TEMPORAL RELATIONS IN TEXTS AND TIME LOGICAL INFERENCES 
J0rgen Kunze 
Central Institute of Linguistics 
Academy of Sciences of GDR 
DDR-1100 Berlin 
Abstract: A calculus is presented which 
allows an efficient treatment of the follow- 
ing components: Tenses, temporal conjunc- 
tions, temporal adverbials (of "definite" 
type), temporal quantifications and phases. 
The phases are a means for structuring the 
set of time-points t where a certain propo- 
sition is valid. For one proposition there 
may exist several "phase"-perspectives. The 
calculus has integrative properties, i. e. 
all five components are represented by the 
same formal means. This renders possible a 
rather easy combination of all informations 
and conditions coming from the aforesaid 
components. 
I. Prerequisits 
We assume that propositions are replaced by 
phase-sets: A proposition R is something 
which is true or false at each point t of the 
time axis U: 
Value(R,t) = T(rue) or F(alse). 
A phase is an interval (span or moment) p 
on the time axis, which a truth value 
(denoted by q(p)) is assigned to: 
q(p) = T: p is an affirmative (main-)phase. 
q(p) = F: p is a declining (secondary) phase. 
A phase-set P is a pair \[P~,q\]: P~ is a set 
of intervals p and q is an evaluation func- 
tion assigning a truth value to each p6 P~. 
The substitution of propositions R by 
phase-sets P is not unequivocal, but also 
not arbitrary. Some necessary conditions 
for the relationship between R and its 
"surrogate" P have been introduced and 
discussed elsewhere (Kunze 1986). One 
essential point is that the simple "moment 
logic" becomes an "interval logic". 
This is also connected with questions as 
expressed by the different definitions of 
HOLD, OCCUR and OCCURING in Allen 1984. 
Another fact connected with phases is the 
unsymmetry in the case of a negation: 
(I) The museum is open today. 
+ The museum is open all day today. 
(2) The museum is closed today. 
= The museum is closed all day today. 
The proposition R is supposed to be 
fixed and given. P is considered as variable 
and provides a formal counterpart of 
different phase-perspectives for a certain 
proposition. The German sentence 
(3) Thomas raucht. 
has at least two of them (and consequently 
two meanings): "Thomas is a smoker" and 
"Thomas is smoking". Furthermore the use of 
phases enables us to consider some parts of 
T(R) as unimportant, accidental or excep- 
tional. These parts form declining phases 
of R. The affirmative phases of R need not 
be disjunct, and they need not be contained 
in T(R). It is also possible to introduce 
nested phases, so that rather complicated 
courses may be rcpresented. 
2. Some formal definitions 
• = p~ Let PI \[ i' qi \] (i : 1,2) be two phase- 
sets with P~ : P~. Then PI and P2 may be 
connected by means of sentential logic: 
For any functor "o" (e. g. "... and ..." 
and "if , then . ") ....... one defines 
P~ o P2 D~f \[P~, ql o q21 with P~ : p~ : p~. 
Phase-operators connect arbitrary phase- 
sets. As an example we take the phase- 
operator OCC: 
\[P~q\] = P : OCC(PI,P 2) means P~ = PI and I 
T, if q1(p) = T and there is a p26P2 
q(p)= with q2(P2 ) = T and p ~ P2 % @' 
F otherwise. 
" " " ~ P2" If one replaces pnp2 # ~ by p = , 
one gets the definition of P = PER(PI,P2) . 
P = OCC(PI,P 2) means "P2 happens during PI"' 
P = PER(PI'P2) "P2 happens throughout PI" 
The important point is that these relation- 
ships between PI and P2 are not represented 
by a Yes-No-decision, but again by a phase- 
set P: OCC(PI,P2) selects from the T-phases 
of the first argument those p for which the 
characteristic condition (= there is a P2 
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with q2(P2 ) = T and p ~ P2 % @) is fulfilled. 
The phase-operator OCC is not the same 
thing as OCCUR or OCCURING in Allen 1984. 
There are at least three differences: OCCUR 
is a Yes-No-predicate, has as first argument 
an event and as second an intervalland the 
arguments are no sets as in our case. It 
makes at any rate difficulties to generalize 
such a Yes-No-property for sets as arguments. 
This is one reason for our definitions. More 
important is that e. g. OCC(PI,P 2) may be 
used as argument in another phase-operator. 
This enables us to express quite easily the 
essential time relation in "In July there are 
evening-planes on Tuesday and Friday.". 
One needs some other operations: 
Given P = \[P~,q\], then alt(P) contains 
exactly those phases which one gets by join- 
ing all phases of P which are not seperated 
from each other and have the same q-value 
(inductively understood). 
If one designates by U ° the phase-set 
consisting only of U as interval (with 
q(U) : T) , then alt(P) = U O means that the 
union of all T-phases of P covers the 
time axis U, i. e. "P is always true". 
In sect° I. we already sketched how to 
represent propositions R by phase-sets P. 
We write P = <R>. Now we have to explain 
the same for temporal adverbials: <tuesday> 
is a phase--set P, whose intervals p are the 
days, and exactly the Tuesdays have the 
