# DAML-Time

From: Jerry Hobbs (hobbs@isi.edu)
Date: 03/05/03

• Next message: pat hayes: "Re: DAML-Time"

Attached is the latest writeup on the DAML-Time ontology.  The
principal differences from previous versions are as follows:

1.  A more careful treatment of the two approaches to infinite time --
points at infinity or not.  Some changes elsewhere to accomodate
that.

2.  Selection of options in the ontology is now accomplished simply by
asserting a single proposition -- e.g., Total-Order() to get total
ordering.  The axioms for that option are conditionalized on these
propositions.

3.  There is a treatment of granularity.  Comments solitcited.

4.  There is a treatment of temporal aggregates.  Comments solicited.

The things that remain to be done in the abstract ontology are as
follows:

1.  Temporal arithmetic.

2.  Rates.

3.  Deictics like "now", "today", "ago", etc.

4.  Vague terms, like "short", "soon", "recent", etc.

The aim in the last two areas is to arrive at an ontology that will
support useful annotation of natural language in web pages.

You might also look at the DAML-Time web page:

http://www.cs.rochester.edu/~ferguson/daml/

-- Jerry

A DAML Ontology of Time

Jerry R. Hobbs

with contributions from

George Ferguson, James Allen, Richard Fikes, Pat Hayes,
Drew McDermott, Ian Niles, Adam Pease, Austin Tate, Mabry Tyson,
and Richard Waldinger

November 2002

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 one-dimensional
entity and a zero-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.

<axiom id="2.1-1">
instant(t) --> temporal-entity(t)
</axiom>

<axiom id="2.1-2">
interval(T) --> temporal-entity(T)
</axiom>

These are the only two subclasses of temporal entities.
<axiom id="2.1-3">
(A T)[temporal-entity(T) --> [instant(T) v interval(T)]]
</axiom>
(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.)

_begins_ and _ends_ are relations between instants and temporal
entities.
<axiom id="2.1-4">
begins(t,T) --> instant(t) & temporal-entity(T)
</axiom>

<axiom id="2.1-5">
ends(t,T) --> instant(t) & temporal-entity(T)
</axiom>

For convenience, we can say that the beginning and end of an instant is
itself.
<axiom id="2.1-6">
instant(t) --> begins(t,t)
</axiom>

<axiom id="2.1-7">
instant(t) --> ends(t,t)
</axiom>

The beginnings and ends of temporal entities, if they exist, are
unique.
<axiom id="2.1-8">
temporal-entity(T) & begins(t1,T) & begins(t2,T) --> t1=t2
</axiom>

<axiom id="2.1-9">
temporal-entity(T) & ends(t1,T) & ends(t2,T) --> t1=t2
</axiom>
As will be seen in Section 2.4, in one approach to infinite intervals,
a positively infinite interval has no end, and a negatively infinite
interval has no beginning.  Hence, we use the relations "begins" and
"ends" in the core ontology, rather than defining functions
"beginning-of" and "end-of", since the functions would not be total.
They can be defined in an extension of the core ontology that posits
instants at positive and negative infinity.

_inside_ is a relation between an instant and an interval.

<axiom id="2.1-10">
inside(t,T) --> instant(t) & interval(T)
</axiom>

This concept of inside is not intended to include beginnings and ends of
intervals, as will be seen below.

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 beginning of the interval.
<axiom id="2.1-11">
(A t,T)[begins-or-in(t,T) <--> [begins(t,T) v inside(t,T)]]
</axiom>

time-between is a relation among a temporal entity and two
instants.
<axiom id="2.1-12">
time-between(T,t1,t2)
--> temporal-entity(T) & instant(t1) & instant(t2)
</axiom>
The two instants are the beginning and end points of the temporal entity.
<axiom id="2.1-13">
time-between(T,t1,t2) <--> begins(t1,T) & ends(t2,T)
</axiom>

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 core ontology is silent about whether intervals are uniquely
determined by their beginnings and ends.  This issue is dealt with in
Section 2.4.

We can define a proper-interval as one whose beginning and end are not
identical.
<axiom id="2.1-14">
(A T)[proper-interval(T)
<--> interval(T)
& (A t1,t2)[begins(t1,T) & ends(t2,T) --> t1 =/= t2]]
</axiom>
A half-infinite or infinite interval, by this definition, is proper.

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 beginning of T2.
Thus, before can be considered to be basic to instants and derived for
intervals.

<axiom id="2.2-1">
(A T1,T2)[before(T1,T2)
<--> (E t1,t2)[ends(t1,T1) & begins(t2,T2) & before(t1,t2)]]
</axiom>

The before relation is anti-reflexive, anti-symmetric and transitive.

<axiom id="2.2-2">
before(T1,T2) --> T1 =/= T2
</axiom>
<axiom id="2.2-3">
before(T1,T2) --> ~before(T2,T1)
</axiom>
<axiom id="2.2-4">
before(T1,T2) & before(T2,T3) --> before(T1,T3)
</axiom>

The end of an interval is not before the beginning of the interval.
<axiom id="2.2-5">
interval(T) & begins(t1,T) & ends(t2,T) --> ~before(t2,t1)
</axiom>

The beginning of a proper interval is before the end of the interval.
<axiom id="2.2-6">
proper-interval(T) & begins(t1,T) & ends(t2,T)
--> before(t1,t2)
</axiom>

The converse of this is a theorem.

begins(t1,T) & ends(t2,T) & before(t1,t2)
--> proper-interval(T)

If one instant is before another, there is a time between them.

<axiom id="2.2-7">
instant(t1) & instant(t2) & before(t1,t2)
--> (E T) time-between(T,t1,t2)
</axiom>

The ontology is silent about whether there is a time from t to t.

If an instant is inside a proper interval, then the beginning of the
interval is before the instant, which is before the end of the
interval.  This is the principal property of "inside".
<axiom id="2.2-8">
inside(t,T) & begins(t1,T) & ends(t2,T)
--> before(t1,t) & before(t,t2)
</axiom>

The converse of this condition is called Convexity and is discussed in
Section 2.4.

The relation "after" is defined in terms of "before".

<axiom id="2.2-9">
after(T1,T2) <--> before(T2,T1)
</axiom>

The basic 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.  This issue
is discussed further in Section 2.4.

The basic 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.  This
issue is discussed further in Section 2.4.

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 relatively
straightforward fashion in terms of "before" and identity on the
beginning and end points.  It is a bit more complicated than the
reader might at first suspect, because allowance has to be made for
the possibility of infinite intervals.  Where one of the intervals
could be infinite, the relation between the end points has to be
conditionalized on their existence.

The standard interval calculus assumes all intervals are proper, and
we will do that here too. The definitions of the interval relations in
terms of "before" relations among their beginning and end points, when
they exist, are given by the following axioms.  In these axioms, t1
and t2 are the beginning and end of interval T1; t3 and t4 are the
beginning and end of T2.

