**From:** Jerry Hobbs (*hobbs@ai.sri.com*)

**Date:** 05/02/02

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Principal changes over previous version: I added instants at plus and minus infinity, as in SUMO. I rearranged things in the section on "before" (2.2) and made the property of Convexity explicit. I augmented the treatment of the interval calculus in ways suggested by George Ferguson, and added an esoteric note comparing Convexity, Total Ordering, and "Extensional Collapse". I redid the treatment of Hath (3.2), breaking it up into treatments of concatenation and Hath. I eliminated the attempt to build granularity into the treatment of Hath. I've clarified the section on time zones along the lines of my response to Pat Hayes. A DAML Ontology of Time 1. Introduction A number of sites, DAML contractors and others, have developed ontologies of time (e.g., DAML-S, Cycorp, CMU, Kestrel, Teknowledge). A group of us have decided to collaborate to develop a representative ontology of time for DAML, which could then be used as is or elaborated on by others needing such an ontology. It is hoped that this collaboration will result in an ontology that will be adopted much more widely than any single site's product would be. We envision three aspects to this effort: 1. An abstract characterization of the concepts and their properties, expressed in first-order predicate calculus. 2. A translation of the abstract ontology into DAML code, to whatever extent possible given the current state of DAML expressivity. 3. Mappings between the DAML ontology and individual sites' ontologies. DAML is under development and is thus a moving target, and that is why separating 1 and 2 is desirable. Level 1 can stabilize before DAML does. A mapping in 3 may be an isomorphism, or it may be something more complicated. The reason for 3 is so DAML users can exploit the wide variety of resources for temporal reasoning that are available. Moreover, it will aid the widespread use of the ontology if it can be linked easily to, for example, the temporal portion of Teknowledge's IEEE Standard Upper Ontology effort or to Cycorp's soon-to-be widely used knowledge base. The purposes of the temporal ontology are both for expressing temporal aspects of the contents of web resources and for expressing time-related properties of web services. The following document outlines the principal features of a representative DAML ontology of time. It is informed by ontology efforts at a number of sites and reflects but elaborates on a tentative consensus during discussions at the last DAML meeting. The first three areas are spelled out in significant detail. The last three are just sketches of work to be done. There are a number of places where it is stated that the ontology is silent about some issue. This is done to avoid controversial choices in the ontolgy where more than one treatment would be reasonable and consistent. Often these issues involve identifying a two-dimensional entity and a one-dimensional entity with one another. In general, functions are used where they are total and have a unique value; predicates are used otherwise. The order of arguments usually follows the subject-object-object of preposition order in the most natural use in an English sentence (except for "Hath", where topicalization applies). A note on notation: Conjunction (&) takes precedence over implication(-->) and equivalence (<-->). Formulas are assumed to be universally quantified on the variables appearing in the antecedent of the highest-level implication. Thus, p1(x) & p2(y) --> q1(x,y) & q2(y) is to be interpreted as (A x,y)[[p1(x) & p2(y)] --> [q1(x,y) & q2(y)]] At the end of each section there is a subsection on MAPPINGS. These are sketches of the relations between some highly developed temporal ontologies and the one outlined here. 2. Topological Temporal Relations 2.1. Instants and Intervals: There are two subclasses of temporal-entity: instant and interval. instant(t) --> temporal-entity(t) interval(T) --> temporal-entity(T) (In what follows, lower case t is used for instants, upper case T for intervals and for temporal-entities unspecified as to subtype. This is strictly for the reader's convenience, and has no formal significance.) start-of and end-of are functions from temporal entities to instants. temporal-entity(T) --> instant(start-of(T)) temporal-entity(T) --> instant(end-of(T)) For convenience, we can say that the start and end of an instant is itself. instant(t) --> start-of(t) = t instant(t) --> end-of(t) = t inside is a relation between an instant and an interval. inside(t,T) --> instant(t) & interval(T) This concept of inside is not intended to include starts and ends of intervals, as will be seen below. Infinite and half-infinite intervals can be handled by positing time instants at positive and negative infinity -- *PosInf* and *NegInf* -- and using them as start and end points. It will be useful in characterizing clock and calendar terms to have a relation between instants and intervals that says that the instant is inside or the start of the interval. in-interval(t,T) <--> start-of(T) = t v inside(t,T) time-between is a relation among a temporal entity and two instants. time-between(T,t1,t2) --> temporal-entity(T) & instant(t1) & instant(t2) The two instants are the start and end points of the temporal entity. time-between(T,t1,t2) <--> start-of(T) = t1 & end-of(T) = t2 The ontology is silent about whether the time from t to t, if it exists, is identical to the instant t. The ontology is silent about whether intervals _consist of_ instants. The ontology is silent about whether intervals are uniquely determined by their starts and ends. We can define a proper-interval as one whose start and end are not identical. proper-interval(t) <--> interval(t) & start-of(t) =/= end-of(t) The ontology is silent about whether there are any intervals that are not proper intervals. 