the nagation is larger than 
that of the iteration, and in the latter 
case, the scope of the negation is smaller 
than that of the iteration. 
(23) N@3:I1 est venu deux fois 
~(N@5) or rather ~3:I1 n'est 
jamais venu. (Total negation) 
(~N)@3:I1 n'est pas venu deux fois. 
(En effe%, il n'est venu qu'une lois.) 
(Partial negation) 
N(~@3): I1 n'esz pas venu deux fois. 
D4j~ deux fois il n'est pas venu. 
F~3:I1 sortait trois fois par 
semalns. 
~(F~3) or rather ~@3:I1 n'est 
jamais sorti. (Total negation) 
(~F)@3:I1 ne sortait pas trois fois 
par semaine: en effet il ne sortait 
que deux fois par semaine. (partial 
negation) 
F(~3): Trois jours par semaine, il 
ne sortait pas. 
It depends on which stage of the opera- 
tions the negation is applied. 
10 INTERPRETATION AND CONCORDANCE RULES 
Several kinds of interpretation 
rules are in view. The interpretation 
rules of the first category are those 
which give adequate interpretations to 
N@I, F~ I etc, in consideration of the 
context on the pragmatic level. N@I gains 
usually an interpretation of N~2, and F~I 
that of F@2. For example, "I walked 
three times this week" can be interpreted 
as: "I walke@ three times from the house 
to the station this week." 
The second interpretation rules are 
concordance rules, which connect diverse 
expressions with one same event. 
Different expressions in appearence or 
different means of expressions are inter- 
connected by these rules. Eventually, 
the distinction of the background from 
the event can be effectuated by certain 
rules. 
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Tedeschi, Ph. & A. Zaenan (eds) 13-29 
Carlson,L. 1981: Aspect and Quantifica- 
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Freed,A.F 1972: The S~mantics of English 
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Imbs,P. 1960: L'emploi des temps verbaux 
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Mourelatos,A.P.D. 1981: Events, Processes 
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peut appliquer la distinction entre nom 
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