<axiom id="2.3-1">
(A T1,T2)[int-equals(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (A t1)[begins(t1,T1) <--> begins(t1,T2)]
& (A t2)[ends(t2,T1) <--> ends(t2,T2)]]]
</axiom>

<axiom id="2.3-2">
int-before(T1,T2) <--> proper-interval(T1) & proper-interval(T2)
& before(T1,T2)
</axiom>

<axiom id="2.3-3">
(A T1,T2)[int-meets(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (E t)[ends(t,T1) & begins(t,T2)]]]
</axiom>

<axiom id="2.3-4">
(A T1,T2)[int-overlaps(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (E t2,t3)[ends(t2,T1) & begins(t3,T2)
& before(t3,t2)
& (A t1)[begins(t1,T1) --> before(t1,t3)]
& (A t4)[ends(t4,T2) --> before(t2,t4)]]]]
</axiom>

<axiom id="2.3-5">
(A T1,T2)[int-starts(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (E t2)[ends(t2,T1)
& (A t1)[begins(t1,T1) <--> begins(t1,T2)]
& (A t4)[ends(t4,T2) --> before(t2,t4)]]]]
</axiom>

<axiom id="2.3-6">
(A T1,T2)[int-during(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (E t1,t2)[begins(t1,T1) & ends(t2,T1)
& (A t3)[begins(t3,T2) --> before(t3,t1)]
& (A t4)[ends(t4,T2) --> before(t2,t4)]]]]
</axiom>
<axiom id="2.3-7">
(A T1,T2)[int-finishes(T1,T2)
<--> [proper-interval(T1) & proper-interval(T2)
& (E t1)[begins(t1,T1)
& (A t3)[begins(t3,T2) --> before(t3,t1)]
& (A t4)[ends(t4,T2) <--> ends(t4,T1))]]]]
</axiom>

The inverse interval relations can be defined in terms of these relations.
<axiom id="2.3-8">
int-after(T1,T2) <--> int-before(T2,T1)
</axiom>
<axiom id="2.3-9">
int-met-by(T1,T2) <--> int-meets(T2,T1)
</axiom>
<axiom id="2.3-10">
int-overlapped-by(T1,T2) <--> int-overlaps(T2,T1)
</axiom>
<axiom id="2.3-11">
int-started-by(T1,T2) <--> int-starts(T2,T1)
</axiom>
<axiom id="2.3-12">
int-contains(T1,T2) <--> int-during(T2,T1)
</axiom>
<axiom id="2.3-13">
int-finished-by(T1,T2) <--> int-finishes(T2,T1)
</axiom>

In addition, it will be useful below to have a single predicate for
"starts or is during".  This is called "starts-or-during".

<axiom id="2.3-14">
starts-or-during(T1,T2)
<--> [int-starts(T1,T2) v int-during(T1,T2)]
</axiom>

It will also be useful to have a single predicate for intervals
intersecting in at most an instant.

<axiom id="2.3-15">
nonoverlap(T1,T2)
<--> [int-before(T1,T2) v int-after(T1,T2) v int-meets(T1,T2)
v int-met-by(T1,T2)]
</axiom>

So far, the concepts and axioms in the ontology of time would be
appropriate for scalar phenomena in general.

2.4.  Optional Extensions:

In the basic ontology we have tried to remain neutral with respect to
controversial issues, while producing a consistent and useable
axiomatization.  In specific applications one may want to have
stronger properties and thus take a stand on some of these issues.  In
this section, we describe some options, with the axioms that would
implement them.  These axioms and any subsequent theorems depending on
them are prefaced with a 0-argument proposition that says the option
is being exercised.  Thus the axiom for total ordering is prefaced by
the proposition

Total-Order() -->

Then to adopt the option of total ordering, one merely has to assert

Total-Order()

Total or Linear Ordering: In many applications, if not most, it will
be useful to assume that time is linearly or totally ordered.  The
axiom that expresses this is as follows:

<axiom id="2.4-1">
Total-Order() -->
(A t1,t2)[instant(t1) & instant(t2)
--> [before(t1,t2) v t1 = t2 v before(t2,t1)]]
</axiom>

This eliminates models of time with branching futures and other
conflations of time and possibility or limited knowledge.

Infinity: There are two common ways of allowing infinitely long
intervals. Both are common enough that it is worth a little effort to
construct the time ontology in a way that accommodates both.  The
statements of the axioms have been complicated modestly in order to
localize the difference between the two approaches to the choice
between two pairs of simple existence axioms, which are themselves
conditioned on 0-argument propositions indicating the choice of that
option.

In the first approach, one posits time instants at positive and
negative infinity.  Half-infinite intervals are then intervals that
have one of these as an endpoint.  Rather than introduce constants for
these in the core ontology, we will have two predicates -- "posinf"
and "neginf" -- which are true of only these points.  The 0-argument
proposition corresponding to the choice of this approach will be

Pts-at-Inf()

In the second approach, there are intervals that have no beginning
and/or end.  "posinf-interval(T)" says that T is a half-infinite
interval with no end.  "neginf-interval(T)" says that T is a
half-infinite interval with no beginning.  The 0-argument proposition
corresponding to this option will be

No-Pts-at-Inf()

In the first approach, "posinf-interval" and "neginf-interval" will
not be true of anything.  In the second approach "posinf" and "neginf"
will not be true of anything.

The axioms that specify the properties of posinf, neginf,
posinf-interval, and neginf-interval will be conditioned on the
existence of such temporal entities.  Thus, if an approach does not
include them, the condition will never be satisfied.  These axioms
can thus be part of the core theory.

Axioms in the core theory will make the two approaches mutually
exclusive.

Then which approach one takes amounts on which of two pairs of
existence axioms
one uses.  This choice is further localized to the decision between
asserting "Pts-at-Inf()" and asserting "No-Pts-at-Inf()".

The arguments of the predicates "posinf" and "neginf", if they exist,
are instants.
<axiom id="2.4-2">
posinf(t) --> instant(t)
</axiom>
<axiom id="2.4-3">
neginf(t) --> instant(t)
</axiom>

The principal property of the point at positive infinity is that every
other instant is before it.
<axiom id="2.4-4">
(A t,t1)[instant(t1) & posinf(t) --> [before(t1,t) v t1 = t]]
</axiom>

The next axiom entails that there are infinitely many instants after
any given instant other than the point at positive infinity.
<axiom id="2.4-5">
(A t1)[instant(t1) & ~posinf(t1)
--> (E t2)[instant(t2) & before(t1,t2)]]
</axiom>
Note that these two axioms are valid in an approach that does not
admit a point at positive infinity; the antecedent of Axiom 2.4-4 will
never be satisfied, and the second conjunct in the antecedent of Axiom
2.4-5 will always be satisfied, guaranteeing that after every instant
there will be another instant.

The principal property of the point at negative infinity is that it is
before every other instant.
<axiom id="2.4-6">
(A t,t1)[instant(t1) & neginf(t) --> [before(t,t1) v t1 = t]]
</axiom>

The next axiom entails that there are infinitely many instants before
any given instant other than the point at negative infinity.
<axiom id="2.4-7">
(A t1)[instant(t1) & ~neginf(t1)
--> (E t2)[instant(t2) & before(t2,t1)]]
</axiom>
Likewise these axioms are valid in an approach that does not admit
a point at negative infinity.

In the second approach instants at positive and negative infinity are
not posited, but intervals can have the properties "posinf-interval"
and "neginf-interval".  Because of Axiom 2.1-4, if an interval has an
end, it is not a positive infinite interval.  Thus, a positive
infinite interval cannot have an end.

An instant inside a postive half-infinite interval has infinitely many
instants after it.
<axiom id="2.4-8">
(A t1,T)[posinf-interval(T) & inside(t1,T)
--> (E t2)[before(t1,t2) & inside(t2,T)]]
</axiom>
This axiom is valid in the first approach as well, since
"posinf-interval" will never be true and the antecedent will never be
satisfied.

Because of Axiom 2.1-3, if an interval has a beginning, it is not a
negative infinite interval.  Thus, a negative infinite interval cannot
have a beginning.

Corresponding to Axiom 2.4-8 is the following axiom for
"neginf-interval":
<axiom id="2.4-9">
(A t1,T)[neginf-interval(T) & inside(t1,T)
--> (E t2)[before(t2,t1) & inside(t2,T)]]
</axiom>

It may be useful to have two more predicates.  An interval is (at
least) a half-infinite interval if either "posinf-interval" or
"neginf-interval" is true of it.
<axiom id="2.4-10">
(A T)[halfinf-interval(T)
<--> [posinf-interval(T) v neginf-interval(T)]]
</axiom>
An interval is an infinite interval if it is both positively and
negatively infinite.
<axiom id="2.4-11">
(A T)[inf-interval(T)
<--> [posinf-interval(T) & neginf-interval(T)]]
</axiom>
Again these axioms are valid in the first approach because the
antecedents will never be true.