2.2. Before: There is a before relation on temporal entities, which gives directionality to time. If temporal-entity T1 is before temporal-entity T2, then the end of T1 is before the start of T2. Thus, before can be considered to be basic to instants and derived for intervals. before(T1,T2) <--> before(end-of(T1),start-of(T2)) The before relation is anti-reflexive, anti-symmetric and transitive. before(T1,T2) --> T1 =/= T2 before(T1,T2) --> ~before(T2,T1) before(T1,T2) & before(T2,T3) --> before(T1,T3) Negative infinity is before every other instant and positive infinity is after every other instant. t =/= *NegInf* --> before(*NegInf*,t) t =/= *PosInf* --> before(t,*PosInf*) The end of an interval is not before the start of the interval. interval(T) --> ~before(end-of(T),start-of(T)) The start of a proper interval is before the end of the interval. proper-interval(T) --> before(start-of(T),end-of(T)) The converse of this is a theorem. If one instant is before another, there is a time between them. instant(t1) & instant(t2) & before(t1,t2) --> (E T) time-between(T,t1,t2) The ontology is silent about whether there is a time from t to t. If an instant is inside a proper interval, then the start of the interval is before the instant, which is before the end of the interval. This is the principal property of "inside". inside(t,T) --> before(start-of(T),t) & before(t,end-of(T)) The converse of this condition is called Convexity and may be stronger than some users will want if they are modeling time as a partial ordering. (See Esoteric Note in Section 2.3.) before(start-of(T),t) & before(t,end-of(T)) --> inside(t,T) In the rest of this development we will point it out whenever any concept or property depends on Convexity. Convexity implies that intervals are contiguous with respect to the before relation, in that an instant between two other instants inside an interval is also inside the interval. before(t1,t2) & before(t2,t3) & inside(t1,T) & inside(t3,T) --> inside(t2,T) The relation "after" is defined in terms of "before". after(T1,T2) <--> before(T2,T1) The ontology is silent about whether time is linearly ordered. Thus it supports theories of time, such as the branching futures theory, which conflate time and possibility or knowledge. The ontology is silent about whether time is dense, that is, whether between any two instants there is a third instant. Thus it supports theories in which time consists of discrete instants. 2.3. Interval Relations: The relations between intervals defined in Allen's temporal interval calculus (Allen, 1984; Allen and Kautz, 1985; Allen and Hayes, 1989; Allen and Ferguson, 1997) can be defined in a straightforward fashion in terms of before and identity on the start and end points. The standard interval calculus assumes all intervals are proper, and we will do that here. proper-interval(T1) & proper-interval(T2) --> [int-equals(T1,T2) <--> start-of(T1) = start-of(T2) & end-of(T1) = end-of(T2)] proper-interval(T1) & proper-interval(T2) --> [int-before(T1,T2) <--> before(T1,T2) proper-interval(T1) & proper-interval(T2) --> [int-after(T1,T2) <--> after(T1,T2) proper-interval(T1) & proper-interval(T2) --> [int-meets(T1,T2) <--> end-of(T1) = start-of(T2) proper-interval(T1) & proper-interval(T2) --> [int-met-by(T1,T2) <--> int-meets(T2,T1)] proper-interval(T1) & proper-interval(T2) --> [int-overlaps(T1,T2) <--> before(start-of(T1),start-of(T2)) & before(start-of(T2),end-of(T1)) & before(end-of(T1),end-of(T2))] proper-interval(T1) & proper-interval(T2) --> [int-overlapped-by(T1,T2) <--> int-overlaps(T2,T1)] proper-interval(T1) & proper-interval(T2) --> [int-starts(T1,T2) <--> start-of(T1) = start-of(T2) & before(end-of(T1),end-of(T2)] proper-interval(T1) & proper-interval(T2) --> [int-started-by(T1,T2) <--> int-starts(T2,T1)] proper-interval(T1) & proper-interval(T2) --> [int-during(T1,T2) <--> (before(start-of(T2),start-of(T1)) & before(end-of(T1),end-of(T2))] proper-interval(T1) & proper-interval(T2) --> [int-contains(T1,T2) <--> int-during(T2,T1)] proper-interval(T1) & proper-interval(T2) --> [int-finishes(T1,T2) <--> before(start-of(T2),start-of(T1)) & end-of(T1) = end-of(T2)] proper-interval(T1) & proper-interval(T2) --> [int-finished-by(T1,T2) <--> int-finishes(T2,T1)] The constraints on the arguments of these relations is that they be proper intervals. We only need to state these for half the relations, as the other half follow from the