-
Artificial Intelligence 166 (2005)
136www.elsevier.com/locate/artint
Abstrac
A fraglast is designs tosatisfactiis showncomplexconstruc 2005
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Keywords
1. Intr
Cons
(1) An(2) The
This pSystems,the EPSRSteedmanthis paper
E-mai
0004-370doi:10.10Temporal prepositions and their logic
Ian Pratt-Hartmann
School of Computer Science, University of Manchester, Manchester
M13 9PL, UKReceived 20 April 2004; accepted 9 April 2005
Available online 1 June 2005
t
ment of English featuring temporal prepositions and the
order-denoting adjectives first andfined by means of a context-free
grammar. The phrase-structures which this grammar as-the sentences
it recognizes are viewed as formulas of an interval temporal logic,
whoseon-conditions faithfully represent the meanings of the
corresponding English sentences. Itthat the satisfiability problem
for this logic is NEXPTIME-complete. The computational
ity of determining logical relationships between English
sentences featuring the temporaltions in question is thus
established.lsevier B.V. All rights reserved.
: Natural language; Temporal prepositions; Interval temporal
logic; Computational complexity
oduction
ider the following sentences:
interrupt was received during every cyclemain process ran after
the last cycle
aper was written during a visit by the author to the Institute
for Communicating and CollaborativeDivision of Informatics,
University of Edinburgh. The hospitality of the ICCS and the
support ofC (grant reference GR/S22509) are gratefully
acknowledged. The author would like to thank Markand David Bre for
valuable discussions and the anonymous referees for their helpful
comments on
.
l address: [email protected] (I. Pratt-Hartmann).
2/$ see front matter 2005 Elsevier B.V. All rights
reserved.16/j.artint.2005.04.003
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2 I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
(3) While the main process ran, an interrupt was received before
loop 1 was executed forthe first time.
These sThe priand theis the cencodin
Thisexecutiowhich dhigh cothat of ttences ssemantiresearchpower
aof temphaving vtypes anthe recesituatioing theof limit
Thetemporagrammament TassigneintervalT PL. Ssatisfiab
Theval to bof intering oveand [c,and fin(J I iJ ,
andsymbollation. F(Russelentences speak of events and their
temporal locations: of what happened and when.ncipal devices they
employ to encode this information are temporal
prepositionsadjectives first and last. The aim of this paper is to
answer the question: What
omputational complexity of determining logical relationships
between sentencesg temporal information using such devices?question
is of theoretical interest, because the events mentioned in
(1)(3)cycles,ns of processes, receipts of interruptsare extended in
time; and temporal logicseal with extended eventsso-called interval
temporal logicstypically exhibit
mputational complexity. Given that the syntax of these logics
has little affinity withemporal expressions in English, it is
natural to ask whether the meanings of sen-uch as (1)(3) can be
captured in a computationally manageable logic. The formalcs of
temporal constructions in English have been investigated by a
succession ofers [4,6,12,13,15,20,21]. Yet in none of these
accounts are the issues of expressivend computational complexity to
the fore. Indeed, many treatments of the semanticsoral
constructions in English represent sentence-meanings in a
first-order languageariables which range over time-intervals and
predicates which correspond to event-d temporal order-relationsa
logic which is easily shown to be undecidable. Givennt surge of
interest in logical fragments of limited computational complexity,
thisn is unsatisfactory. There are evident practical and
theoretical reasons for present-semantics of natural language
constructions, where possible, using formal systemsed expressive
power.plan of this paper is as follows. Section 2 outlines the
semantics of the Englishl constructions considered in this paper.
Section 3 then uses a simple context-freer to define a fragment of
English featuring these constructions; we call this frag-PE , short
for temporal preposition English. We show how the
phrase-structuresd to T PE-sentences by this grammar can in fact be
viewed as expressions in an
temporal logic, which we call T PL. Section 4 presents formal
semantics forections 5 and 6 provide matching upper and lower
complexity-bounds for T PL-ility, showing that this problem is
NEXPTIME-complete.
following terminology and notation will be used throughout. We
take a (time) inter-e a closed, bounded, convex (non-empty) subset
of the real line. We denote the setvals by I , and we use the
(possibly decorated) letters I , J , . . . , as variables rang-r I
. Observe that intervals may be punctual. If I and J denote the
intervals [a, b]d], respectively, with a, b, c, d R and a c d b, we
let the terms init(J, I )J, I ) denote the intervals [a, c] and [d,
b], respectively. In other words, whenevers true, we take init(J, I
) to denote the initial segment of I up to the beginning offin(J, I
) to denote the final segment of I from the end of J . More
standardly, the always denotes the strict subset relation, and the
corresponding non-strict re-inally, we occasionally employ the
definite quantifier x(,) with the standard
lian) semantics.
-
I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136 3
2. Semantics
In thfragmenlow motaking cfragmenSection
2.1. Tem
Cons
(4) An(5) An(6) An(7) Afte
pro
Sentencintervalthat it iswe may
(8) J0Notice tby the fing a trmeanin
Sentcycle ocunary pcycle oc
(9) J1The nophrase dinterruptical toby a qural prepabstractis
section, we consider the semantics of the temporal constructions
featured in thet of English defined belowprincipally, the temporal
prepositions. Here, we fol-
dern usage and count temporal subordinating conjunctions as
temporal prepositionslausal (rather than nominal) complements. We
defer a formal specification of thet in question to Section 3, and
the algorithmic derivation of sentence-meanings to4.
poral preposition-phrases: basic semantics
ider the following sentences:
interrupt was receivedinterrupt was received during every
cycleinterrupt was received during every cycle until the main
process ranr the initialization phase, an interrupt was received
during every cycle until the main
cess ran.
e (4) asserts that, within some contextually specified interval
of interest, there is anover which an interrupt was received.
Interpreting the unary predicate int-rec sosatisfied by all and
only those time intervals over which an interrupt was received,thus
represent the meaning of (4) by the formula
(int-rec(J0) J0 I ).
hat the temporal context to which the quantification in (4) is
limited is representedree variable I in (8). That is: the meaning
of (4) is a temporal abstract, receiv-uth-value (in an
interpretation) only relative to a time interval. Viewing
sentencegs in this way greatly simplifies the semantics of temporal
preposition-phrases.ence (5) asserts that, within the given
temporal context, every interval over which acurs includes some
interval over which an interrupt was received. Interpreting the
redicate cyc so that it is satisfied by all and only those time
intervals over which acurs, we may thus represent the meaning of
(5) by the formula
(cyc(J1) J1 I J0(int-rec(J0) J0 J1)).
rmal type in (9) indicates the material contributed by the
temporal preposition-uring every cycle, and the light type the
material contributed by the sentence An
t was received, which it modifies. Observe that this material in
light type is iden-the formula (8), except that the free temporal
context variable has been boundantifier introduced by the temporal
preposition-phrase. On this view, the tempo-osition-phrase
functions semantically as a modal operator, mapping one temporalto
another.
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4 I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
Sentences (6)(7) can now be treated analogously. Making use of
the notation intro-duced at the end of Section 1, and helping
ourselves to a suitable signature of unarypredicates of intervals,
we may plausibly represent these sentences truth-conditions
as,respecti
(10) J
(11)J
We pasdefinitemal typpreposiby thesuccess
the temquantifiand is d
Theto statesextendeexistentthe quamay befound inhoweve
We drepresena propoevent-tyfeaturesI if anddual ofmore co
(12) D(13) [D
It is obvbe treat
Seveknownvely,
2(main(J2) J2 I,J1( cyc(J1) J1 init(J2, I ) J0(int-rec(J0) J0
J1)))
3(init-phase(J3) J3 I,J2(main(J2) J2 fin(J3, I ),
J1(cyc(J1) J1 init(J2,fin(J3, I )) J0(int-rec(J0) J0 J1)))).
s over the usual issues as to the faithfulness of the Russellian
interpretation ofquantification (either expressed or implied) in
these sentences. Again, the nor-
e in (10) and (11) indicates the material contributed by the
newly-added temporaltion-phrases in (6) and (7) respectively, and
the light type the material contributedsentences they modify.