Finally for the core ontology, we probably want to stipulate that one
either uses the "posinf" approach or the "posinf-interval" approach.
This is accomplished by the following axiom.
<axiom id="2.4-12">
[(E t) posinf(t)] <--> ~[(E T) posinf-interval(T)]
</axiom>
Similarly,
<axiom id="2.4-13">
[(E t) neginf(t)] <--> ~[(E T) neginf-interval(T)]
</axiom>
Note that one could use one approach for negative infinity and
the other for positive infinity, although this development does not
support it.

This completes the treatment of infinite time in the core ontology.

The following two axioms give points at infinity if we are using the
first approach, indicated by the proposition "Pts-at-Inf()" in the
antecedent.
<axiom id="2.4-14a">
Pts-at-Inf() --> (E t) posinf(t)
</axiom>
<axiom id="2.4-15a">
Pts-at-Inf() --> (E t) neginf(t)
</axiom>
That is, there are instants out at positive and negative infinity,
respectively, when the Points at Infinity approach is taken.  Again,
to adopt this approach, simply assert

Pts-at-Inf()

When one adopts this approach, one can also, for convenience, extend
the language to include the two constants, PositiveInfinity and
NegativeInfinity, where

posinf(PositiveInfinity)
neginf(NegativeInfinity)

One can also extend the language to include the functions
"beginning-of" and "end-of", defined as follows:

beginning-of(T) = t <--> begins(t,T)
end-of(T) = t <--> ends(t,T)

We stipulated the uniqueness of "begins" and "ends" in Section 2.1,
and Axioms 2.4-12 and 2.4-13 rule out intervals with no beginnings or
ends, so the functions will be total.

The following two axioms guarantee the existence of half infinite
intervals if one takes the "No Points at Infinity" approach.
<axiom id="2.4-14b">
No-Pts-at-Inf() -->
(A t)[instant(t)
--> (E T)[posinf-interval(T) & begins(t,T)]]
</axiom>
<axiom id="2.4-15b">
No-Pts-at-Inf() -->
(A t)[instant(t)
--> (E T)[neginf-interval(T) & ends(t,T)]]
</axiom>
To specify that we are using the second approach, we would assert

No-Pts-at-Inf()

Suppose we wish to map between the two ontologies.  Suppose the
predicates and constants in the theory using the first approach are
subscripted with 1 and the predicates in the theory using the second
approach are subscripted with 2.  The domains of the two theories are
the same.  All predicates and functions of the two theories are
equivalent with the exception of "begin", "ends", "beginning-of",
"end-of", "posinf", "neginf", "posinf-interval", and
"neginf-interval".  These are related by the following two
articulation axioms.

posinf1(end-of1(T)) <--> posinf-interval2(T)

neginf1(beginning-of1(T)) <--> neginf-interval2(T)

Density: In some applications it is useful to have the property of
density, that is, the property that between any two distinct instants
there is a third distinct instant.  The axiom for this is as follows,
where the 0-argument predicate indicating the exercising of this
option is "Dense()":

<axiom id="2.4-16">
Dense() -->
(A t1,t2)[instant(t1) & instant(t2) & before(t1,t2)
--> (E t)[instant(t) & before(t1,t) & before(t,t2)]]
</axiom>

This is weaker than the mathematical property of continuity, which we
will not axiomatize here.  If time is totally ordered and continuous,
it is isomorphic to the real numbers.

Convexity:  In Section 2.2 we gave the axiom 2.2-8:

inside(t,T) & begins(t1,T) & ends(t2,T)
--> before(t1,t) & before(t,t2)

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 below.)  To choose the option of
Convexity, simply assert the 0-argument proposition "Convex()".

<axiom id="2.4-17">
Convex() -->
[begins(t1,T) & ends(t2,T) & before(t1,t) & before(t,t2)
--> inside(t,T)]
</axiom>

In the rest of this development anny property that depends on
Convexity will be conditioned on the proposition "Convex()".

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.

Convex() -->
[before(t1,t2) & before(t2,t3) & inside(t1,T) & inside(t3,T)
--> inside(t2,T)]

Extensional Collapse: 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
beginning and end points.  We can call this the property of
Extensional Collapse, and indicate it by the 0-argument proposition
"Ext-Collapse()".

<axiom id="2.4-18">
Ext-Collapse() --> (A T1,T2)[int-equals(T1,T2) --> T1 = T2]
</axiom>

If we think of different intervals between the end points as being
different ways the beginning 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 beginning and end nodes and all the
arcs between the beginning 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.

The time ontology links to other things in the world through four
predicates -- at-time, during, holds, and time-span.  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 term
"eventuality" in this document is only an expositional convenience and
has no formal role in the time ontology.

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.

<axiom id="2.5-1">
at-time(e,t) --> instant(t)
</axiom>

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.

<axiom id="2.5-2">
during(e,T) --> interval(T)
</axiom>

If an eventuality obtains during an interval, it obtains at every
instant inside the interval.

<axiom id="2.5-3">
during(e,T) & inside(t,T) --> at-time(e,t)
</axiom>

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(e,T)" would say that e holds at
instant T or during interval T.  The predicate "holds" would be part
of the event ontology, not part of the time ontology, although its
second argument would be be provided by the time ontology.  The
designers of the event ontology may or may not want to relate "holds"
to "at-time" and "during" by axioms such as the following:

holds(e,t) & instant(t) <--> at-time(e,t)
holds(e,T) & interval(T) <--> during(e,T)

Similarly, the event ontology may provide other ways of linking events
with times, for example, by including a time parameter in
predications.

p(x,t)

The 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 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.  In Section 6 we will develop a treatment of discontinuous
temporal sequences, and it will be useful to remain open to having
these as time spans of eventualities as well.

<axiom id="2.5-4">
time-span(T,e) --> temporal-entity(T) & tseq(T)
</axiom>
<axiom id="2.5-5">
time-span(T,e) & interval(T) --> during(e,T)
</axiom>
<axiom id="2.5-6">
time-span(t,e) & instant(t) --> at-time(e,t)
</axiom>
<axiom id="2.5-7">
time-span(T,e) & interval(T) & ~inside(t,T)
& ~begins(t,T) & ~ends(t,T)
--> ~at-time(e,t)
</axiom>
<axiom id="2.5-8">
time-span(t,e) & instant(t) & t1 =/= t --> ~at-time(e,t1)
</axiom>

Whether the eventuality obtains at the beginning 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" 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-begins(t,e) <--> time-span(T,e) & begins(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 beginning of the heating at the beginning 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, beginning-of is BeginFn, and
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 beginning-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.  Because of infinite intervals, the range must
also include Infinity.

minutes: Intervals --> Reals U {Infinity}
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 U {Infinity}
duration([5:14,5:17), *Minute*) = 3

The two approaches are interdefinable:

<axiom id="3.1-1">
seconds(T) = duration(T,*Second*)
</axiom>
<axiom id="3.1-2">
minutes(T) = duration(T,*Minute*)
</axiom>
<axiom id="3.1-3">
hours(T) = duration(T,*Hour*)
</axiom>
<axiom id="3.1-4">
days(T) = duration(T,*Day*)
</axiom>
<axiom id="3.1-5">
weeks(T) = duration(T,*Week*)
</axiom>
<axiom id="3.1-6">
months(T) = duration(T,*Month*)
</axiom>
<axiom id="3.1-7">
years(T) = duration(T,*Year*)
</axiom>

Ordinarily, the first is more convenient for stating specific facts
about particular units.  The second is more convenient for stating

The constraints on the arguments of duration are as follows:

<axiom id="3.1-8">
duration(T,u) --> proper-interval(T) & temporal-unit(u)
</axiom>

The temporal units are as follows:

<axiom id="4.2-17">
temporal-unit(*Second*) & temporal-unit(*Minute*)
& temporal-unit(*Hour*) & temporal-unit(*Day*)
& temporal-unit(*Week*) & temporal-unit(*Month*)
& temporal-unit(*Year*)
</axiom>

The aritmetic relations among the various units are as follows:

<axiom id="3.1-9">
seconds(T) = 60 * minutes(T)
</axiom>
<axiom id="3.1-10">
minutes(T) = 60 * hours(T)
</axiom>
<axiom id="3.1-11">
hours(T) = 24 * days(T)
</axiom>
<axiom id="3.1-12">
days(T) = 7 * weeks(T)
</axiom>
<axiom id="3.1-13">
months(T) = 12 * years(T)
</axiom>

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 begins-or-in covers only
beginning-of and inside.)