q-value T. In <day> all intervals (: day) 
have the q-value T. <1982> is a phase-set 
with years as intervals, but only one 
(:"1982") has the q-value T. Obviously 
x<tuesday> is a single unspecified Tuesday, 
x<day> an unspecified day. 
3. Examples 
Now we are ready to give some examples. 
Let be R = "John goes to see Mary". <R> : P 
is obviously the set of all visits of John 
to Mary. Then we have: 
(4) In 1982 John want to see Mary every 
Tuesday. 
is represented by the following condition 
(,,~" for "if ..., then ...") : 
(5) alt(<1982> ~ PER (<year> , ... 
alt(<tuesday> -~ OCC(<day>,P)))) : U ° 
This has to be read as: It is true 
. (a\]t(...) = U°), that 1982 is a year, durinq 
which (<1982> ~ PER(.cyear>, ...) it was/ 
is/will be always the case (alt(...)) that 
every Tuesday is a day, when it occurred/ 
occurs/will occur (<tuesday> ~ OCC(<day>, 
...)) that R happens. It should be noted 
that (5) has no reference to tenses\[ 
Whereas (4) represents something with the 
logical status of being true or false, (6) 
defines a certain phase-set: 
(6) The Tuesdays when John want to see Mary 
The corresponding expression is 
(7) OCC (<tuesday> ,P) . 
This time the additional, condition is not 
a\]t(...) = U O as before, but alt(o..) + ~ U ° 
("~" as sign for negation, ~ U ° the phase- 
set containing only U as interval with 
g (U) : ~') : 
(8) alt(OCC(<tuesday:>,P) : ~ U ° 
This means: 
(9) There is at \].east one Tuesday when R 
happened. 
In this case it is possible to apply the 
x-operation (to (7)): 
(10) xOCC(<tuesday>,\]?) 
This can be paraphrased as 
(I I) A Tuesday when John went to see Mary 
Behind these examples stand some genera\] 
questions: \]'he two condition alt(...) = U ° 
and alt(...) # ~ U ° have the status of 
truth-conditions. They refer to the two 
cases, where a phase-set is considered as a 
Yes-No-property and where it is the basis 
for a determined (or defined) time, which is 
again a phase-set. This becomes clear by 
(I 2) As long as John went to see Mary every 
Tuesday (she believed in his promise of 
marriage) . 
These spans (there may be more!) have to be 
represented by 
(13) alt(<tuesday> ~ OCC(<day>,P)) 
with truth-condition alt(...) % ~ U ° (for 
112) becomes inacceptable, if there is no 
such Tuesday at all!). Is R = "Mary believes 
in John's promise of marriage" and <R> = P, 
so 
351 
(14) alt(PER(alt(<tuesday> 
OCC(<day>,P)),P)) = U ° 
is the corresponding expression for (12). If 
we take (13) as ~, (5) becomes 
(15) alt(<1982> ~ PER(<year>,~)) = U ° 
and (14) becomes 
(16) alt(PER(~,P)) = U °. 
Using the definition of PER one gets 
(17) alt(<~982> ~ PER(<year>,P)) = U °, 
which can be paraphrased as 
(18) During 1982 Mary believed in John's 
promise of marriage. 
This answers a second general question: Time 
logical inferences may be based on these 
expressions which represent phase-sets. 
Another question concerns quantification. 
The expressions avoid the (always trouble- 
some) quantification and render it possible 
to perform the inferences rather simply. 
The quantifications are "hidden" in the 
following sense: The expression 
(19) Vx3y alt(OCC(XPl,YP2)) # ~ U ° 
(for every T-phase Pl of Pl there is a T- 
phase P2 of P2 such that P2 happens during 
pl ) is equivalent to 
(20) alt(P I ~ OCC(PI,P2)) = U ° 
(an expression without formal quantifica- 
tion!). It can be proved, that for every ex- 
pression with (linguistically reasonable) 
quantification there is an equivalent ex- 
pression without explicit quantification. 
The expressions reflect in fact a 
structure of texts. The constituents of 
this structure belong to two categories: 
"propositional" and "temporal", where the 
second includes some quantifications 
(ever~ Tuesday, ~ on Tuesdays), 
frequencies (three times), measures (for 
three days), (21) gives a simplified 
version of this structure for (12): 
John want to see Mary p. 
every Tuesday t. p 
she believed ... p. 
So we have three types of structures (if we 
restrict ourselves to the sentence-level): 
(a) the syntactic structure (e.g. a 
dependency tree), 
352 
(b) the macrostructure as in (21), which has 
some features of a constituent tree, but 
reminds more of categorial grammar, if 
one considers the problem thoroughly, 
(c) the structure of the expression (14) for 
(12) . 
They may be used as interface structures for 
two steps of analysis. The step from (b) to 
(c) has to apply rules, which we already 
used for (5): 
(22) P every Tuesday 
alt(<tuesday> -, OCC(<day>,P)) 
(23) as long as P , P ~ alt(PER(P,P)) 
etc. It should be noted, that the three 
essential temporal parts in (21) are ex- 
pressed by totally different means: 
Tuesday : phase-set 
every : ... ~ PO(...,...) 
(PO = variable phase-operator) 
as long as : phase-operator 
Another example is 
(24) P only on Tuesdays ~ 
alt(OCC(<day>,P) ~tuesday> 

References: 

James F. Allen, Towards a General Theory of 
Action and Time; Artificial Intelligence 23 
(1984), p. 123 - 154 

JQrgen Kunze, Probleme der Selektion und 
Semantik, to appear 1986 in Studia Grammatica, 
Berlin 