definitions. int-equals(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-before(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-meets(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-overlaps(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-starts(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-during(T1,T2) --> proper-interval(T1) & proper-interval(T2) int-finishes(T1,T2) --> proper-interval(T1) & proper-interval(T2) In addition, it will be useful below to have a single predicate for "starts or is during". This is called "int-in". int-in(T1,T2) <--> [int-starts(T1,T2) v int-during(T1,T2)] It will also be useful to have a single predicate for intervals intersecting in at most an instant. int-disjoint(T1,T2) <--> [int-before(T1,T2) v int-after(T1,T2) v int-meets(T1,T2) v int-met-by(T1,T2)] So far, the concepts and axioms in the ontology of time would be appropriate for scalar phenomena in general. In the standard development of interval calculus, it is assumed that any intervals that are int-equals are identical. That is, intervals are uniquely determined by their start and end points. We can call this the property of Extensional Collapse. int-equals(T1,T2) --> T1 = T2 If we think of different intervals between the end points as being different ways the start can lead to the end, then Extensional Collapse can be seen as collapsing all these into a single "before" relation. In the rest of this development we will point it out whenever any concept or property depends on Extensional Collapse. Esoteric Note: Convexity, Extensional Collapse, and Total Ordering are independent properties. This can be seen by considering the following four models based on directed graphs, where the arcs define the before relation: 1. An interval is any subset of the paths between two nodes. (For example, time is partially ordered and an interval is any path from one node to another.) 2. An interval is the complete set of paths between two nodes. 3. An interval consists of the start and end nodes and all the arcs between the start and end nodes but no intermediate nodes. So inside(t,T) is never true. (This is a hard model to motivate.) 4. The instants are a set of discrete, linearly ordered nodes. There are multiple arcs between the nodes. The intervals are paths from one node to another, including the nodes. (For example, the instants may be the successive states in the situation calculus and the intervals sequences of actions mapping one state into the next. Different actions can have the same start and end states.) Model 1 has none of the three properties. Model 2 has Convexity and Extensional Collapse, but is not Totally Ordered. Model 3 is Totally Ordered and has Extensional Collapse but not Convexity. Model 4 is Totally Ordered and Convex, but lacks Extensional Collapse. 2.4. Linking Time and Events: The time ontology links to other things in the world through four predicates -- at-time, during, holds, and time-span-of. We assume that another ontology provides for the description of events -- either a general ontology of event structure abstractly conceived, or specific, domain-dependent ontologies for specific domains. The term "eventuality" will be used to cover events, states, processes, propositions, states of affairs, and anything else that can be located with respect to time. The possible natures of eventualities would be spelled out in the event ontologies. The predicate at-time relates an eventuality to an instant, and is intended to say that the eventuality holds, obtains, or is taking place at that time. at-time(e,t) --> instant(t) The predicate during relates an eventuality to an interval, and is intended to say that the eventuality holds, obtains, or is taking place during that interval. during(e,T) --> interval(T) If an eventuality obtains during an interval, it obtains at every instant inside the interval. during(e,T) & inside(t,T) --> at-time(e,t) Whether a particular process is viewed as instantaneous or as occuring over an interval is a granularity decision that may vary according to the context of use, and is assumed to be provided by the event ontology. Often the eventualities in the event ontology are best thought of as propositions, and the relation between these and times is most naturally called "holds". holds can be defined in terms of at-time and during: holds(e,t) & instant(t) <--> at-time(e,t) holds(e,T) & interval(T) <--> during(e,T) The event ontology may provide other ways of linking events with times, for example, by including a time parameter in predications. p(x,t) This time ontology provides ways of reasoning about the t's; their use as arguments of predicates from another domain would be a feature of the ontology of the other domain. The predicate time-span-of relates eventualities to instants or intervals. For contiguous states and processes, it tells the entire instant or interval for which the state or process obtains or takes place. time-span-of(T,e) --> temporal-entity(T) time-span-of(T,e) & interval(T) --> during(e,T) time-span-of(t,e) & instant(t) --> at-time(e,t) time-span-of(T,e) & interval(T) & ~inside(t1,T) & ~start-of(t1,T) & ~end-of(t1,T) --> ~at-time(e,t1) time-span-of(t,e) & instant(t) & t1 =/= t --> ~at-time(e,t1) time-span-of is a predicate rather than a function because until the time ontology is extended to aggregates of temporal entities, the function would not be defined for noncontiguous