<axiom id="3.2-1">
concatenation(x,S)
<--> proper-interval(x)
& (A z)[begins-or-in(z,x)
--> (E y)[member(y,S) & begins-or-in(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 nonoverlap(y1,y2)]]
</axiom>

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.

Convex() -->
[concatenation(x,S)
--> (A y1)[member(y1,S)
--> [int-finishes(y1,x)
v (E! y2)[member(y2,S) & int-meets(y1,y2)]]]]

Convex() -->
[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 nonoverlap.

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:

<axiom id="3.2-2">
Hath(N,u,x)
<--> (E S)[card(S) = N
& (A z)[member(z,S) --> duration(z,u) = 1]
& concatenation(x,S)]
</axiom>

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:

<axiom id="3.2-3">
Hath(N,u,x) --> integer(N) & temporal-unit(u) & proper-interval(x)
</axiom>

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.

<axiom id="3.3-1">
second(T) <--> seconds(T) = 1
</axiom>
<axiom id="3.3-2">
minute(T) <--> minutes(T) = 1
</axiom>
<axiom id="3.3-3">
hour(T) <--> hours(T) = 1
</axiom>
<axiom id="3.3-4">
day(T) <--> days(T) = 1
</axiom>
<axiom id="3.3-5">
week(T) <--> weeks(T) = 1
</axiom>
<axiom id="3.3-6">
month(T) <--> months(T) = 1
</axiom>
<axiom id="3.3-7">
year(T) <--> years(T) = 1
</axiom>

We are now in a position to state the relations between successive
temporal units.

<axiom id="3.3-8">
minute(T) --> Hath(60,*Second*,T)
</axiom>
<axiom id="3.3-9">
hour(T) --> Hath(60,*Minute*,T)
</axiom>
<axiom id="3.3-10">
day(T) --> Hath(24,*Hour*,T)
</axiom>
<axiom id="3.3-11">
week(T) --> Hath(7,*Day*,T)
</axiom>
<axiom id="3.3-12">
year(T) --> Hath(12,*Month*,T)
</axiom>

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.  A delay is the interval between the instants
at which two events occur.

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.  (The BCE era is similarly time zone
dependent, although this will almost never be 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 beginning 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:

minit(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:

minitFn(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:

<axiom id="4.2-1">
sec(y,n,x) <--> secFn(n,x) = y
</axiom>
<axiom id="4.2-2">
sec(y,n,x) <--> clock-int(y,n,*Second*,x)
</axiom>
<axiom id="4.2-3">
minit(y,n,x) <--> minitFn(n,x) = y
</axiom>
<axiom id="4.2-4">
minit(y,n,x) <--> clock-int(y,n,*Minute*,x)
</axiom>
<axiom id="4.2-5">
hr(y,n,x)  <--> hrFn(n,x) = y
</axiom>
<axiom id="4.2-6">
hr(y,n,x)  <--> clock-int(y,n,*Hour*,x)
</axiom>
<axiom id="4.2-7">
da(y,n,x)  <--> daFn(n,x) = y
</axiom>
<axiom id="4.2-8">
da(y,n,x)  <--> cal-int(y,n,*Day*,x)
</axiom>
<axiom id="4.2-9">
mon(y,n,x) <--> monFn(n,x) = y
</axiom>
<axiom id="4.2-10">
mon(y,n,x) <--> cal-int(y,n,*Month*,x)
</axiom>
<axiom id="4.2-11">
yr(y,n,x)  <--> yrFn(n,x) = y
</axiom>
<axiom id="4.2-12">
yr(y,n,x)  <--> cal-int(y,n,*Year*,x)
</axiom>

Weeks and months are dealt with separately below.

The am/pm designation of hours is represented by the function hr12.

<axiom id="4.2-13">
hr12(y,n,*am*,x) <--> hr(y,n,x)
</axiom>
<axiom id="4.2-14">
hr12(y,n,*pm*,x) <--> hr(y,n+12,x)
</axiom>

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

<axiom id="4.2-15">
cal-int(y,n,u,x) <--> clock-int(y,n-1,u,x)
</axiom>

Note that the Common Era is a calendar interval in this sense, since
it begins with 1 C.E. and not 0 C.E.

The type constraints on the arguments of cal-int are as follows:

<axiom id="4.2-16">
cal-int(y,n,u,x) --> interval(y) & integer(n) & temporal-unit(u)
& interval(x)
</axiom>

We allow x to be any interval, not just a calendar interval.  When x
does not begin at the beginning of a calendar unit of type u, we take
y to be the nth _full_ interval of type u in x.  Thus, the first year
of World War II, in this sense, is 1940, the first full year, and not
1939, the year it began.  The first week of the year will be the first
full week.  We can express this constraint as follows:
<axiom id="4.2-17">
cal-int(y,n,u,x) --> starts-or-during(y,x)
</axiom>

Each of the calendar intervals is that unit long; for example, a
calendar year is a year long.
<axiom id="4.2-18">
cal-int(y,n,u,x) --> duration(y,u) = 1
</axiom>

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.

<axiom id="4.2-19">
cal-int(y,n,u,x) & Hath(N,u,x)  --> 0 < n <= N
</axiom>

There is a 1st small interval, and it starts the large interval.

<axiom id="4.2-20">
Hath(N,u,x) --> (E! y) cal-int(y,1,u,x)
</axiom>
<axiom id="4.2-21">
Hath(S,N,u,x) & cal-int(y,1,u,x) --> int-starts(y,x)
</axiom>

There is an Nth small interval, and it finishes the large interval.

<axiom id="4.2-22">
Hath(N,u,x) --> (E! y) cal-int(y,N,u,x)
</axiom>
<axiom id="4.2-23">
Hath(N,u,x) & cal-int(y,N,u,x) --> int-finishes(y,x)
</axiom>

All but the last small interval have a small interval that succeeds
and is met by it.

<axiom id="4.2-24">
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)]
</axiom>

All but the first small interval have a small interval that precedes
and meets it.

<axiom id="4.2-25">
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)]
</axiom>

4.3.  Weeks

A week is any seven consecutive days.  A calendar week, by contrast,
according to a commonly adopted convention, starts at midnight,
Saturday night, and goes to the next midnight, Saturday night.  There
are 52 weeks in a year, but there are not usually 52 calendar weeks in
a year.

Weeks are 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.

<axiom id="4.3-1">
wk(y,n,x)  <--> wkFn(n,x) = y
</axiom>
<axiom id="4.3-2">
wk(y,n,x)  <--> cal-int(y,n,*Week*,x)
</axiom>

As it happens, the n and x arguments will often be irrelevant, when we
only want to say that some period is a calendar week.

The day of the week is a calendar interval of type *Day*.  The nth
day-of-the-week in a week is the nth day in that interval.

<axiom id="4.3-3">
dayofweek(y,n,x) <--> day(y,n,x) & (E n1,x1) wk(x,n1,x1)
</axiom>

The days of the week have special names in English.

<axiom id="4.3-4">
dayofweek(y,1,x) <--> Sunday(y,x)
</axiom>
<axiom id="4.3-5">
dayofweek(y,2,x) <--> Monday(y,x)
</axiom>
<axiom id="4.3-6">
dayofweek(y,3,x) <--> Tuesday(y,x)
</axiom>
<axiom id="4.3-7">
dayofweek(y,4,x) <--> Wednesday(y,x)
</axiom>
<axiom id="4.3-8">
dayofweek(y,5,x) <--> Thursday(y,x)
</axiom>
<axiom id="4.3-9">
dayofweek(y,6,x) <--> Friday(y,x)
</axiom>
<axiom id="4.3-10">
dayofweek(y,7,x) <--> Saturday(y,x)
</axiom>

For example, Sunday(y,x) says that y is the Sunday of week x.

