Temporal Prepositions and Their Logic

Artificial Intelligence 166 (2005) 136www.elsevier.com/locate/artint

Abstrac

A fraglast is designs tosatisfactiis showncomplexconstruc 2005 E

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

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.


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


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 ).


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


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


(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.


(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


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.


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/


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.


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/(


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.


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


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 } ,


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] | {} | {}> | {} | {}


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.


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}


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.


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 .


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,


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 .


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


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 .


Fig. 8.

SupposI I J satisa1i+2, anincludesatisfiedJ Kdepicted

Let qof the fo

(59)[b[b

Suppos(1 i li, {li}


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


(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 }


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 }


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


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


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}


(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 }


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/

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Temporal Prepositions and Their Logic