eventualities. Whether the eventuality obtains at the start and end points of its time span is a matter for the event ontology to specify. The silence here on this issue is the reason time-span-of is not defined in terms of necessary and sufficient conditions. The event ontology could extend temporal functions and predicates to apply to events in the obvious way, e.g., ev-start-of(e) = t <--> time-span-of(T,e) & start-of(T) = t This would not be part of the time ontology, but would be consistent with it. Different communities have different ways of representing the times and durations of states and events (processes). In one approach, states and events can both have durations, and at least events can be instantaneous. In another approach, events can only be instantaneous and only states can have durations. In the latter approach, events that one might consider as having duration (e.g., heating water) are modeled as a state of the system that is initiated and terminated by instantaneous events. That is, there is the instantaneous event of the start of the heating at the start of an interval, that transitions the system into a state in which the water is heating. The state continues until another instantaneous event occurs---the stopping of the heating at the end of the interval. These two perspectives on events are straightforwardly interdefinable in terms of the ontology we have provided. This is a matter for the event ontology to specify. This time ontology is neutral with respect to the choice. MAPPINGS: Teknowledge's SUMO has pretty much the same ontology as presented here, though the names are slightly different. An instant is a TimePoint, an interval is a TimeInterval, start-of is BeginFn, ans so on. SUMO implements the Allen calculus. Cyc has functions #startingPoint and #endingPoint that apply to intervals, but also to eventualities. Cyc implements the Allen calculus. Cyc uses a holdIn predicate to relate events to times, but to other events as well. Cyc defines a very rich set of derived concepts that are not defined here, but could be. For instant Kestral uses Time-Point, for interval they use Time-Interval, for start-of they use start-time-point, and so on. PSL axiomatizes before as a total ordering. 3. Measuring Durations 3.1. Temporal Units: This development assumes ordinary arithmetic is available. There are at least two approaches that can be taken toward measuring intervals. The first is to consider units of time as functions from Intervals to Reals, e.g., minutes: Intervals --> Reals minutes([5:14,5:17)) = 3 The other approach is to consider temporal units to constitute a set of entities -- call it TemporalUnits -- and have a single function _duration_ mapping Intervals x TemporalUnits into the Reals. duration: Intervals x TemporalUnits --> Reals duration([5:14,5:17), *Minute*) = 3 The two approaches are interdefinable: seconds(T) = duration(T,*Second*) minutes(T) = duration(T,*Minute*) hours(T) = duration(T,*Hour*) days(T) = duration(T,*Day*) weeks(T) = duration(T,*Week*) months(T) = duration(T,*Month*) years(T) = duration(T,*Year*) Ordinarily, the first is more convenient for stating specific facts about particular units. The second is more convenient for stating general facts about all units. The constraints on the arguments of duration are as follows: duration(T,u) --> proper-interval(T) & temporal-unit(u) The aritmetic relations among the various units are as follows: seconds(T) = 60 * minutes(T) minutes(T) = 60 * hours(T) hours(T) = 24 * days(T) days(T) = 7 * weeks(T) months(T) = 12 * years(T) The relation between days and months (and, to a lesser extent, years) will be specified as part of the ontology of clock and calendar below. On their own, however, month and year are legitimate temporal units. In this development durations are treated as functions on intervals and units, and not as first class entities on their own, as in some approaches. In the latter approach, durations are essentially equivalence classes of intervals of the same length, and the length of the duration is the length of the members of the class. The relation between an approach of this sort (indicated by prefix D-) and the one presented here is straightforward. (A T,u,n)[duration(T,u) = n <--> (E d)[D-duration-of(T) = d & D-duration(d,u) = n]] At the present level of development of the temporal ontology, this extra layer of representation seems superfluous. It may be more compelling, however, when the ontology is extended to deal with the combined durations of noncontiguous aggregates of intervals. 