Since a day of the week is also a calendar day, it is a theorem that
it is a day long.

dayofweek(y,n,x) --> day(y)

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.

<axiom id="4.3-11">
(A z)(E x) Tuesday(dayFn(1,monFn(1,yrFn(2002,CE(z)))),x)
</axiom>

We can define weekdays and weekend days as follows:

<axiom id="4.3-12">
weekday(y,x) <--> [Monday(y,x) v Tuesday(y,x) v Wednesday(y,x)
v Thursday(y,x) v Friday(y,x)]
</axiom>
<axiom id="4.3-13">
weekendday(y,x) <--> [Saturday(y,x) v Sunday(y,x)]
</axiom>

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.

<axiom id="4.4-1">
mon(y,1,x) <--> January(y,x)
</axiom>
<axiom id="4.4-2">
mon(y,2,x) <--> February(y,x)
</axiom>
<axiom id="4.4-3">
mon(y,3,x) <--> March(y,x)
</axiom>
<axiom id="4.4-4">
mon(y,4,x) <--> April(y,x)
</axiom>
<axiom id="4.4-5">
mon(y,5,x) <--> May(y,x)
</axiom>
<axiom id="4.4-6">
mon(y,6,x) <--> June(y,x)
</axiom>
<axiom id="4.4-7">
mon(y,7,x) <--> July(y,x)
</axiom>
<axiom id="4.4-8">
mon(y,8,x) <--> August(y,x)
</axiom>
<axiom id="4.4-9">
mon(y,9,x) <--> September(y,x)
</axiom>
<axiom id="4.4-10">
mon(y,10,x) <--> October(y,x)
</axiom>
<axiom id="4.4-11">
mon(y,11,x) <--> November(y,x)
</axiom>
<axiom id="4.4-12">
mon(y,12,x) <--> December(y,x)
</axiom>

The number of days in a month have to be spelled out for individual
months.

<axiom id="4.4-13">
January(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-14">
March(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-15">
April(m,y) --> Hath(30,*Day*,m)
</axiom>
<axiom id="4.4-16">
May(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-17">
June(m,y) --> Hath(30,*Day*,m)
</axiom>
<axiom id="4.4-18">
July(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-19">
August(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-20">
September(m,y) --> Hath(30,*Day*,m)
</axiom>
<axiom id="4.4-21">
October(m,y) --> Hath(31,*Day*,m)
</axiom>
<axiom id="4.4-22">
November(m,y) --> Hath(30,*Day*,m)
</axiom>
<axiom id="4.4-23">
December(m,y) --> Hath(31,*Day*,m)
</axiom>

The definition of a leap year is as follows:

<axiom id="4.4-24">
(A z)[leap-year(y)
<--> (E n,x)[year(y,n,CE(z))
& [divides(400,n) v [divides(4,n) & ~divides(100,n)]]]]
</axiom>

We leave leap seconds to specialized ontologies.

Now the number of days in February can be specified.

<axiom id="4.4-25">
February(m,y) & leap-year(y) --> Hath(29,*Day*,m)
</axiom>
<axiom id="4.4-26">
February(m,y) & ~leap-year(y) --> Hath(28,*Day*,m)
</axiom>

A reasonable approach to defining month as a unit of temporal measure
would be to specify that the beginning and end points have to be on the
same days of successive months.  The following rather ugly axiom
captures this.

<axiom id="4.4-27">
month(T)
<--> (E t1,t2,d1,d2,n,m1,m2,n1,y1,y2,n2,e)
[begins(t1,T) & ends(t2,T)
[begins-or-in(t1,d1) & begins-or-in(t2,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)]]]
</axiom>

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)]

Holidays like Easter can be defined in terms of this ontology coupled
with an ontology of the phases of the moon.

Other calendar systems could be axiomatized similarly.  and the BCE
era could also be axiomatized in this framework.  These are left as
exercises for interested developers.

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.

<axiom id="4.5-1">
time-of(t,y,m,d,h,n,s,z)
<--> begins-or-in(t,secFn(s,minFn(n,hrFn(h,daFn(d,
monFn(m,yrFn(y,CE(z))))))))
</axiom>

Alternatively (and not assuming Extensional Collapse),

<axiom id="4.5-2">
time-of(t,y,m,d,h,n,s,z)
<--> (E s1,n1,h1,d1,m1,y1,e)
[begins-or-in(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)]
</axiom>

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)[begins-or-in(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.

The predicate "time-of" has 8 arguments.  It will be convenient in
cases where exact times are not known or don't need to be specified to
have functions that identify each of the slots of a time description.
These functions are also useful for partial implementations of the
time ontology in versions of DAML based on description logic, such as
DAML+OIL.  We introduce functions that allow "time-of" to be expressed
as a collection of values of the functions: "second-of", "minute-of",
etc.  However, these functions cannot be applied to instants directly,
since an instant can have many "time-of" predications, one for each
time zone and alternate equivalent descriptions involving, for
example, 90 minutes versus 1 hour and 30 minutes..  Thus, we need an
intervening "temporal description".  An instant can have many temporal
descriptions, and each temporal description has a unique value for
"second-of", "minute-of", etc.Thus, "time-of(t,2002,2,6,17,14,35,PST)",
or 5:14:35pm PST, February 6, 2002, would be expressed by asserting of
the instant t the property "temporal-description(d,t)", meaning that d
is a temporal description of t, and asserting for d the properties
"year-of(d) = 2002", "month-of(d) = 2", etc. Coarser granularities on
times can be expressed by leaving the finer-grained units unspecified.

These functions can be defined by the following axiom:
<axiom id="4.5-3">

(A t,y,m,d,h,n,s,z)[time-of(t,y,m,d,h,n,s,z)
<--> (E d1)[temporal-description(d1,t)
& year-of(d1) = y & month-of(d1) = m
& day-of(d1) = d & hour-of(d1) = h
& minute-of(d1) = n & second-of(d1) = s
& time-zone-of(d1) = z]]

</axiom>
The domain of the functions is an entity of type "temporal
description".  The range of the functions is inherited from the
constraints on the arguments of "time of".
<axiom id="4.5-4">
(A d1,y)[year-of(d1) = y --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-5">
(A d1,m)[month-of(d1) = m --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-6">
(A d1,d)[day-of(d1) = d --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-7">
(A d1,h)[hour-of(d1) = h --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-8">
(A d1,n)[minute-of(d1) = n --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-9">
(A d1,s)[second-of(d1) = s --> (E t)[temporal-description(d1,t)]]
</axiom>
<axiom id="4.5-10">
(A d1,z)[time-zone-of(d1) = z --> (E t)[temporal-description(d1,t)]]
</axiom>

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. Temporal Granularity

Useful background reading for this note includes Bettini et
al. (2002), Fikes and Zhou, and Hobbs (1985).

Very often in reasoning about the world, we would like to treat an
event that has extent as instantaneous, and we would like to express
its time only down to a certain level of granularity.  For example, we
might want to say that the election occurs on November 5, 2002,
without specifying the hours, minutes, or seconds.  We might want to
say that the Thirty Years' War ended in 1648, without specifying the
month and day.

For the most part, this can be done simply by being silent about the
more detailed temporal properties.  In Section 2.5 we introduced the predication
"time-span(T,e)" relating events to temporal entities, the relation
"temporal-description(d,t)" relating a temporal entity to a
description of the clock and calendar intervals it is included in, and
the functions "second-of(d)", "minute-of(d)", "hour-of(d)",
"day-of(d)", "month-of(d)", and "year-of(d)".  Suppose we know that an
event occurs on a specific day, but we don't know the hour, or it is
inappropriate to specify the hour.  Then we can specify the day-of,
month-of, and year-of properties, but not the hour-of, minute-of, or
second-of properties.  For example, for the election e, we can say

time-span(t,e), temporal-description(d,t), day-of(d) = 5,
month-of(d) = 11, year-of(d) = 2002

and no more.  We can even remain silent about whether t is an instant
or an interval.