Again, this colouring scheme highlights the fact that theive
temporal preposition phrases function semantically as modal
operators, bindingporal context variables associated with the
sentences they modify. This cascadingcation, typical of iterated
temporal preposition phrases, was pointed out in [18],iscussed
further in [25].fragment of temporal English considered here deals
only with events, as opposedthat is, only with telic as opposed to
atelic eventualities ([22]; see [19] for and discussion). The
thesis that all simple, event-reporting sentences are
implicitlyially quantified was proposed in [5], and is defended in
[17]. These authors takentification in question to be over events
rather than time intervals; but this issueignored for present
purposes. A recent collection of papers on this topic can be[10].
One could doubtless quibble about whether the in (8)(11) should be
;
r, the operative concepts seem too vague for this issue to admit
of resolution.rew attention above to the fact that the formulas
(8)(11) feature a free variableting a temporal context. This
naturally suggests an alternative representation using
sitional modal logic in which formulas are evaluated relative to
time-intervals, andpes are represented by propositional variables.
Suppose, for example, such a logicthe modal operator D, where D is
taken to be true at an interval of evaluationonly if, for some
proper subinterval J of I , is true at J ; and let [D] be the
modalD. Then the 1-place first-order formulas (8) and (9) can be
equivalentlyandmpactlyre-written as the propositional modal
formulas
int-rec](cyc Dint-rec).
ious that, with the aid of appropriate modal operators, formulas
(10) and (11) coulded analogously.ral such logics have in fact been
proposed in the literature, of which the best-are the systems
usually referred to as CDT [24] and HS ([9]; see also [23]).
The
-
I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136 5
logic CDT is strictly less expressive than the first-order
language employed in (8)(11);and the logic HS is in turn strictly
less expressive than CDT. Despite its sthetic ap-peal, however, a
reformulation along the lines of (12)(13) yields no useful
information onthe comlanguagporal flosee [8].intervalpoint-evof
[3,14they arelanguag
Onequantifisatisfyinthe predits dualout restthey lacwhich
wmeanin
2.2. Co
It isof the Eforegointo the elish prein partic
We b
(14) An
to be trduringusage isare beininterruprender t
(15) J
Noticemain prputational complexity of the logic generated by
temporal constructions in naturale. Halpern and Shoham [9] showed
thatHS is undecidable over all interesting tem-ws; and still very
little is known about its decidable fragments. (For a
discussion,
) In fact, the most commonly encountered way to ensure
decidability for modaltemporal logics is to impose the restriction
that the proposition-letters representents. This move leads
naturally to various well-known systems, for example, those,16].
While these logics are of considerable theoretical interest in
their own right,of little use for representing the meanings of
temporal constructions in natural
e.striking characteristic of formulas (8)(11) is the
quasi-guarded nature of thecation they feature. Thus, for example,
(8) existentially quantifies over intervalsg the predicate int-rec;
likewise, (9) universally quantifies over intervals satisfyingicate
cyc; and so on. By contrast, the modal operator D suggested above
(and
) quantify over all proper subintervals of the current interval
of evaluation with-riction; corresponding remarks apply to all the
modal operators of CDT and HS:k the quasi-guarded character of
formulas (8)(11). It is precisely this featuree shall exploit in
our search for a computationally manageable logic to capture
the
gs of temporal expressions in English.
mplications
impossible, within the space of a few pages, to do full justice
to the complexitiesnglish constructions featured in this paper.
Nevertheless, some elaboration of theg account is required; we
confine ourselves to those features of greatest relevance
nsuing computational analysis. For a comprehensive guide to the
grammar of Eng-positions, see [11, Chapter 7]; for an account of
the English temporal prepositionsular, see, e.g., [1].egin with
some remarks on the temporal preposition before. We take the
sentence
interrupt was received before the main process ran
ue in a temporal context I when there is a unique running of the
main processI , and an interrupt is received over some subinterval
of I prior thereto. Ordinaryvague as to whether it is the
beginning- or end-times of the events in question thatg compared.
To resolve any uncertainty, we simply take (14) to require that
somet-event finished before the run of the main process began. We
therefore propose tohe meaning of (14) by
1(main(J1) J1 I, J0(int-rec(J0) J0 init(J1, I ))).
that these truth-conditions impose no limit on how long before
the running of theocess the interrupt was received (except that
imposed by the temporal context I ).
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6 I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
That is: before is here used in the sense of some time before.
Sometimes, however, beforeis taken to mean just before or shortly
before (The tablets are to be taken before dinner).This latter
sense reflects the possibility of adding a time-measure as a
specifier, as in thephraseincorpothe sem
Actuhere asprovideoccurrinwhich texistent
(16) An
We takemous w
contextuniversa
As foing thequantifithe univuntil masemantition ofit is
nathere.
Theis downsubinteruniversamonoto
(17) ? A(18) ? A
Thus, omonotovided bdifferento truth-ward mThe expand untfive
minutes before. In this paper, we ignore this latter sense of
before entirely:rating it into our account would involve us in a
discussion of either vagueness orantics of temporal
measure-phrases, both of which we choose to avoid.ally, the
previous paragraph is misleading in glossing the sense of before
assumedsome time before. For the existential quantification in the
meaning (15) of (14) is notd by the before-phrase at all, but
rather by the sentence An interrupt was receivedg in its scope; the
before-phrase serves merely to specify a temporal context to
hat quantification is restricted. In fact, there is no reason
this quantification need beial at all, thus:
interrupt was received during every cycle before the main
process ran.
(16) to have the meaning (10); that is, we take it to be
(truth-conditionally) synony-ith (6). Here again, the before-phrase
in (16) serves merely to identify a temporalto which the
quantification in its scope is restricted; in particular, it
provides nol quantification of its own.r before, so for until:
until-phrases serve only to create temporal contexts restrict-
quantification provided by the sentences in their scope; but
they do not provide thatcation. This is most apparent by
considering the pair of sentences (5) and (6), whereersal
quantification evidently arises from the determiner every. This
treatment ofy surprise readers familiar with so-called
until-operators in temporal logic, whosecs do typically contribute
universal quantification. Apparently, there is an associa-until
with universal quantification, at least in the minds of temporal
logicians; andural to ask how this apparent association can be
reconciled with the view adopted
answer is as follows. Sentence (5), which the until-phrase in
sentence (6) modifies,ward monotonic: if it is true over some
interval I , then it is also true over allvals of I . (Downward
monotonicity is, of course, characteristic of sentences whichlly
quantify over subintervals.) It transpires that until-phrases
require a downward-
nic scope, as witnessed by the anomalous
n interrupt was received until the main process rann interrupt
was received during some cycle until the main process ran.
n our account, the universal quantificationor more accurately,
downwardnicityis not provided by until; but the presence of until
requires it to be pro-y something else. Before imposes no such
requirement, as we have seen. Thus, thece between before (in the
sense adopted here) and until lies not in their
contributionconditions, but merely in the situations in which they
can be used. Actually, down-
onotonicity is not always sufficient for applicability of
until-phrases (see, e.g., [26]).loration of this issueand indeed of
the myriad other differences between before
illies outside the scope of the present enquiry. We note in
passing that until, like
-
I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136 7
before, also allows nominal complements. However, in the case of
until, these complementsmust clearly denote an event or a time:
(19) Anma
(20) ? A
Thecate pro
(21) An
Sentencbetweenon thethe prestreat (21
(22) An
and give
(23) J
Our excapproxifrom ou
We hmove toral prepcation tthe occuin (9)(
Clautifier; anquantifian -opeover whto counit wouldbefore tour
frag
Theplemeninterrupt was received during every cycle until 5
oclock/the first execution of thein processn interrupt was received
during every cycle until the main process.
preposition when creates another sort of difficulty. When serves
primarily to indi-ximity between the events identified in its scope
and complement, thus:
interrupt was received when the main process ran.
es such as (21) in fact impose remarkably loose constraints on
the temporal relationthe events in question, as various writers
have noted. But whatever the final verdict
nature of those constraints, we cannot usefully treat the
associated vagueness inent paper, and some further regimentation is
necessary. To simplify matters, we) as synonymous with
interrupt was received while the main process ran,
it the semantics
1(main(J1) J1 I, J0(int-rec(J0) J0 J1)).
use for doing so is simply that inclusion is an easier relation
to work with thanmate collocation. Readers who find this expedient
too brutal can simply omit whenr fragment.ave so far discussed
quantification in the scope of temporal prepositions; we nowthe
issue of quantification in their complements. Nominal complements
of tempo-
ositions typically include determiners; and these determiners
contribute quantifi-o the meanings of sentences containing them.