3.2. Concatenation and Hath: The multiplicative relations above don't tell the whole story of the relations among temporal units. Temporal units are _composed of_ smaller temporal units. A larger temporal unit is a concatenation of smaller temporal units. We will first define a general relation of concatenation between an interval and a set of smaller intervals. Then we will introduce a predicate "Hath" that specifies the number of smaller unit intervals that concatenate to a larger interval. Concatenation: A proper interval x is a concatenation of a set S of proper intervals if and only if S covers all of x, and all members of S are subintervals of x and are mutually disjoint. (The third conjunct on the right side of <--> is because in-interval covers only start-of and inside.) concatenation(x,S) <--> proper-interval(x) & (A z)[in-interval(z,x) --> (E y)[member(y,S) & in-interval(z,y)]] & (A z)[end-of(x) = z --> (E y)[member(y,S) & end-of(y) = z] & (A y)[member(y,S) --> [int-starts(y,x) v int-during(y,x) v int-finishes(y,x)]] & (A y1,y2)[member(y1,S) & member(y2,S) --> [y1=y2 v int-disjoint(y1,y2)]] The following properties of "concatenation" can be proved as theorems: There are elements in S that start and finish x: concatenation(x,S) --> (E! y1)[member(y1,S) & int-starts(y1,x)] concatenation(x,S) --> (E! y2)[member(y2,S) & int-finishes(y2,x)] Except for the first and last elements of S, every element of S has elements that precede and follow it. These theorems depend on the property of Convexity. concatenation(x,S) --> (A y1)[member(y1,S) --> [int-finishes(y1,x) v (E! y2)[member(y2,S) & int-meets(y1,y2)]]] concatenation(x,S) --> (A y2)[member(y2,S) --> [int-starts(y2,x) v (E! y1)[member(y1,S) & int-meets(y1,y2)]]] The uniqueness (E!) follows from int-disjoint. Hath: The basic predicate used here for expressing the composition of larger intervals out of smaller temporal intervals of unit length is "Hath", from statements like "30 days hath September" and "60 minutes hath an hour." Its structure is Hath(N,u,x) meaning "N proper intervals of duration one unit u hath the proper interval x." That is, if Hath(N,u,x) holds, then x is the concatenation of N unit intervals where the unit is u. For example, if x is some month of September then "Hath(30,*Day*,x)" would be true. "Hath" is defined as follows: Hath(N,u,x) <--> (E S)[card(S) = N & (A z)[member(z,S) --> duration(z,u) = 1] & concatenation(x,S)] That is, x is the concatenation of a set S of N proper intervals of duration one unit u. The type constraints on its arguments can be proved as a theorem: N is an integer (assuming that is the constraint on the value of card), u is a temporal unit, and x is a proper interval: Hath(N,u,x) --> integer(N) & temporal-unit(u) & proper-interval(x) **** Due to popular demand, I'm giving up trying to build in granularity at this point. If you want an interval to be at the finest granularity you are dealing with, you will not specify that it is concatenated out of smaller intervals. **** This treatment of concatenation will work for scalar phenomena in general. This treatment of Hath will work for measurable quantities in general. 3.3. The Structure of Temporal Units: We now define predicates true of intervals that are one temporal unit long. For example, "week" is a predicate true of intervals whose duration is one week. second(T) <--> seconds(T) = 1 minute(T) <--> minutes(T) = 1 hour(T) <--> hours(T) = 1 day(T) <--> days(T) = 1 week(T) <--> weeks(T) = 1 month(T) <--> months(T) = 1 year(T) <--> years(T) = 1 We are now in a position to state the relations between successive temporal units. minute(T) --> Hath(60,*Second*,T) hour(T) --> Hath(60,*Minute*,T) day(T) --> Hath(24,*Hour*,T) week(T) --> Hath(7,*Day*,T) year(T) --> Hath(12,*Month*,T) The relations between months and days are dealt with in Section 4.4. ***** MAPPINGS: Teknowledge's SUMO has some facts about the lengths of temporal units in terms of smaller units. Cyc reifies durations. Cyc's notion of time covering subsets aims at the same concept dealt with here with Hath. Kestrel uses temporal units to specify the granularity of the time representation. PSL reifies and axiomatizes durations. PSL includes a treatment of delays between events. This is not treated here, but could be. 4. Clock and Calendar 4.1. Time Zones: What hour of the day an instant is in is relative to the time zone. This is also true of minutes, since there are regions in the world, e.g., central Australia, where the hours are not aligned with GMT hours, but are, e.g., offset half an hour. Probably seconds are not relative to the time zone. Days, weeks, months and years are also relative to the time zone, since, e.g., 2002 began in the Eastern Standard time zone three hours before it began in the Pacific Standard time zone. Thus, predications about all clock and calendar intervals except seconds are relative to a time zone. This can be carried to what seems like a ridiculous extreme, but turns out to yield a very concise treatment. The Common Era (C.E. or A.D.) is also relative to a time zone, since 2002 years ago, it began three hours earlier in what is now the Eastern Standard time zone than in what is now the Pacific Standard time zone. What we think of as the Common Era is in fact 24 (or more) slightly displaced half-infinite intervals. (We leave B.C.E. to specialized ontologies.) The principal functions and predicates will specify a clock or calendar unit interval to be the nth such unit in a larger interval. The time zone need not be specified in this predication if it is already built into the nature of the larger interval. That means that the time zone only needs to be specified in the largest interval, that is, the Common Era; that time zone will be inherited by all smaller intervals. Thus, the Common Era can be considered as a function from time zones (or "time standards", see below) to intervals. CE(z) = T Fortunately, this counterintuitive