Sometimes it may be necessary to talk explicitly about the granularity
at which we are viewing the world.  For that we need to become clear
about what a granularity is, and how it functions in a reasoning
system.

A granularity G on a set of entities S is defined by an
indistinguishability relation, or equivalently, a cover of S, i.e. a
set of sets of elements of S such that every element of S is an
element of at least one element of the cover.  We will identify the
granularity G with the cover.

(A G,S)[cover(G,S)
<--> (A x)[member(x,S)
<--> (E s)[member(s,G) & member(x,s)]]]

Two elements of S are indistinguishable with respect to G if they are
in the same element of G.

(A x1,x2,G)[indisting(x1,x2,G)
<--> (E s)[member(s,G) & member(x1,s) & member(x2,s)]]

A granularity can be a partition of S, in which case every element of
G is an equivalence class.  The indistinguishability relation is
transitive in this case.  A common case of this is where the classes
are defined by the values of some given function f.

(A G,S)[G = f-gran(S,f)
<--> [cover(G,S) & (A x1,x2)[indisting(x1,x2,G)
<--> f(x1) = f(x2)]]]

For example, if S is the set of descriptions of instants and f is the
function "year-of", then G will be a granularity on the time line that
does not distinguish between two instants within the same calendar
year.

The granularities defined by Bettini et al. (2002) are essentially of
this nature.  They will be discussed further after we have introduced
temporal aggregates in Section 6 below.

A granularity can also consist of overlapping sets, in which case the
indistinguishability relation is not transitive.  A common example of
this is in domains where there is some distance function d, and any
two elements that are closer than a given distance a to each other are
indistinguishable.  We will suppose d takes two entities and a unit u
as its arguments and returns a real number.

(A G,S)[G = d-gran(S,a)
<--> [cover(G,S) & (A x1,x2)[indisting(x1,x2,G)
<--> d(x1,x2,u) < a]]]

For example, suppose S is the set of instants, d is duration of the
interval between the two instants, the unit u is *Minute*, and a is 1.
Then G will be the granularity on the time line that does not
distinguish between instants that are less than a minute apart.  Note
that this is not transitive, because 9:34:10 is indistinguishable from
9:34:50, which is indistinguishable from 9:35:30, but the first and
last are more than a minute apart and are thus distinguishable.

Both of these granularities are uniform over the set, but we can
imagine wanting variable granularities.  Suppose we are planning a
robbery.  Before the week preceeding the robbery, we may not care what
time any events occur.  All times are indistinguishable.  The week
preceeding the robbery we may care only what day events take place on.
On the day of the robbery we may care about the hour in which an event
occurs, and during the robbery itself we may want to time the events
down to ten-second intervals.  Such a granularity could be defined as
above; the formula would only be more complex.

The utility of viewing the world under some granularity is that the
are possible in the world at large can be ignored in the task.  One
way of cashing this out in a theorem-proving framework is to treat the
relevant indistinguishability relation as equality.  This in effect
reduces the number of entities in the universe of discourse and makes
available rapid theorem-proving techniques for equality such as
paramodulation.

We can express this assumption with the axiom

(5.1)   (A x1,x2)[indisting(x1,x2,G) --> x1 = x2]

for the relevant G.  For a temporal ontology, if 0-length intervals
are instants, this axiom has the effect of collapsing some intervals
into instants.

There are several nearly equivalent ways of viewing the addition of
such an axiom -- as a context shift, as a theory mapping, or an an
extra antecedent condition.

Context shift: In some formalisms, contexts are explicitly
represented.  A context can be viewed as a set of sentences that are
true in that context.  Adding axiom (5.1) to that set of sentences
shifts us to a new context.

Theory mapping: We can view each granularity as coinciding with a
theory.  Within each theory, entities that are indistinguishable with
respect to that granularity are viewed as equal, so that, for example,
paramodulation can replace equals with equals.  To reason about
different granularities, there would be a "mediator theory" in which
all the constant, function and predicate symbols of the granular
theories are subscripted with their granularities.  So equality in a
granular theory G would appear as the predicate "=_G" in the mediator
theory.  In the mediator theory paramodulation is allowed with "true"
equality, but not with the granular equality relations =_G.  However,
invariances such as

if x =_G y, then [p_G(x) implies p_G(y)]

hold in the mediator theory.

Extra antecedent condition: Suppose we have a predicate
"under-granularity" that takes a granularity as its one argument and
is defined as follows:

(A g)[under-granularity(g)
<--> (A x1,x2)[indisting(x1,x2,g) --> x1 = x2]]

Then we can remain in the theory of the world at large, rather than
moving to a subtheory.  If we are using a granularity G, rather than
proving a theorem P, we prove the theorem

under-granularity(G) --> P

If the granularity G is transitive, and thus partitions S, adding
axiom (5.1) should not get us into any trouble.  However, if G is not
transitive and consists of overlapping sets, such as the episilon
neighborhood granularity, then contradictions can result.  When we use
(5.1) with such a granularity, we are risking contradiction in the
hopes of efficiency gains.  Such a tradeoff must be judged on a case
by case basis, depending on the task and on the reasoning engine used.

6.  Aggregates of Temporal Entities

6.1.  Describing Aggregates of Temporal Entities

In annotating temporal expressions in newspapers, Laurie Gerber
encountered a number of problematic examples of temporal aggregates,
including expressions like "every 3rd Monday in 2001", "every morning
for the last 4 years", "4 consecutive Sundays", "the 1st 9 months of
1997", "3 weekdays after today", "the 1st full day of competition",
and "the 4th of 6 days of voting".  We have taken these as challenge
problems for the representation of temporal aggregates, and attempted
to develop convenient means for expressing the possible referents of
these expressions.

In this section, we will assume the notation of set theory.  Sets and
elements of sets will be ordinary individuals, and relations such as
"member" will be relations between such individuals. In particular, we
will use the relation "member" between an element of a set and the
set, and the relation "subset" between two sets.  We will use "Phi" to
refer to the empty set.  We will use the notation "{x}" for the
singleton set containing the element x.  We will use the symbol "U" to
refer to the union operation between two sets.  The function "card"
will map a set into its cardinality.

In addition, for convenience, we will make moderate use of
second-order formulations, and quantify over predicate symbols.  This
could be eliminated with the use of an "apply" predicate and axiom
schemas systematically relating predicate symbols to corresponding
individuals, e.g., the axiom schema for unary predicates p,

(A x)[apply(*p*,x) <--> p(x)]

It will be convenient to have a relation "ibefore" that generalizes
over several interval and instant relations, covering both
"int-before" and "int-meets" for proper intervals.
<axiom id="6.1-1">

(A T1,T2)[ibefore(T1,T2)
<--> [before(T1,T2)
v [proper-interval(T1) & proper-interval(T2)
& int-meets(T1,T2)]]]

</axiom>

It will also be useful to have a relation "iinside" that generalizes
over all temporal entities and aggregates.  We first define a
predicate "iinside-1" that generalizes over instants and intervals and
covers "int-starts", "int-finishes" and "int-equals" as well as
"int-during" for intervals.  We break the definition into several
cases.
<axiom id="6.1-2">

(A T1,T2)[iinside-1(T1,T2)
<--> [T1=T2
v [instant(T1) & proper-interval(T2) & inside(T1,T2)]
v [(E t) begins(t,T1) & ends(t,T1)
& proper-interval(T2) & inside(t,T2)]
v [proper-interval(T1) & proper-interval(T2)
& [int-starts(T1,T2) v int-during(T1,T2)
v int-finishes(T1,T2) v int-equals(T1,T2)]]]]

</axiom>
The third disjunct in the definition is for the case of 0-length
intervals, should they be allowed and distinct from the corresponding
instants.