This is evident, for example, withrrences of during every cycle in
(5)(7), which contribute the universal quantifiers
11).sal complements of temporal prepositions, by contrast,
typically lack an overt quan-d the question therefore arises as to
how the variables in these complements get
ed. The answer is that they are (almost always) definitely
quantifiedi.e. bound byrator. Thus, until the main process ran in
(6) is interpreted as until the unique timeich the main process
ran, as reflected by the -operator in (10). It may seem harsht (6)
as false if there are two runs of the main process within the
temporal context;
perhaps be fairer to interpret the relevant until-phrase as
picking out the periodhe first time over which the main process
ran. But since this facility is available inment anyway, as
discussed in Section 2.3, the issue need not detain us.obvious
exception to the rule that temporal prepositions interpret their
clausal com-ts as definitely quantified is whenever. Thus, we
take
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8 I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
(24) Whenever the main process ran, an interrupt was
received
to have the truth-conditions
(25) J
That is:quantifiof tempNote thtemporarepresenwith va
Soming discin Janulent toof in anlish is uinto
thisproximacompleered her
Theprefersments w
(26) An(27) ? A
(Note thIn addit
(28) Th
Finally,clausaltences.the gram
In so
(29) An(30) An1(main(J1) J1 I J0(int-rec(J0) J0 J1)).
the variable contributed by the complement of the
whenever-phrase is universallyed. In the sequel, we shall assume
that all quantification in clausal complementsoral prepositions is
definite, except in the case of whenever, where is it universal.at
we are mimicking our earlier discussion of when in again taking the
operativel relation here to be inclusion rather than approximate
collocation. As before, thists a certain deviation from ordinary
usage; again, however, we cannot sensibly deal
gue truth-conditions here, and so we pass over the issue.e
temporal prepositions have been conspicuous by their absence from
the forego-ussion. The temporal prepositions on and in, in phrases
such as on Mondays or
ary, are specific to certain categories of complements, but are
otherwise equiva-during. Since this detail clearly has no logical
significance, we ignore these usesd on, and confine our attention
to during. The preposition at, which in Eng-sed in conjunction with
clock-times (and some religious festivals) may also fallcategory,
though there are further complications here concerning its inherent
ap-teness. The prepositions for and in, in phrases such as for/in
five minutes, take as
ments temporal measure-phrases. These lie outside the scope of
the logic consid-e.preposition by, in its temporal sense, functions
analogously to until, except that itupward-monotonic sentences in
its scope; moreover, like until, it dislikes comple-hich are not
explicitly temporal, thus:
interrupt was received by 5 oclockn interrupt was received by
the first cycle.
at (37) has a perfectly natural reading in which by is
interpreted non-temporally.)ion, by exhibits interesting
interactions with aspect:
e main process ran/had run/was running by 5 oclock.
we observe that by occurs frequently in the construction by the
time . . . with acomplement, again with the same preference for
qualifying upward-monotonic sen-Dealing with the rather difficult
behaviour of by in our fragment would complicate
mar without adding anything of logical interest, and so we
ignore it.me respects, the mirror-image of both until and by is
since:
interrupt has been received since the main process raninterrupt
has been received during every cycle since the main process
ran.
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136 9
(When used in its temporal sense, since requires the sentence in
its scope to have perfectaspect.) Unlike until and by, however,
since resists embedding in contexts established byquantification,
as we see by comparing
(31) Du(32) ? D
Becauseincludemirror ithe frag
2.3. Fir
Our
(33) An(34) An
Suppthat is, aintervalof J . Awhen thduring idiagramsignific
It is unclegislate
We tmany evJ be thand assusubset Jelementproper sphrase
tdepictedlast e, wring every cycle, an interrupt did not occur
until the main process ranuring every cycle, an interrupt has/had
not occurred since the main process ran.
of these complications, we do not include since in our fragment.
However, we doafter, which we take (again, ignoring some linguistic
subtleties) to function as amage of before. Given the inclusion of
after, our omission of since does not affectments
(truth-conditional) expressive power.
st and last
fragment will also contain sentences such as
interrupt was received during the first cycleinterrupt was
received before the main process ran for the last time.
ose that, in the relevant temporal context I , there is an
unambiguously first cycle:cycle which begins and ends before all
the others. Then (33) asserts that, if J is theover which this
cycle occurs, then an interrupt was received over some
subintervalcorresponding account can of course be given for (34).
Problems arise, however,ere is no unambiguously first cycle within
I . Suppose, for example, cycles occurntervals J1, J2, and nowhere
else, in either of the following arrangements. (In suchs,
left-to-right arrangement depicts temporal order; vertical
arrangement has no
ance.)I
cycleJ1
cycleJ2
I
cycleJ2
cycleJ1
lear what the truth-value of (33) should be in such cases.
Apparently, we need to.ake the mathematically simplest way out.
Since we may assume that only finitelyents of any given type e
occur within a given interval I , we proceed as follows. Lete
collection of all proper subintervals of I over which an event of
type e occurs,me J is nonempty. Since J is by hypothesis finite, we
can select the (non-empty) whose elements have the (unique)
earliest end-point. Now select the uniqueJ J whose start-point is
latest. Thus, J is the smallest of the earliest-endingubintervals
of I over which an e-event occurs. In the sequel, then, we
interpret thehe first e, within a temporal context I , to pick out
this interval. (In the situations
above, these are the intervals marked J1.) Similarly, we
interpret the phrase theithin a temporal context I including at
least one occurrence of e, to pick out the
-
10 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
smallest of the latest-beginning proper subintervals of I over
which an e-event occurs. Tore-iterate, we are simply legislating
here in the most convenient way in cases where native-speaker
intuition returns no clear verdict; if readers prefer to say that
the relevant sentenceslack truThe onl
3. A fr
Thewritingstructurcosmetithe tem
3.1. De
We b
(35) An(36) An
For presture. Acgramma
S
Moreovis, we wobtaine
S
This exlogicalto (35) aint-rec athe corr
Temgory PN
(37) du(38) aft(39) beth-values in such cases, then the results
obtained below apply unproblematically.y point at which we appeal
to this legislation is in Lemma 3 of Section 5.
agment of temporal English
task of this section is to define a fragment of temporal
English. We do this bya context-free grammar to recognize its
sentences. The grammar assigns phrase-es to these sentences in the
familiar way, and we shall see that, following somec
re-arrangement, the phrase-structures in question can be regarded
as formulas ofporal logic T PL defined in Section 4.
lineating the fragment
egin with the simplest sentences in our fragment:
interrupt was receivedinterrupt was not received.
ent purposes, sentence (35) is taken as atomic: that is, we
ignore its internal struc-cordingly we treat such sentences as
vocabulary items, of class S0, and write ther rules:
S0 S0 an interrupt was received/int-rec.er, the only property of
sentence (36) which concerns us is its relation to (35): thatish to
ignore other aspects of its structure. Accordingly, we pretend that
(36) is
d by simply prefixing the word not to (35), and write the
grammar rules Neg,S0 Neg not/.pedient removes needless clutter from
our grammar, while affecting nothing ofsubstance. (It is a simple
exercise to restore the clutter.) Thus, our grammar assignsnd (36)
the phrase-structures shown in Fig. 1. These diagrams feature the
symbolsnd , as specified in the grammar rules. These symbols are
simply mnemonics foresponding vocabulary items, which will be used
later.poral prepositions with nominal complements belong in our
grammar to the cate-, and occur in phrases such as
ring every cycleer the initialization phasefore the first
interrupt.