conceptualization will usually be invisible and, for example, will not be evident in the most useful expressions for time, in Section 4.5 below. In fact, the CE predication functions as a good place to hide considerations of time zone when they are not relevant. Esoteric Aside: Strictly speaking the use of CE as a function depends on Extensional Collapse. If we don't want to assume that, then we can use a corresponding predicate -- CEPred(e,z) -- to mean era e is the Common Era in time zone z. We have been refering to time _zones_, but in fact it is more convenient to work in terms of what we might call the "time standard" that is used in a time zone. That is, it is better to work with *PST* as a legal entity than with the *PST* zone as a geographical region. A time standard is a way of computing the time, relative to a world-wide system of computing time. For each time standard, there is a zone, or geographical region, and a time of the year in which it is used for describing local times. Where and when a time standard is used have to be axiomatized, and this involves interrelating a time ontology and a geographical ontology. These relations can be quite complex. Only the entities like *PST* and *EDT*, the time standards, are part of the _time_ ontology. If we were to conflate time zones (i.e., geographical regions) and time standards, it would likely result in problems in several situations. For example, the Eastern Standard zone and the Eastern Daylight zone are not identical, since most of Indiana is on Eastern Standard time all year. The state of Arizona and the Navajo Indian Reservation, two overlapping geopolitical regions, have different time standards -- one is Pacific and one is Mountain. Time standards that seem equivalent, like Eastern Standard and Central Daylight, should be thought of as separate entities. Whereas they function the same in the time ontology, they do not function the same in the ontology that articulates time and geography. For example, it would be false to say those parts of Indiana shift in April from Eastern Standard to Central Daylight time. In this treatment it will be assumed there is a set of entities called time standards. Some relations among time standards are discussed in Section 4.5. 4.2. Clock and Calendar Units: The aim of this section is to explicate the various standard clock and calendar intervals. A day as a calender interval begins at and includes midnight and goes until but does not include the next midnight. By contrast, a day as a duration is any interval that is 24 hours in length. The day as a duration was dealt with in Section 3. This section deals with the day as a calendar interval. Including the start but not the end of a calendar interval in the interval may strike some as arbitrary. But we get a cleaner treatment if, for example, all times of the form 12:xx a.m., including 12:00 a.m. are part of the same hour and day, and all times of the form 10:15:xx, including 10:15:00, are part of the same minute. It is useful to have three ways of saying the same thing: the clock or calendar interval y is the nth clock or calendar interval of type u in a larger interval x. This can be expressed as follows for minutes: min(y,n,x) If the property of Extensional Collapse holds, then y is uniquely determined by n and x, it can also be expressed as follows: minFn(n,x) = y For stating general properties about clock intervals, it is useful also to have the following way to express the same thing: clock-int(y,n,u,x) This expression says that y is the nth clock interval of type u in x. For example, the proposition "clock-int(10:03,3,*Minute*,[10:00,11:00))" holds. Here u can be a member of the set of clock units, that is, one of *Second*, *Minute*, or *Hour*. In addition, there is a calendar unit function with similar structure: cal-int(y,n,u,x) This says that y is the nth calendar interval of type u in x. For example, the proposition "cal-int(12Mar2002,12,*Day*,Mar2002)" holds. Here u can be one of the calendar units *Day*, *Week*, *Month*, and *Year*. The unit *DayOfWeek* will be introduced below in Section 4.3. The relations among these modes of expression are as follows: sec(y,n,x) <--> secFn(n,x) = y <--> clock-int(y,n,*sec*,x) min(y,n,x) <--> minFn(n,x) = y <--> clock-int(y,n,*min*,x) hr(y,n,x) <--> hrFn(n,x) = y <--> clock-int(y,n,*hr*,x) da(y,n,x) <--> daFn(n,x) = y <--> cal-int(y,n,*da*,x) mon(y,n,x) <--> monFn(n,x) = y <--> cal-int(y,n,*mon*,x) yr(y,n,x) <--> yrFn(n,x) = y <--> cal-int(y,n,*yr*,x) Weeks and months are dealt with separately below. The am/pm designation of hours is represented by the function hr12. hr12(y,n,*am*,x) <--> hr(y,n,x) hr12(y,n,*pm*,x) <--> hr(y,n+12,x) Each of the calendar intervals is that unit long; a calendar year is a year long. sec(y,n,x) --> second(y) min(y,n,x) --> minute(y) hr(y,n,x) --> hour(y) da(y,n,x) --> day(y) mon(y,n,x) --> month(y) yr(y,n,x) --> year(y) A distinction is made above between clocks and calendars because they differ in how they number their unit intervals. The first minute of an hour is labelled with 0; for example, the first minute of the hour [10:00,11:00) is 10:00. The first day of a month is labelled with 1; the first day of March is March 1. We number minutes for the number