A temporal aggregate is first of all a set of temporal entities, but
it has further structure.  The relation "ibefore" imposes a natural
order on some sets of temporal entities, and we will use the
predicate "tseq" to describe those sets.
<axiom id="6.1-3">

(A s)[tseq(s) <--> (A t)[member(t,s) --> temporal-entity(t)]
& (A t1,t2)[member(t1,s) & member(t2,s)
--> [t1 = t2 v ibefore(t1,t2)
v ibefore(t2,t1)]]]

</axiom>
That is, a temporal sequence is a set of temporal entities totally
ordered by the "ibefore" relation.  A temporal sequence has no
overlapping temporal entities.

It will be useful to have the notion of a temporal sequence whose
elements all have a property p.
<axiom id="6.1-4">

(A s,p)[tseqp(s,p) <--> tseq(s) & (A t)[member(t,s) --> p(t)]]

</axiom>

A uniform temporal sequence is one all of whose members are of equal
duration.
<axiom id="6.1-5">

(A s)[uniform-tseq(s)
<--> (A t1,t2,u)[member(t1,s) & member(t2,s) & temporal-unit(u)
--> duration(t1,u) = duration(t2,u)
</axiom>

The same temporal aggregate can be broken up into a set of intervals
in many different ways.  Thus it is useful to be able to talk about
temporal sequences that are equivalent in the sense that they cover
the same regions of time.
<axiom id="6.1-6">

(A s1,s2)[equiv-tseq(s1,s2)
<--> tseq(s1) & tseq(s2)
& (A t)[temporal-entity(t)
--> [(E t1)[member(t1,s1) & iinside-1(t,t1)]
<--> (E t2)[member(t2,s2) & iinside-1(t,t2)]]]]
</axiom>
That is, s1 and s2 are equivalent temporal sequences when any temporal
entity inside one is also inside the other.

A minimal temporal sequence is one that is minimal in that its
intervals are maximal, so that the number of intervals in minimal.  We
can view a week as a week or as 7 individual successive days; the
first would be minimal.  We can go from a nonminimal to a minimal
temporal sequence by concatenating intervals that meet.
<axiom id="6.1-7">

(A s)[min-tseq(s)
<--> (A t1,t2)[member(t1,s) & member(t2,s)
--> [t1 = t2
v (E t)[ibefore(t1,t) & ibefore(t,t2)
& ~member(t,s)]]]]

</axiom>
That is, s is a minimal temporal sequence when any two distinct
intervals in s have a temporal entity not in s between them.

A temporal sequence s1 is a minimal equivalent temporal sequence to
temporal sequence s if s1 is minimal and equivalent to s.
<axiom id="6.1-8">

(A s1,s)[min-equiv-tseq(s1,s)
<--> min-tseq(s1) & equiv-tseq(s1,s)]

</axiom>

We can now generalize "iinside-1" to the predicate "iinside", which
covers both temporal entities and temporal sequences.  A temporal
entity is "iinside" a temporal sequence if it is "iinside-1" one of
the elements of its minimal equivalent temporal sequence.
<axiom id="6.1-9">

(A t,s)[iinside(t,s)
<--> [temporal-entity(t) & temporal-entity(s)
& iinside-1(t,s)]
v [temporal-entity(t) & tseq(s)
& (E s1,t1)[min-equiv-tseq(s1,s) & member(t1,s1)
& iinside-1(t,t1)]]]

</axiom>

We can define a notion of "isubset" on the basis of "iinside".
<axiom id="6.1-10">

(A s,s0)[isubset(s,s0)
<--> [tseq(s) & tseq(s0)
& (A t)[member(t,s) --> iinside(t,s0)]]]

</axiom>
That is, every element of temporal sequence s is inside some element
of the minimal equivalent temporal sequence of s0.

We can also define a relation of "idisjoint" between two temporal
sequences.
<axiom id="6.1-11">

(A s1,s2)[idisjoint(s1,s2)
<--> [tseq(s1) & tseq(s2)
& ~(E t,t1,t2)[member(t1,s1) & member(t2,s2)
& iinside(t,t1) & iinside(t,t2)]]]

</axiom>
That is, temporal sequences s1 and s2 are disjoint if there is no
overlap between the elements of one and the elements of the other.

The first temporal entity in a temporal sequence is the one that is
"ibefore" any of the others.
<axiom id="6.1-12">

(A t,s)[first(t,s)
<--> [tseq(s) & member(t,s)
& (A t1)[member(t1,s) --> [t1 = t v ibefore(t,t1)]]]]

</axiom>
The predicate "last" is defined similarly.
<axiom id="6.1-13">

(A t,s)[last(t,s)
<--> [tseq(s) & member(t,s)
& (A t1)[member(t1,s) --> [t1 = t v ibefore(t1,t)]]]]

</axiom>

More generally, we can talk about the nth element of temporal
sequence.
<axiom id="6.1-14">

(A t,s)[nth(t,n,s)
<--> [tseq(s) & member(t,s) & natnum(n)
& (E s1)[(A t1)[member(t1,s1)
<--> [member(t1,s) & ibefore(t1,t)]]
& card(s1) = n-1]]]

</axiom>
That is, the nth element of a temporal sequence has n-1 elements
before it.

It is a theorem that the first is the nth when n is 1, and that the
last is the nth when n is the cardinality of s.

(A t,s)[first(t,s) <--> nth(t,1,s)]
(A t,s)[last(t,s) <--> nth(t,card(s),s)]

Later in this development it will be convenient to have a predicate
"nbetw" that says there are n elements in a sequence between two given
elements.
<axiom id="6.1-15">

(A t1,t2,s,n)[nbetw(t1,t2,s,n)
<--> (E s1)[card(s1) = n
& (A t)[member(t,s1)
<--> ibefore(t1,t) & ibefore(t,t2)
& member(t,s)]]]

</axiom>

It may sometimes be of use to talk about the convex hull of a temporal
sequence.
<axiom id="6.1-16">

(A t,s)[convex-hull(t,s)
<--> [tseq(s) & interval(t)
& (A t1)[first(t1,s) --> int-starts(t1,t)]
& (A t2)[last(t2,s) --> int-finishes(t2,t)]]]

</axiom>
Note,however, that we cannot simply dispense with temporal sequences
and talk only about their convex hulls.  "Every Monday in 2003" has as
its convex hull the interval from January 6 to December 29, 2003, but
if we use that interval to represent the phrase, we lose all the
important information in the notice "The group will meet every Monday
in 2003."

The predicate "ngap" will enable us to define "everynthp" below.
Essentially, we are after the idea of a temporal sequence s containing
every nth element of s0 for which p is true.  The predicate "ngap"
holds between two elements of s and says that there are n-1 elements
between them that are in s0 and not in s for which p is true.
<axiom id="6.1-17">

(A t1,t2,s,s0,p,n)
[ngap(t1,t2,s,s0,p,n)
<--> [member(t1,s) & member(t2,s) & tseqp(s,p) & tseq(s0)
& isubset(s,s0) & natnum(n)
& (E s1)[card(s1) = n-1 & idisjoint(s,s1)
& (A t)[member(t,s1)
<--> [iinside(t,s0) & p(t)
& ibefore(t1,t) & ibefore(t,t2)]]]]]

</axiom>

The predicate "everynthp" says that a temporal sequence s consists of
every nth element of the temporal sequence s0 for which property p is
true.  It will be useful in describing temporal aggregates like "every
third Monday in 2001".
<axiom id="6.1-18">

(A s,s0,p,n)[everynthp(s,s0,p,n)
<--> [tseqp(s,p) & tseq(s0) & natnum(n)
& (E t1)[first(t1,s)
& ~(E t)[iinside(t,s0) & ngap(t,t1,s,s0,p,n)]]
& (E t2)[last(t2,s)
& ~(E t)[iinside(t,s0) & ngap(t2,t,s,s0,p,n)]]
& (A t1)[last(t1) v (E t2) ngap(t1,t2,s,s0,p,n0)]]]

</axiom>
That is, the first element in s has no p element n elements before it
in s0, the last element in s has no p element n elements after it, and
every element but the last has a p element n elements after it.