-
I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
11
Nomina(lexicalAgain, w
N
We allojectivesproducerules
PNDP
where tour grabefore,of the o
Thecategoriinvolvinthe obsea first in
Ourporal prrequiremwhich c
(40) ? A
For befosense of(again:
(41) An(42) AnFig. 1. The structure of sentences (35)(36).
l expressions such as cycle, initialization phase and interrupt
are taken to be of) category N0 and to denote event-types in the
same way as items of category S0.
e regard them as structureless:0 cycle/cyc N0 initialization
phase/init N0 interrupt/int-rec.w these expressions to be
optionally modified (once) by the order-specifying ad-first and
last, resulting in a phrase which in turn combines with a
determiner tothe complement of a temporal preposition. Accordingly,
we write the grammar
P PN,D , NPD NPD DetD , N1D N1D N01! OAdj, N0 OAdj first/f OAdj
last/let every/[ ] Det some/ Det! the/{ }N,D during/= PN,!
after/> PN,! before/
-
12 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
to be eqof beforsense inquantifiof befo
(43) An(44) An
Indeed,complexadmittinnegationfor notasimplycle.
TemPS, and
(45) be(46) wh(47) whFig. 2. Structures of preposition-phrases
with nominal complements.
uivalent in contexts where there is a unique first reset point,
as our assumed sensee would require. We conclude that the term
before can only have the shortly-before(41), and so we banish that
sentence from our fragment. Admittedly, existentially
ed complements with these prepositions sound better, even with
our chosen sensere:
interrupt occurred before some reset pointinterrupt occurred
during every cycle until some reset point.
such sentences could be admitted into our fragment without
compromising theity-theoretic results derived below. However,
banning sentences such as (41) whileg those such as (43) would
generate a logical fragment not fully closed under; and, while such
fragments are unproblematic in principle, they tend to maketional
and conceptual clutter. For simplicity, therefore, we duck the
issue, and
decree that these temporal prepositions require complements with
the definite arti-
poral prepositions with clausal complements belong in our
grammar to the categoryoccur in phrases such as
fore the main process ranenever the main process ranile the main
process ran for the last time.
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
13
Unmodmar perfirst/lastfirst/last
PSS
thus asswhenevment. Tthan { }in Secti
We atype, as
SSince wgrammaof provaddress
Finasentenc
SFig. 4 sof frontFig. 3. Structures of preposition-phrases with
sentential complements.
ified clausal complements are taken to be atomic, again of
category S0. Our gram-mits modification (once) of these clausal
complements by the adverbials for thetime, analogous to the
modification of nominal complements by the adjectives
. Accordingly, we write the grammar rulesP PS,D , S1D PS,!
while/(=, { }) OAdv for the first time/f1! S0, OAdv PS,!
before/(
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14 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
that thisEnglishEnglish
3.2. Re
In Selanguagthe satisthe meathe presarrangerecursivFig. 4.
Structures of sentences (4)(6).
defect can easily be rectified. This completes our explanation
of the fragment ofstudied in this paper. We dub this fragment T PE
, short for temporal preposition
; the full list of grammar rules is given in Appendix A.
-writing phrase-structures
ction 4, we show how phrase-structures in T PE can be treated as
formulas in ae for which a recursive semantics can be given in the
style due to Tarski. Moreover,faction-conditions thus associated
with T PE-sentences convincingly systematizenings proposed for the
various examples considered in Section 2. To facilitateentation, we
first subject T PE phrase-structures to some minor geometrical
re-ment, which we now proceed to describe. We have three base cases
and threee cases to consider.
-
I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
15
First base case: Any structure of the forms depicted in Fig. 1
will be re-written morecompactly as follows:
S
S
e
(Here aright of
Secondfollows
N
N
Third ba
S
S
First recture know hfollows
S
where =.0 e=
S
Neg S0
e e=.
nd in the sequel, we have replaced all terminal nodes with the
mnemonics to thethe obliques: this simply unclutters the
diagrams.)
base case: Any structure of category N1 will be re-written more
compactly as:
1D
0
e
e
N1!
OAdj N0
f e
efN1!
OAdj N0
l e
el .
se case: Any structure of category S1 will be re-written
analogously:
1D
0
e
e
S1!
S0 OAdv
e f
efS1!
S0 OAdv
e l
el .
ursive case: Consider a structure of category S immediately
dominating a struc-of category S and a PP with a nominal complement
. Assuming that we alreadyow to re-write and , such a structure
will be re-written more compactly as:
S
: PP
PN,D NPD
DetD N1D:
if and ,
denotes any of the bracket-pairs , [ ] or { }, and any of the
symbols or
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16 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
Second recursive case: Consider a structure of category S
immediately dominating astructure of category S and a PP with a
clausal complement . Assuming that we alreadyknow how to re-write
and , such a structure will be re-written more compactly
asfollows
S
where =.
Third recategory or formall
ConsFig. 4. R
iApart frsults ofwith thelook; analong this obvioprocessa
familiwanted.
Therhow PPform [ ] or {quantifief , el , t
ewhere eFinally,instead
e:
S
: PP
PS,D S1D:
(, )
if and ,
denotes either of the bracket-pairs [ ] or { }, and any of the
symbols or
cursive case: Any structure of category S immediately dominating
a node ofConj will be re-written more compactly as an expression
with major connectives
in the obvious way. The details are routine and are left to the
reader to spell outy.ider, for example, the phrase-structures of
the T PE-sentences (4)(6), as drawn ine-writing these
phrase-structures yields the respective expressions
nt-rec=, [cyc]=int-rec=, {main}. However, our grammar imposes
restrictions on thecation in PP-complements ensuring that, if {} or
if has one of the formshen is { }. This cuts down the set of modal
operators to the forms=, [e]=, {e}=, {e} , {e }=, {e } ,corresponds
to a vocabulary item (of category S0 or N0), {} and {f, l}.to avoid
clutter, we may take the =-subscripts as understood, e.g., writing
[e]
of [e]=. Thus, the final collection of operators is, [e], {e},
{e} , {e }, {e } ,
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
17
with e a vocabulary item, {} and {f, l}.Let us take stock. In
Section 2, we proposed truth-conditions for a range of
sentences
involving temporal prepositions and the order-denoting
adjectives first and last. By treatingsentenccould bEnglishthe
phrabe re-arlogic. Oa formatask we
4. The
In th
Definitirelation
It isactly: TT PL+course,
purposeWhe
understT PL+,conditio
Recaboundefin(J, I
DefinitiI E.A(e) fo
Thinevent ofinite seevent-tye-meanings as temporal abstracts, we
showed how temporal preposition-phrasese viewed (semantically) as
modal operators. In this section, we have formalized thefragment we
are working with using a context-free grammar. We observed
thatse-structures which this grammar associates with the sentences
it recognizes canranged as formulas of a language whose syntax
resembles propositional dynamicf course, the point of this
re-arrangement is that the resulting formulas can be givenl
semantics which reproduce the truth-conditions proposed in Section
2. It is to thatnow turn.
temporal logic
e sequel, let E be a fixed infinite set. We refer to elements of
E as event-atoms.
on 1. Let e range over the set E of event-atoms. We define the
categories of event-, T PL-formula and T PL+-formula as follows::=
e | ef | el;:= e | e | e | [e] | {} | {}> | {} | {}
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18 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
We now turn to the interpretation of event-relations. Recalling
our (rather artificial)stipulations about the meanings of the words
first and last applied to event-types of whichthere is no
unambiguously first or last instance, we adopt the following
terminology.