just completed; we number days for the day we are working on. Thus, if the larger unit has N smaller units, the argument n in clock-int runs from 0 to N-1, whereas in cal-int n runs from 1 to N. To state properties true of both clock and calendar intervals, we can use the predicate cal-int and relate the two notions with the axiom cal-int(y,n,u,x) <--> clock-int(y,n-1,u,x) The type constraints on the arguments of cal-int are as follows: cal-int(y,n,u,x) --> interval(y) & integer(n) & temporal-unit(u) & interval(x) The temporal units are as follows: temporal-unit(*sec*), temporal-unit(*min*), temporal-unit(*hr*), temporal-unit(*da*), temporal-unit(*mon*), temporal-unit(*yr*) In addition, from below, temporal-unit(*dayofweek*), temporal-unit(*wk*) There are properties relating to the labelling of clock and calendar intervals. If N u's hath x and y is the nth u in x, then n is between 1 and N. cal-int(y,n,u,x) & Hath(N,u,x) --> 0 < n <= N There is a 1st small interval, and it starts the large interval. Hath(N,u,x) --> (E! y) cal-int(y,1,u,x) Hath(S,N,u,x) & cal-int(y,1,u,x) --> int-starts(y,x) There is an Nth small interval, and it finishes the large interval. Hath(N,u,x) --> (E! y) cal-int(y,N,u,x) Hath(N,u,x) & cal-int(y,N,u,x) --> int-finishes(y,x) All but the last small interval have a small interval that succeeds and is met by it. cal-int(y1,n,u,x) & Hath(N,u,x) & n < N --> (E! y2)[cal-int(y2,n+1,u,x) & int-meets(y1,y2)] All but the first small interval have a small interval that precedes and meets it. cal-int(y2,n,u,x & Hath(N,u,x) & 1 < n --> (E! y1)[cal-int(y1,n-1,u,x) & int-meets(y1,y2)] 4.3. Weeks A calendar week starts at midnight, Saturday night, and goes to the next midnight, Saturday night. It is independent of months and years. However, we can still talk about the nth week in some larger period of time, e.g., the third week of the month or the fifth week of the semester. So the same three modes of representation are appropriate for weeks as well. wk(y,n,x) <--> wkFn(n,x) = y <--> cal-int(y,n,*wk*,x) As it happens, the n and x arguments will often be irrelevant. A calendar week is one week long. wk(y,n,x) --> week(y) The day of the week is a temporal unit (*DayOfWeek*) in a larger interval, so the three modes of representation are appropriate here as well. dayofweek(y,n,x) <--> dayofweekFn(n,x) = y <--> cal-int(y,n,*dayofweek*,x) Whereas it makes sense to talk about the nth day in a year or the nth minute in a day or the nth day in a week, it does not really make sense to talk about the nth day-of-the-week in anything other than a week. Thus we can restrict the x argument to be a calendar week. dayofweek(y,n,x) --> (E n1,x1) wk(x,n1,x1) The days of the week have special names in English. dayofweek(y,1,x) <--> Sunday(y,x) dayofweek(y,2,x) <--> Monday(y,x) dayofweek(y,3,x) <--> Tuesday(y,x) dayofweek(y,4,x) <--> Wednesday(y,x) dayofweek(y,5,x) <--> Thursday(y,x) dayofweek(y,6,x) <--> Friday(y,x) dayofweek(y,7,x) <--> Saturday(y,x) For example, Sunday(y,x) says that y is the Sunday of week x. A day of the week is also a day of the month (and vice versa), and thus a day long. (A y)[[(E n,x) dayofweek(y,n,x)] <--> [(E n1,x1) da(y,n1,x1)]] One correspondance will anchor the cycle of weeks to the rest of the calendar, for example, saying that January 1, 2002 was the Tuesday of some week x. (A z)(E x) Tuesday(dayFn(1,monFn(1,yrFn(2002,CE(z)))),x) We can define weekdays and weekend days as follows: weekday(y,x) <--> [Monday(y,x) v Tuesday(y,x) v Wednesday(y,x) v Thursday(y,x) v Friday(y,x)] weekendday(y,x) <--> [Saturday(y,x) v Sunday(y,x)] As before, the use of the functions wkFn and dayofweekFn depend on Extensional Collapse. 4.4. Months and Years The months have special names in English. mon(y,1,x) <--> January(y,x) mon(y,2,x) <--> February(y,x) mon(y,3,x) <--> March(y,x) mon(y,4,x) <--> April(y,x) mon(y,5,x) <--> May(y,x) mon(y,6,x) <--> June(y,x) mon(y,7,x) <--> July(y,x) mon(y,8,x) <--> August(y,x) mon(y,9,x) <--> September(y,x) mon(y,10,x) <--> October(y,x) mon(y,11,x) <--> November(y,x) mon(y,12,x) <--> December(y,x) The number of days in a month have to be spelled out for individual months. January(m,y) --> Hath(31,*Day*,m) March(m,y) --> Hath(31,*Day*,m) April(m,y) --> Hath(30,*Day*,m) May(m,y) --> Hath(31,*Day*,m) June(m,y) --> Hath(30,*Day*,m) July(m,y) --> Hath(31,*Day*,m) August(m,y) --> Hath(31,*Day*,m) September(m,y) --> Hath(30,*Day*,m) October(m,y) --> Hath(31,*Day*,m) November(m,y) --> Hath(30,*Day*,m) December(m,y) --> Hath(31,*Day*,m) The definition of a leap year is as follows: (A z)[leap-year(y) <--> (E n,x)[year(y,n,(CE(z)) & [divides(400,n) v [divides(4,n) & ~divides(100,n)]]] We leave leap seconds to specialized ontologies. Now the number of days in February can be specified. February(m,y) & leap-year(y) --> Hath(29,*Day*,m) February(m,y) & ~leap-year(y) --> Hath(28,*Day*,m) A reasonable approach to defining month as a unit of temporal measure would be to specify that the start and end points have to be on the same days of successive months. The following rather ugly axiom captures this. month(T) <--> (E d1,d2,n,m1,m2,n1,y1,y2,n2,e) [in-interval(start-of(T),d1) & in-interval(end-of(T),d2) & da(d1,n,m1) & mon(m1,n1,y1) & yr(y1,n2,e) & da(d2,n,m2) & [mon(m2,n1+1,y1) v (E y2)[n1=12 & mon(m2,1,y2) & yr(y2,n2+1,e)]]] The last disjunct takes care of months spaning December and January. So the month as a measure of duration would be related to days as a measure of duration only indirectly, mediated by the calendar. It is possible to prove that months are between 28 and 31 days. To say that July 4 is a holiday in the United States one could write (A d,m,y)[da(d,4,m) & July(m,y) --> holiday(d,USA)] 4.5. Time Stamps: Standard notation for times list the year, month, day, hour, minute, and second. It is useful to define a predication for this. time-of(t,y,m,d,h,n,s,z) <--> in-interval(t,secFn(s,minFn(n,hrFn(h,daFn(d, monFn(m,yrFn(y,CE(z)))))))) Alternatively (and not assuming Extensional Collapse), time-of(t,y,m,d,h,n,s,z) <--> (E s1,n1,h1,d1,m1,y1,e) [in-interval(t,s1) & sec(s1,s,n1) & min(n1,n,h1) & hr(h1,h,d1) & da(d1,d,m1) & mon(m1,m,y1) & yr(y1,y,e) & CEPred(e,z)] For example, an instant t has the time 5:14:35pm PST, Wednesday, February 6, 2002 if the following properties hold for t: time-of(t,2002,2,6,17,14,35,*PST*) (E w,x)[in-interval(t,w) & Wednesday(w,x)] The second line says that t is in the Wednesday w of some week x. The relations among time zones can be expressed in terms of the time-of predicate. Two examples are as follows: h < 8 --> [time-of(t,y,m,d,h,n,s,*GMT*) <--> time-of(t,y,m,d-1,h+16,n,s,*PST*)] h >= 8 --> [time-of(t,y,m,d,h,n,s,*GMT*) <--> time-of(t,y,m,d,h-8,n,s,*PST*)] time-of(t,y,m,d,h,n,s,*EST*) <--> time-of(t,y,m,d,h,n,s,*CDT*) The time-of predicate will be convenient for doing temporal arithmetic. MAPPINGS: Teknowledge's SUMO distinguishes between durations (e.g., HourFn) and clock and calendar intervals (e.g., Hour). Time zones are treated as geographical regions. The treatment of dates and times via functions follows Cyc's treatment. Kestrel's roundabout attempts to state rather straightforward facts about the clock and calendar are an excellent illustration of the lack of expressivity in DAML+OIL. The ISO standard for dates and times can be represented straightforwardly with the time-of predicate or the unitFn functions. 5. Deictic Time Deictic temporal concepts, such as ``now'', ``today'', ``tomorrow night'', and ``last year'', are more common in natural language texts than they will be in descriptions of Web resources, and for that reason we are postponing a development of this domain until the first three are in place. But since most of the content on the Web is in natural language, ultimately it will be necessary for this ontology to be developed. It should, as well, mesh well with the annotation standards used in automatic tagging of text. We expect that the key concept in this area will be a relation _now_ between an instant or interval and an utterance or document. now(t,d) The concept of "today" would also be relative to a document, and would be defined as follows: today(T,d) <--> (E t,n,x)[now(t,d) & in-interval(t,T) & da(T,n,x)] That is, T is today with respect to document d if and only if there is an instant t in T that is now with respect to the document and T is a calendar day (and thus the nth calendar day in some interval x). Present, past and future can be defined in the obvious way in terms of now and before. Another feature of a treatment of deictic time would be an axiomatization of the concepts of last, this, and next on anchored sequences of temporal entities. 6. Aggregates of Temporal Entities A number of common expressions and commonly used properties are properties of sequences of temporal entities. These properties may be properties of all the elements in the sequence, as in ``every Wednesday'', or they may be properties of parts of the sequence, as in ``three times a week'' or ``an average of once a year''. We are also postponing development of this domain until the first three domains are well in hand. This may be the proper locus of a duration arithmetic, since we may want to know the total time an intermittant process is in operation. A reasonable development of this area would be in terms of the following subsections. 6.1. Describing Aggregates of Temporal Entities 6.2. Durations as Entities 6.3. Duration Arithmetic 6.4. Rates 7. Vague Temporal Concepts In natural language a very important class of temporal expressions are inherently vague. Included in this category are such terms as "soon", "recently", and "a little while". These require an underlying theory of vagueness, and in any case are probably not immediately critical for the Semantic Web. This area will be postponed for a little while. References Allen, J.F. (1984). Towards a general theory of action and time. Artificial Intelligence 23, pp. 123-154. Allen, James F., and Henry A. Kautz. 1985. ``A Model of Naive Temporal Reasoning'', {\it Formal Theories of the Commonsense World}, ed. by Jerry R. Hobbs and Robert C. Moore, Ablex Publishing Corp., pp. 251-268. Allen, J.F. and P.J. Hayes (1989). Moments and points in an interval-based temporal logic. Computational Intelligence 5, pp. 225-238. Allen, J.F. and G. Ferguson (1997). Actions and events in interval temporal logic. In Oliveiro Stock (ed.), Spatial and Temporal Reasoning, Kluwer Academic Publishers, pp. 205-245.

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