The variable for the temporal sequence s0 is, in a sense, a context
parameter.  When we say "every other Monday", we are unlikely to mean
every other Monday in the history of the Universe.  The parameter s0
constrains us to some particular segment of time.  (Of course, that
segment could in principle be the entire time line.)

The definition of "everyp" is simpler.
<axiom id="6.1-19">

(A s,s0,p)[everyp(s,s0,p)
<--> (A t)[member(t,s) <--> [iinside(t,s0) & p(t)]]]

</axiom>
It is a theorem that every p is equivalant to every first p.

(A s,s0,p)[everyp(s,s0,p) <--> everynthp(s,s0,p,1)]

We could similarly define "every-other-p", but the resulting
simplification from "everynthp(s,s0,p,2)" would not be sufficient to
justify it.

Now we will consider a number of English expressions for temporal
aggregates and show how they would be represented with the machinery
we have built up.

"Every third Monday in 2001":  In Section 4.3, "Monday" is a predicate
with two arguments, the second being for the week it is in.  Let us
define "Monday1" as describing a Monday in any week.

(A d)[Monday1(d) <--> (E w) Monday(d,w)]

Then the phrase "every third Monday in 2001" describes a temporal
sequence S for which the following is true.

(E y,z)[yr(y,2001,CE(z)) & everynthp(S,{y},Monday,3)]

Note that this could describe any of three temporal sequences,
depending on the offset determining the first element of the set.

"Every morning for the last four years": Suppose "nowfn" maps a
document d into the instant or interval viewed as "now" from the point
of view of that document, and suppose D is the document this phrase
occurs in.  Suppose also the predicate "morning" describes that part
of each day that is designated a "morning".  Then the phrase describes
a temporal sequence S for which the the following is true.

(E T,t)[duration(T,*Year*) = 4 & ends(t1,T) & iinside(t1,nowfn(D))
& everyp(S,{T},morning)]

"Four consecutive Mondays":  This describes a temporal sequence S for
which the following is true.

(E s0)[everyp(S,s0,Monday1) & card(S) = 4]

"The first nine months of 1997": This describes the temporal sequence
S for which the following is true.

(E z)(A m)[member(m,S)
<--> month(m,n,yrFn(1997,CD(z))) & 1 =< n =< 9]

Note that this expression is ambiguous between the set of nine
individual months, and the interval spanning the nine months.  This is
a harmless ambiguity because the minimal equivalent temporal sequence
of the first is the singleton set consisting of the second.

"The first full day of competition": For the convenience of this
example, let us assume an ontology where "competition" is a substance
or activity type and "full" relates intervals to such types.  Then the
phrase describes an interval D for which the following is true.

(E s)[(A d)[member(d,s)
<--> (E n,T)[day(d,n,T) & competition(c) & full(d,c)]]
& first(D,s)]

"Three weekdays after January 10": Suppose the predicate "weekday1"
describes any weekday of any week, similar to "Monday1".  Then this
phrase describes the temporal aggregate S for which the following is
true.

(E d,y,T)[da(d,11,moFn(1,y)) & everyp(S,{T},weekday1)
& int-starts(d,T) & card(S) = 3]

That is, January 11, the day after January 10, starts the interval
from which the three successive weekdays are to be taken.  The last of
these weekdays is the day D for which "last(D,S)" is true.  If we know
that January 10 is a Friday, we can deduce that the end of S is
Wednesday, January 15.

"The fourth of six days of voting":  Let us, for the sake of the
example, say that voting is a substance/activity type and can appear
as the first argument of the predicate "during".  Then a voting day
can be defined as follows:

(A d)[voting-day(d)
<--> (E v,n,T)[da(d,n,T) & voting(v) & during(v,d)]]

Then the phrase describes an interval D for which the following is
true.

(E s,s0)[everyp(s,s0,voting-day) & card(s) = 6 & nth(D,4,s)]

Betti et al.'s (19??) concept of granularity is simply a temporal
sequence in our terminology.  All of the examples they give are
uniform temporal sequences.  For example, their "hour" granularity
within an interval T is the set S such that "everyp(S,T,hr1)", where
"hr1" is to "hr" as "Monday1" is to "Monday".

Their notion of one granularity "grouping into" another can be defined
for temporal sequences.
<axiom id="6.1-20">

(A s1,s2)[groups-into(s1,s2)
<--> tseq(s1) & tseq(s2) & iinside(s1,s2)
& (A t)[member(t,s2)
--> (E s)[subset(s,s1) & min-equiv-tseq({t},s)]]]

</axiom>
That is, temporal sequence s1 groups into temporal sequence s2 if
every element of s2 is made up of a concatenation of elements of s1
and nothing else is in s1.

Betti et al. also define a notion of "groups-periodically-into",
relative to a period characterized by integers r.  Essentially,
every r instances of a granule in the coarser granularity groups
a subset of the same number of granules in the finer granularity.
<axiom id="6.1-21">

(A s1,s2,r)[groups-periodically-into(s1,s2,r)
<--> groups-into(s1,s2) & natnum(r)
& (A t1,t2,s3)[member(t1,s2) & member(t2,s2)
& nbetw(t1,t2,s2,r-1) & subset(s3,s1)
& groups-into(s3,{t1})
--> (E s4)[subset(s4,s1) & groups-into(s4,{t2})
& card(s3) = card(s4)]]]]]

</axiom>

To know the time of an event down to a granularity of one clock hour
(in the context of S0) is to know which element it occurred during in
the set S such that "everyp(S,S0,hr1)".

A transitive granularity, as defined in Section 5, is a temporal
sequence.

6.2.  Durations of Temporal Aggregates

The concept of "duration", defined in Section 3, can be extended to
temporal sequences in a straightforward manner.  If a temporal
sequence is the empty set, Phi, its duration is zero.
<axiom id="6.2-1">

(A u)[temporal-unit(u) --> duration(Phi,u) = 0]

</axiom>
The duration of a singleton set is the duration of the temporal entity
in it.
<axiom id="6.2-2">

(A t,u)[temporal-entity(t) & temporal-unit(u)
--> duration({t},u) = duration(t,u)]

</axiom>
The duration of the union of two disjoint temporal sequences is the
sum of their durations.
<axiom id="6.2-3">

(A s1,s2,u)[tseq(s1) & tseq(s2) & idisjoint(s1,s2) & temporal-unit(u)
--> duration(s1 U s2,u) = duration(s1,u) + duration(s2,u)]

</axiom>
We need to use the predicate "idisjoint" to ensure that there is no
overlap between intervals in s1 and intervals in s2.

The duration of the convex hull of a temporal sequence is of course
not the same as the duration of the temporal sequence.  Sometimes one
notion is appropriate, sometimes the other.  For determining what
workers hired on an hourly basis should be paid, we want to know the
duration of the temporal sequence of the hours that they worked,
whereas for someone on an annual salary, the appropriate measure is
the duration of its convex hull.

It is a theorem that the duration of the convex hull of a temporal
sequence is at least as great as that of the temporal sequence.

(A t,s,u)[convex-hull(t,s) --> duration(t,u) >= duration(s,u)]

6.3.  Duration Arithmetic

five business days after January 8, 2003.

6.4.  Rates

7.  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) & begins-or-in(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.

8.  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.

Claudio Bettini, X. Sean Wang, and Sushil Jajodia, "Solving
multi-granularity temporal constraint networks", Artificial
Intelligence, vol. 140 (2002), pp. 107-152.

Richard Fikes and Qing Zhou, "A Reusable Time Ontology"

Jerry Hobbs, "Granularity", IJCAI-85, or
http://www.ai.sri.com/~hobbs/granularity-web.pdf


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