Definitithat J =J = [asay thatJ = [a
DefinitiA |=I,J(1) A |=(2) A |=(3) A |=
It isthe min
We a
Definitisively a
(1) A |=(2) A |=(3) A |=(4) A |
A |=(5) A |=
;(6) the
If A |=IA |=I ; andwe saysatisfiab
We rI E i
Sincate quesreproduon 3. Let I be an interval and J I , where J
satisfies some property P . We say[a, b] is the minimal-first
subinterval of I satisfying P just in case for every
, b] I satisfying P , either (i) b < b or (ii) b = b and a a.
Likewise, weJ = [a, b] is the minimal-last subinterval of I
satisfying P just in case for every
, b] I satisfying P , either (i) a > a or (ii) a = a and b
b.
on 4. Let be an event-relation, A an interpretation, and I, J I
. We define by cases as follows:
I,J e iff J I and e A(J );I,J e
f iff A |=I,J e and J is the minimal-first such interval;I,J
e
l iff A |=I,J e and J is the minimal-last such interval.
obvious that, since A is finite, if there exists any J I such
that J, e A, thenimal-first and minimal-last such J exist and are
unique.re now ready to give the satisfaction-conditions for
formulas in T PL+.
on 5. Let be a formula, A an interpretation, and I I . We define
A |=I recur-s follows:
I e iff for some J , A |=I,J e and A |=J ;I [e] iff for all J ,
A |=I,J e implies A |=J ;I {} iff there is a unique J I such that A
|=I,J , and for that J , A |=J ;
=I {} iff there is a unique J I such thatA |=I,J , and for that
J ,A |=fin(J,I )
usual rules for , , and .
, we say that is true at I in A. For any set of formulas , we
write A |=I iffor all . If, for all A and I , A |=I implies A |=I ,
we say that entails
if is the sole element in , we say that entails . If and entail
each other,they are logically equivalent and write . A set of
formulas is said to bele if, for some A and I , A |=I .
emark that the condition in Definition 2 that interpretations
are finite subsets ofs significant. For example, the T PL-formula e
[e]e is unsatisfiable.e any T PL-formula is just the
phrase-structure of a T PE-sentence, the immedi-tion is whether the
satisfaction-conditions assigned to in Definition 5 correctlyce the
meanings proposed in Section 2.
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
19
A little thought shows that they do. For example, the grammar of
Section 3 assigns tothe sentences (4)(6), which we repeat here for
convenience as
(48) An(49) An(50) An
the resp
(51) in(52) [cy(53) {m
From Dexactly
(54) J(55) J(56) J
But thesin Sectimodal oin theirsee howpreposiand l a
Thisincorposentencwith sathem, mparticulity of
asecondproblem
5. Upp
TheT PL) iof satisfinterrupt was receivedinterrupt was received
during every cycleinterrupt was received during every cycle until
the main process ran,
ective phrase-structures
t-recc]int-recain}
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20 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
Lemma 1. For all e E, T PL+, {} and {f, l}:e [e] [e] e
Proof.
Lemmain subfo
Proof.ties, allo
Definitithere exdepth to
Thenever co
a desceshown iHowevecomprothe righ
LemmaA |=I A A
Proof.involvesevery ev
L
L{e} {e} {e} {e } [e] {e }{e} {e} {e} {e } [e] {e }.
Trivial.
2. Every T PL+-formula is logically equivalent to one in which
appears onlyrmulas of the forms {e} and (= ).
The logical equivalences of Lemma 1, together with familiar
propositional validi-w negations to be moved successively inwards
until the desired form is reached.
on 6. Let A = be an interpretation. The depth of A is the
greatest m for whichist J1 Jm with A(Ji) = for all i (1 i m). If A
is empty, we take itsbe 0.
next lemma shows that, in determining satisfiability of T
PL+-formulas, we neednsider very deep interpretations. To
illustrate the basic idea, let I1 I4 be
nding chain of intervals, and let A be the interpretation {Ii, a
| 1 i 4}, asn the left-hand diagram in Fig. 5. Evidently, for any I
I1, A |=I a {a}.r, it is clear that we can remove the occurrence of
a at I1 (indeed, also at I2) withoutmising this fact. Thus, if A is
the interpretation {Ii, a | 2 i 4} depicted int-hand diagram of
Fig. 5, we still have, for any I I1, A |=I a {a}.
3. Let be a T PL+-formula, A an interpretation and I an interval
such that. Denote the number of symbols in by ||. Then there exists
an interpretationwith depth at most O(||2) such that A |=I .
We may assume that has the form guaranteed by Lemma 2, and
further, that Ano event-atoms not mentioned in . Let be the set of
subformulas of . For
ent-atom e mentioned in and every interval J , define
(J ) = { |A |=J }e(J ) = L(J ) \
{L(K) | K J,K A(e)}.
Fig. 5. Two interpretations making a {a} true at any I I1.
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
21
Thus, Le(J ) records which subformulas of are true at an
interval J , ignoring thosesubformulas which are true at some
proper subinterval of J satisfying e. Say that a pairJ, e A is
redundant if Le(J ) = and there exist K,K A(e) such that K K J .Now
se
ATo illuspicted iLa(I1)as depicdoes no
Retuand Le(m is thethus suf
We p
,,If thereOtherwpairs JredundaredundaEither w
Thee, ef o
For theand A |, so th = {e}this J ,By induremainiimplieson the
ris no un
Theoreinterpre
Proof.LemmaAs befomajor cS(, I)
see thatt =A \ {J, e | J, e is redundant}.trate, suppose for the
moment that is a{a} and A the interpretation de-n the left-hand
diagram of Fig. 5. It is routine to check that L(I1) = L(I2),
whence= . On the other hand, La(I2), La(I3) and La(I4) are all
non-empty, so that A isted in the right-hand diagram of Fig. 5. As
we observed, the reduction of A to At affect the truth-value of at
any interval I I1.rning to the general case, it is obvious that, if
J J with J,J A(e), then Le(J )J ) are disjoint. It follows that the
depth of A is bounded by m(m + 2), wherenumber of event-atoms
occurring in and m the number of subformulas of . It
fices to show that, for all I and all , A |=I implies A |=I
.roceed by induction on the complexity of . The base cases are of
the forms ={e}. The first two of these are trivial. For the case =
{e}, suppose A |=I .is no J I with J A(e), then since A A, we
certainly have A |=I .
ise, there exist J I and J I with J = J and J,J A(e). If neither
of the, e and J , e is redundant, then J,J A(e). On the other hand,
if J, e isnt, there must exist K K J such that the pairs K,e and K
, e are non-nt elements of A, whence K,K A(e); and similarly if J ,
e is redundant.ay, then, A |=I .
recursive cases are of the forms = [e] , e , {} , {} , where is
of the formsr el , and {}. For the case = [e] , we need only
observe that A A.case = e , suppose A |=I . Then there exists J I
such that J A(e)
=J . By the finiteness of A, choose such a J which is minimal
under the orderat J A(e). By inductive hypothesis, A |=J ; hence A
|=I . For the case , suppose A |=I . Then there exists a unique J I
such that J A(e); and forA |=J . In particular, there is no K J
such that K A(e), whence J A(e).ctive hypothesis and the fact that
A A, we then easily have A |=I . The
ng cases are dealt with exactly as for = {e} , noting, in
particular, thatA |=I,J efA |=I,J ef and A |=I,J el implies A |=I,J
el . (This is the point at which we relyather artificial choice of
semantics for ef and el in Definition 4 in cases where
thereambiguous first or last e-interval.)
m 1. Let be a formula of T PL+. If is satisfiable, then is
satisfied in antation of size bounded by 2p(||), for some fixed
polynomial p.
Suppose that A |=I0 . We may assume that has the form guaranteed
by2; and by Lemma 3, we may assume that the depth of A is at most
of order ||2.re, let be the set of subformulas of . Say that a
formula is basic if theonnective of is neither nor . For any
interval I and any , denote bythe set of all maximal basic
subformulas of such that A |=I . It is easy to
, for any and I I with A |=I , S(, I) entails .
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22 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
begin tree(, I0)Choose some object v0, and setQ = V = {v0}; (v0)
= I0; L(v0) = S(, I0); E = .
until QSelectfor eve
(1) IS
(2) IQ
(3) I
E
(4) I(5) I
SV
end foend unt
end tree
begin unfor evexists
Seend fo
end univ
Wechoosintree(bellingsA usingand thinval. Thensure,
ded calwitnessis mainalso thaproof oties.
We cterminasteps 1point in= dov Q, set I := (v), and set Q := Q
\ {v}.ry L(v), dof = e , let J be such that A |=I,J e and A |=J .
Select w / V and set (w) := J ; L(w) :=(, J ); Q := Q {w}; V := V
{w}; E := E {(v,w)}. Execute univ(w).
f = {} , let J be such that A |=I,J . Select w / V and set (w)
:= J ; L(w) := S(, J ); Q := {w}; V := V {w}; E := E {(v,w)}.
Execute univ(w).
f = {} , proceed symmetrically.f is {e}, and there exist J I , J
I with J = J and J,J A(e), choose any such J,J .elect w,w / V and
set (w) := J ; (w) := J ; L(w) := ; L(w) := ; Q := Q {w,w}; V :=
{w,w}; E := E {(v,w), (v,w)}. Execute univ(w) and univ(w).
r everyil
iv(u)ery formula [e] such that (u), e A and thereL (u) with A
|=L [e] do
t L(u) := L(u) S(,(u)).r every
Fig. 6. Construction of small interpretations in T PL+ .
now construct a sub-interpretation A of A, starting with the
interval I0 andg witnesses, tableau-style, for formulas in . More
specifically, the procedure, I0) in Fig. 6 grows a labelled tree
with nodes V , edges E, and the two la- :V I and L :V P(); the
interpretation A will then be extracted fromthis labelled tree. For
v V , think of (v) as the interval represented by v,
k of L(v) as some collection of formulas which must all be true
at this inter-e variable Q is simply a queue of nodes in V awaiting
processing. Steps 15roughly, that existential formulas in have
witnesses as required; the embed-ls to univ(u) ensure that
universal formulas in are not falsified by thesees. A
straightforward check shows that the invariant A |=(v) L(v) for all
v Vtained by tree(, I0). Note that the function is not required to
be 11. Notet the individual steps in tree(, I0) need not be
effective: all we require for thef the theorem is the existence of
the interpretation A with the advertised proper-
laim that tree(, I0) terminates after finitely many iterations,
and that, upontion, the tree (V ,E) satisfies the size bound of the
Theorem. By inspection of5, whenever an edge (v,w) is added to E,
we have (w) (v). Therefore, at anythe execution of tree(, I0), if
the tree (V ,E) contains a path v0 vm,
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
23
then (v0) (vm). Consider those values of i (0 i < m) for
which the call touniv(vi+1) adds material to L(vi+1). By inspection
of univ, this can certainly happenonly if, for at least one
event-atom e, e A((vi+1)). Therefore, it can happen for atmost
Dadds atL(vi+1)order |bound oin V is
Nowbound oinductioDenoteThe cas / L(is no JJ,J of treand
(wstraight
(1) SupandhypBy
(2) SupCertheto uwhe
(3) The
Corolla
Proof.operatowhose station Aand forfin(J0, Jwhich sto
checkcorrect,different values of i, where D is the depth of A.
Moreover, any call to univ(vi+1)most ||2 symbols to L(vi+1); and if
the call to univ(vi+1) adds no material to, then L(vi+1) contains
strictly fewer symbols than L(vi). Since D is at most of|2, the
length of the path v0 vm is therefore at most of order ||4. Then
the eventual size of V then follows from the fact that the
out-degree of any nodebounded by 2||.letA = {J, e A | for some v V
, J = (v)}. Evidently, |A| satisfies the sizef the theorem; it thus
suffices to show thatA |=I0 . In fact, we show by structuraln that,
for any node v V and any formula , L(v) implies A |=(v) .(v) by I .
(Hence A |=I L(v).) The base cases are of the forms = ,,{e}.e = is
trivial. For the case = , the fact that A |=I L(v) ensures thatv).
For the case = {e}, if L(v), A |=I L(v) ensures that either (i)
there I such that J A(e) or (ii) there exist J I , J I with J = J
such thatA(e). In the former case, since A A, then A |=I . In the
latter case, step 5e(, I0) ensures that, for some such J,J , we
have w,w V with (w) = J) = J ; hence J,J A(e) and A |=I . The
inductive cases are almost asforward:
pose is e . If L(v), then, by step 1 of tree(, I0), there exists
w VJ I such that (w) = J , S(, J ) L(w), J, e A, andA |=J . By
inductiveothesis, A |=J S(, J ), and since A |=J , S(, J ) entails
, whence A |=J .construction, J, e A. Hence, L(v) implies A |=I
.pose is [e] . If L(v), then A |=I . Consider any J I with J
A(e).tainly, then, J A(e); hence A |=J , so that S(, J ) entails .
Moreover, byconstruction of A there exists w V with (w) = J , in
which case the callniv(w) ensures that S(, J ) L(w). By inductive
hypothesis, A |=J S(, J ),nce A |=J . Hence, L(v) implies A |=I
.remaining cases are handled similarly to Case 1, or are
trivial.
ry 1. The satisfiability problem for T PL+ is in NEXPTIME.
Let be a formula of T PL+, and let d be the maximum depth of
nesting of modalrs in . By Theorem 1, if is satisfiable, then it is
satisfiable in an interpretationize is bounded by some fixed
exponential function of ||. Guess such an interpre-and an interval
I . Let J0 be the set of intervals mentioned in A together with I
,
any i 0, let Ji+1 be Ji together with all intervals expressible
as init(J0, J ) or), where J0 J0 and J Ji . Now, for all i (0 i d)
and all J Jdi , guess
ubformulas of having modal depth i are true at J in A. It is
then straightforward, in time bounded by some fixed exponential
function of ||, that these guesses areand thence to determine
whether A |=I .
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24 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
The proofs of Lemma 3 and Theorem 1 thus make essential use of
the quasi-guardednature of T PL+, which we observed in Section 2.1,
together with the assumption that onlyfinitely many events occur in
a bounded time-interval. Note that the construction employedin the
pprocedu(non-terinterpreapproaclinear tiover a bsuch tab
6. Low
In thNEXPTtiling pV are bV the(C,H,V
is positi (0 2n 1);(0,2nof unit stop leftlist
whiproblemdiscussi
To shtial tilintime, asmain tethat thea givenroutine.
6.1. Fix
Let matoms. Talwaysis to enT PL+.roof of Theorem 1 does not, as
formulated there, constitute a tableau decisionre for T PL+,
because the steps are not necessarily effective. We remark that
aminating) tableau procedure has been devised for the interval
temporal logic CDT,ted over branching-time structures [7]. It is
not immediately clear whether such anh could be adapted to yield a
terminating procedure for T PL+, interpreted over ame flow, and
incorporating the assumption that only finitely many events can
occurounded time-interval. However, the results of the next section
indicate that anyleau method is likely to require extensive
backtracking.
er complexity bound
is section, we show that the satisfiability problem for T PL
(and hence T PL+) isIME-hard. Denote by Nn the natural numbers less
than n. Define an exponentialroblem to be a triple (C,H,V ), where
C = {c0, . . . , cM1} is a set and H andinary relations over C. We
call the elements of C colours, and we call H andhorizontal
constraints and the vertical constraints, respectively. An instance
of) is a list c0, . . . , cn1 of elements of C (repetitions
allowed). Such an instance
ive if there exists a function :N2n N2n C such that: (i) (i,0) =
ci for alli n 1); (ii) (i, j), (i + 1, j) H for all i, j (0 i <
2n 1,0 j (iii) (i, j), (i, j + 1) V for all i, j (0 i 2n 1,0 j <
2n 1); and (iv) 1) = c0. We refer to as a tiling. Intuitively, the
elements of C represent coloursquare tiles which must be arranged
so as to fill a grid of 2n 2n squares, with the
-hand square required to have the colour c0. The constraints H
(respectively, V )ch colours are allowed to go to the right of
(respectively, above) which others. The
instance c0, . . . , cn1 lists the colours of the first n tiles
in the bottom row. For aon of exponential tiling problems, see [2,
Section 6.1.1].ow that a problem P is NEXPTIME-hard, it suffices to
show that, for any exponen-g problem (C,H,V ), any instance of
(C,H,V ) may be encoded, in polynomialan instance of P . We now
proceed to do this where P is T PL-satisfiability. The
chnical challenge is to encode, using a succinct formula of T
PL, the informationre are exactly 22n pairwise disjoint intervals
satisfying some event-atom t withininterval I . We begin by
tackling this problem; the remainder of the reduction is
ing a large number of tiles
2 and let a0, a01, . . . , a0m+1, a11, . . . , a1m+1 and z be
pairwise distinct event-o simplify the notation, we write a0
alternatively as a00 or a
10 . The event-atom z will
function as a harmless dummy; it occurs in subformulas z whose
only purposesure that we remain inside the temporal logic T PL,
rather than the more generalThe following terminology will be used
to aid readability. Where an interpretation
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
25
A is clealternat
Definh 1:
(57) {a
Let A bdefine a
(1) I is(2) J i
theI sa
Given tha0i+1 anexactlyalternatin genein A.
Thesubinteris to wrI satis
Let btinct anpreventthe evengether.and 0
(58) [a[bFig. 7. Arrangement of i-witnesses (0 i m).
ar from context, we say that an interval I satisfies an
event-atom e if I, e A;ively, we say that I is an e-interval.e 1 to
be the conjunction of the following formulas, where 0 i m and 0
0}z, [ahi ]{a0i+1}>a1i+1, [ahi ]{a1i+1}z.
e an interpretation and I an interval such that A |=I 1. For all
i (0 i m),n i-witness inductively as follows:
a 0-witness if and only if I is the unique proper subinterval of
I satisfying a0.s an (i + 1)-witness if and only if there exists an
i-witness I such that J is eitherunique proper subinterval of I
satisfying a0i+1 or the unique proper subinterval oftisfying
a1i+1.
at A |=I 1, each i-witness I properly includes exactly one
interval J satisfyingd exactly one interval J satisfying a1i+1,
with J preceding J . Thus, there are2i i-witnesses for all i (1 i
m); moreover, these are pairwise disjoint and
e between intervals satisfying a0i and a1i , as depicted in Fig.
7. Note however that,ral, the i-witnesses will be a subset of the
subintervals of I satisfying a0i or a1i
formula 1 thus provides a succinct way of guaranteeing that at
least 2m1 propervals of I satisfy a0m in Aviz, every other
m-witness. A much greater challengeite a succinct collection of
formulas ensuring that no other proper subintervals offy a0m. This
task occupies the remainder of Section 6.1.1, . . . , bm, p
00, . . . , p
0m1 and p10, . . . , p1m1 be new event-atoms (i.e., pairwise
dis-
d distinct from z, a0 and the ahi ). Intuitively, the
event-atoms bi will be used toadditional a0i -events and a
1i -events slipping in between successive i-witnesses;
t-atoms p0j and p1j will function as nails, holding the whole
rickety structure to-Let 2 be the conjunction of the following
formulas, where 0 i < m, 0 h 1h 1:
hi ]{bi+1}z, [ah
i ]{phi }z, [ahi+1]phi ,
i+1]phi , [phi ]a1hi+2 .
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26 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
Fig. 8.
SupposI I J satisa1i+2, anincludesatisfiedJ Kdepicted
Let qof the fo
(59)[b[b
Suppos(1 i li, {li}
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28 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
Further, for all i (1 i m), let Li be the unique proper
subinterval of I satisfying li .Then the conjuncts in the second
row of (60) ensure that Ji ends before Li begins, and,moreover, Ji
is the only subinterval of I satisfying either a0 or a1 which ends
before Libegins.
Symjunction
{a{r
Let 4
Claim 3or a1i ca
We alet 1, .1 i
(61) [a
ClaimproperFig. 7. H
Proof.that noand J sproperlyJ endsbefore tand 3 rua1i are i
As asecutive(1 i meral isLet 6
(62) [a
Claim 5these ini i
In particular, no subinterval of I satisfying either a0i or a1i
ends before Ji ends.metrically, let r1, . . . , rm, r 1, . . . , r
m+1 be new event-atoms, and let r4 be the con-of the following
collection of formulas, where i (1 i m):
0}r 1, {r i}{a1i }r i+1,i}{a1i }{a1i }z, {ri}>a0i .be l4 r4 .
We thus have:
. If A |=I 1 4 and 1 i m, then no subinterval of I satisfying
either a0in end before the first i-witness ends or begin after the
last i-witness begins.
re now ready to achieve the main task of Section 6.1. Fix n >
0. Set m = 2n + 1,. . , 4 be as above, and let 5 be the conjunction
of the following formulas, wherem, 0 h 1 and 0 h 1:
hi ]ah
i .
4. Let A |=I 1 5. Then, for all i (0 i m), there exist exactly
2isubintervals of I satisfying either a0i or a1i . These intervals
are arranged as in
ence, there are exactly 22n proper subintervals of I satisfying
a0m.
Suppose 0 i m. Certainly, there are exactly 2i i-witnesses. It
suffices to showother proper subinterval of I satisfies a0i or a1i
. Suppose, for contradiction, J I atisfies ahi , but J is not an
i-witness. By 5, J neither properly includes nor is
included in any i-witness. Hence, the following possibilities
are exhaustive: (i)before the first i-witness ends; (ii) J begins
after one i-witness begins and endshe next one ends; and (iii) J
begins after the last i-witness begins. But Claims 2le out all
these possibilities. Hence, all proper subintervals of I satisfying
a0i or
-witnesses.
final trick, we show how the 22n a0m-intervals identified in
Claim 4 can be con-ly numbered. Let d01 , . . . , d
0m1, d11 , . . . , d1m1 be new event-atoms. Think of d
hi
m 1, 0 h 1) as stating that the ith digit in a certain (m
1)-digit binary nu-h, where the first digit is the most significant
and the (m1)th the least significant.
be the conjunction of the following formulas, where 1 i < m
and 0 h 1:hi ][a0m]dhi , [a0m](d0i d1i ).
. Suppose A |=I 1 6, and consider the 22n m-witnesses which
satisfy a0m. Lettervals be numbered in order of temporal precedence
as J0, . . . , J22n1. For all k
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
29
(0 k < 22n), and all i (1 i 2n) denote the ith digit in the
2n-digit binary numeralfor k (counting the most significant as the
first) by k[i]. Then we have:
k[{
Proof.
Let umore su
J0, . . . ,index bthe 2n-d
6.2. Or
Groublock asJ in thigrid inensurinthat J i
Contlet 7 b
(63) [a
Fig. 9 il1 witness
Nowor f 10 , a1 and 0
(64) f[fi] = 1 iff A |=Jk d1i
0 iff A |=Jk d0i .
By formula 6 and inspection of Fig. 7.
s refer to the 22n a0m-intervals identified in Claim 4 as tiles,
and let us write a0mggestively as t . We continue to denote the
tiles in order of temporal precedence asJ22n1, and we say that Jk
(0 k < 22n) has index k. If J is any tile, denote itsy kJ . In
that case, Claim 5 lets us read A |=J dhi as saying that the ith
digit inigit binary representation of kJ is h.
ganizing the tiles into a grid
p the 22n tiles into 2n blocks, each containing 2n consecutive
tiles. Regarding eacha row gives us a 2n 2n grid. If J and J are
tiles, then J lies immediately above
s grid in case kJ = kJ + 2n; similarly, J lies immediately to
the right of J in thecase kJ = kJ + 1 and the last n bits of kJ are
not all 1s. We now write formulasg that, for all tiles J , J such
that kJ = kJ + 2n, we can identify an interval L suchs the first
tile included in L and J is the last.inuing to write m for 2n+ 1,
let g01, . . . , g0m, g11, . . . , g1m, be new event-atoms, ande
the conjunction of the following formulas, where 0 i < m and 0 h
1:hi ]{g0i+1}>a0i+1, [ahi ]{g1i+1}a1i+1.
lustrates how the g0i+1- and g1i+1-intervals are arranged under
an i-witness ifA |=I7. It helps to think of the ghi -intervals as
short intervals separating consecutive i-es.let f0, f 01 , . . . ,
f
02n, f
11 , . . . , f
12n be new event-atoms, write f
0 alternatively as f 00nd let 8 be the conjunction of the
following formulas, where 0 i < 2n, 0 h h 1:
0, [f h2n]ah2n, [f h2n]{(ah2n)f }
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30 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
To motdistribuof everyequal tointerval
Suppproper sahi -interintervalthe 2n dL1 all i (1intervalk[i]
and[f hi ]{(astart ofintervalof K , anthat K iKi K
Claim 6that L iis Jk .
Proof.cluded i
In th 8each ofJk , wheFig. 10. Arrangement of f hi
- and f hi+1-intervals.
ivate this construction, it helps to imagine the f hi -intervals
guaranteed by 8 asted similarly to the corresponding ahi -intervals
in Fig. 7, except that the end-pointf hi -interval is shifted right
by a large fixed amountspecifically, an amountthe time occupied by
2n consecutive tiles. Fig. 10 illustrates how f hi - and f
hi+1-
s are arranged in such an interpretation.ose A |=I 1 8. Ignoring
for the moment all intervals which are notubintervals of I , any f
hi -interval (1 i 2n, 0 h 1) properly includes anval; and any f hi
-interval (1 i < 2n, 0 h 1) properly includes a unique f
0i+1-and a unique f 1i+1-interval. Now let k be an integer with 0 k
< 2n, and denoteigits of k by k[i] (1 i 2n) as in Claim 5. Then
we can form a chain of intervals L2n such that, for all i (1 i 2n),
Li is an f k[i]i -interval. Moreover, for i 2n), Li properly
includes some ak[i]i -interval; so let Ki be the first such. We
claim that K1 K2n. To see this, suppose 1 i < 2n, and write h
forh for k[i + 1]. Let K be the unique ahi+1-interval properly
included in Ki . From
hi )
f }
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
31
To aid rinto log
As athe follo
(65) [t]
The purintegers(i) for afor anysuch thai < j formula
(66)
If A |=Isatisfyinthe firstment ofto estabare anal
6.3. En
We a
TheoreFig. 11. Arrangement of event-atoms indicating vertical
neighbourhood in the grid.
eadability, we occasionally employ T PL+-formulas in the sequel;
their conversionically equivalent T PL-formulas is completely
routine.preliminary, let d, d1 , . . . , d2n be new event-atoms,
and 9 be the conjunction ofwing formulas, where 1 i n:(d 1jnd0j ),
[t](di (d0i i
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32 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
Proof. Let (C,H,V ) be any exponential tiling problem and c0, .
. . , cn1 an instance ofsize n. Construct the formulas 1, . . . ,14
as above. If C = {c0, . . . , cM1}, take the cj(0 j
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
33
that no linguistically motivated tightening of our fragment T PE
could affect the abovecomplexity result.
At first sight, this seems an impossible demand, since we cannot
know in advancewhat reproof stawkwarcrease tmust guresult. Athat
proV andthey casentencincludethe NEXening o
The
{aBut thes
(67) Du(68) Du(69) Du
of(70) Du
For addthis faci
Form
{lthese fo
(71) Du(72) Du(73) a1i(74) Be
Form
[athese arfinements might be made to our English grammar.
However, it turns out that therategy employed above yields an easy
solution. Obviously, eliminating marginal ord sentences from T PE
can only cause the fragment to contract, and so cannot in-he
computational complexity of its satisfiability problem. The only
possibility weard against is that such a contraction might
invalidate the NEXPTIME-hardnessnd this is where the details of the
proof of that result come to our rescue. For
of depends on the encoding of tiling problems by the formulas
114, T , H ,I . All we need do then is examine these formulas one
by one and check that
n be generated, using the grammar presented above, by good,
idiomatic Englishes. If so, we know that any linguistically
motivated restrictions on T PL will stillthese sentences, and will
assign them the advertised satisfaction-conditions.
Thus,PTIME-completeness result will still apply to any
linguistically motivated tight-
f the grammar.formulas 13, given in (57)(59), consist of
conjuncts of the forms0}z, [ahi+1]phi , [ahi ]{a0i+1}>a1i+1,
[ahi ]{a1i+1}z.e formulas express the meanings of the
unobjectionable T PE-sentences
ring the occurrence of a0, z occurredring every occurrence of
ahi+1, p
hi occurred
ring every occurrence of ahi , a1i+1 occurred after the
occurrence
a0i+1ring every occurrence of ahi , z occurred during the
occurrence of a
1i+1.
ed naturalness, we have fronted one preposition-phrase in each
of these sentences;lity could easily be incorporated into our
grammar, of course.ula l4, given in (60), additionally involves
conjuncts of the formsi}{a0i }li+1, {li}{a0i }>li, {li}
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34 I. Pratt-Hartmann / Artificial Intelligence 166 (2005)
136
(75) During every occurrence of ahi , ah
i did not occur(76) During every occurrence of ahi , dhi
occurred during every occurrence of a0m(77) During every occurrence
of a0 , either d0 did not occur or d1 did not occur.
For addincorpo
Formaddition
fthese ar
(78) f0(79) Du(80) Du
In thexpress
[tfor vari
(81) Du
(82) Du
These snot to brelevantthree diH , Vtightenithe satipaper is
7. Con
In thing intecompleT PE aductionof the em i i
ed naturalness, we have helped ourselves to the word either,
which could be easilyrated into our grammar.ula 7, given in (63),
presents no new difficulties. Formula 8, given in (64),ally
involves conjuncts of the forms0, [f hi ]{(ahi )f }
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I. Pratt-Hartmann / Artificial Intelligence 166 (2005) 136
35
temporal prepositionsput at speakers disposal. By the standards
of most interval tem-poral logics, T PL has low complexity. In the
search for logics of limited expressivepower, fragments owing their
salience to the syntax of natural language are a good placeto
look.
We ea reason
aims arbiologymaticaldelicatespeakerin whatthe lingremarkahas
beeto coveeffectslogic, re
Append
Syntax
S S,S S,S S0S NeS1D SS1! S
Closed-
Det Det Det! Neg Conj Conj ndeavoured throughout to be faithful
to the facts of English usage while retainingably perspicuous
formal system, amenable to mathematical analysis. These two
e to some extent antagonistic, of course. Natural languages are
products of humanand human civilization, and as such do not always
admit of a comfortable mathe-description. Thus, even the simple
fragment of English considered here skirts manyissues of syntax,
and includes sentences about whose exact semantics even native
s are uncertain. In this situation, we have occasionally had to
legislate, sometimesever way is mathematically most convenient.
Nevertheless, while faithfulness touistic data is a virtue, it is
all too easy, in pursuit of this virtue, to lose sight of theble
logical regularity of the constructions studied here; and it is
this regularity that
n the focus of our attention. To what extent this analysis can
be usefully extendedr other temporal constructions in English (and
other natural languages), and whatsuch extensions will have on the
complexity of satisfiability in the accompanyingmain open.
ix A. The grammar rules for T PE
PPConj, S
g, S00
0, OAdv
NPD DetD , N1DN1D N0N1! OAdj, N0PP PN,D , NPDPP PS,D , S1D
Open-class lexicon
S0 an interrupt was received/int-recS0 the main process
ran/main. . .
N0 cycle/cycN0 run of the main process/main. . .
class lexicon
every/[ ]some/ the/{ }not/and/or/
OAdj first/fOAdj last/lPN,D during/=
PN,! until/