2/12/01 Professor Richard Fikes Representing Time Computer Science Department Stanford University CS222 Winter 2001 Knowledge Systems Laboratory, Stanford

2/12/01

Professor Richard Fikes

Representing Representing TimeTime

Computer Science DepartmentStanford University

CS222Winter 2001

Knowledge Systems Laboratory, Stanford University


2

About TimeAbout Time

A physical dimension (the Time-Dimension)

Time plenum Large temporal space in which all events are located

E.g., “time line”

“temporally possible worlds” Assume time is continuous and linear

> Time line analogous to continuous number line

Duration An amount of time

E.g., “a century”

“25 minutes”

“as long as it takes for the kettle to boil”


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Points and IntervalsPoints and Intervals Time pointTime point

Position on a temporal coordinate systemE.g., “2:22 p.m., February 2, 2000”

Primitive object Analogous to a real number Also represented at varying granularities

E.g., “March 14, 1994”

Time intervalTime interval Set of two or more time pointsE.g., “the 16th century” “10:50 to 11:00 a.m. on May 30, 1993” “noon to 1:00 p.m. every Tuesday in 2000” Primitive object Convex intervalConvex interval analogous to interval on number line Has two distinguished points: Start-Point and End-Point Can be Left-Open, Left-Closed, Right-Open, and/or Right-Closed


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Class Hierarchy of Time OntologyClass Hierarchy of Time Ontology

Calendar-January

. . .. . .. . .

Time-Point Time-Interval

Convex-Time-Interval

Regular- Non-Convex -Time-Interval

Time-Interval-Left-Open

Time-Interval-Right-Open

Time-Interval-Right-Closed

Time-Interval-Left-Open-Right-Open

Time-Interval-Left-Closed-Right-Closed



Calendar-Month Calendar-Day

Non-Convex-Time-Interval

Calendar-Sunday

Calendar-Saturday

Calendar-Day-1

Calendar-Day-31

Calendar-December

Time-Quantity

Time-Interval-Left-Closed


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Class Frame Time-PointClass Frame Time-PointTime-Point

Subclass-Of: Thing *Location-Of

Maximum-Cardinality: 1Value-Type: Time-Quantity

*Year-OfMaximum-Cardinality: 1Value-Type: Integer

*Month-OfMaximum-Cardinality: 1Value-Type: Calendar-Month-Type

*Day-OfMaximum-Cardinality: 1Value-Type: Calendar-Day-Type

*Week-Day-OfMaximum-Cardinality: 1Value-Type: Calendar-Week-Day-Type

*Hour-OfMaximum-Cardinality: 1Value-Type: Integer

*Minute-OfMaximum-Cardinality: 1Value-Type: Integer

*Second-OfMaximum-Cardinality: 1Value-Type: Integer

*Granularity-OfSlot-Cardinality: 1Value-Type: Time-

Granularity


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Location of Time PointsLocation of Time Points

Function Location-OfLocation-Of

Amount of time from “point zero” to the time-point

Value is a time quantity (i.e., a duration)

Point zero assumed to be midnight Jan 1, 1900

Midnight-January-1-1900Instance-Of: Time-PointLocation-Of: Time-Instant

Time-InstantInstance-Of: Time-Quantity

(=> (Time-Unit ?u) (Quantity-Magnitude Time-Instant ?u 0))


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Year-Of, Month-Of, Day-Of, etc.Year-Of, Month-Of, Day-Of, etc. Derivable from Location-Of Value of Year-OfYear-Of, Hour-OfHour-Of, Minute-OfMinute-Of, & Second-OfSecond-Of is an integer

(=> (Time-Point ?p)

(= (Year-Of ?p)

(+ (LINLT (Quantity-Magnitude (Location-Of ?p) Year-Unit)) 1900)))

(=> (Time-Point ?p)

(= (Hour-Of ?p)

(Mod (LINLT (Quantity-Magnitude (Location-Of ?p) Hour-Unit)) 24)))

[Note: LINLT means “Largest Integer Less Than”.]

Value of Month-OfMonth-Of is a Calendar-Month-Type Calendar-Month-Type is a class whose instances are the 12 subclasses of

Calendar-Month (e.g., Calendar-January)

Value of Day-OfDay-Of is a Calendar-Day-Type Calendar-Day-Type is a class whose 31 instances are classes Calendar-Day-1

through Calendar-Day-31

Value of Week-Day-OfWeek-Day-Of is a Calendar-Week-Day-Type


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Binary Relations on Time-PointsBinary Relations on Time-Points BeforeBefore, AfterAfter, Equal-PointEqual-Point

Defined in terms of Location-Of

(=> (and (Time-Point ?i) (Time-Point ?j)) (<=> (Before ?i ?j) ... (< (Location-Of ?i) (Location-Of ?j)) ... )

(=> (Physical-Quantity ?q1) (Physical-Dimension ?q1 ?d) (Physical-Quantity ?q2) (Physical-Dimension ?q2 ?d) (Unit-Of-Measure ?u) (Unit-Dimension ?u ?d) (<=> (< ?q1 ?q2) (< (Quantity-Magnitude ?q1 ?u) (Quantity-Magnitude ?q2 ?u))))

Point-In-IntervalPoint-In-Interval Primitive relation


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Class Frame Time-IntervalClass Frame Time-IntervalTime-Interval

Instance-Of: Class*Starting-Point:

Value-Type: Time-PointSlot-Cardinality: 1

*Ending-Point:Value-Type: Time-PointSlot-Cardinality: 1

----------------------------------------------------------------------------(=> (Time-Interval ?i) (and (Before (Starting-Point ?i) (Ending-Point ?i)) (=> (Time-Point ?p) (and (=> (Before ?p (Starting-Point ?i)) (not (Point-In-Interval ?p ?i))) (=> (Before (Ending-Point ?i) ?p) (not (Point-In-Interval ?p ?i)))))))


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Functions on Time-IntervalFunctions on Time-Interval Starting-PointStarting-Point and Ending-PointEnding-Point

(=> (and (Time-Point ?s) (Time-Interval ?i))

(<=> (Starting-Point ?i ?s)

(and (not (exists ?j (and (Time-Point ?j)

(Before ?j ?s)

(Point-In-Interval ?j ?i))))

(=> (Time-Point ?p)

(not (exists ?k (and (Time-Point ?k)

(Before ?k ?p)

(Point-In-Interval ?k ?i))))

(or (Equal-Point ?p ?s) (Before ?p ?s))))))) DurationDuration

Convex time interval> Time quantity whose magnitude is the difference between the location of the interval’s ending

point and starting point

Non-convex time interval> Sum of durations of all convex time intervals contained in it


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Relations on Time-IntervalRelations on Time-Interval James Allen defined a “complete” set of relations on time intervals

Precedes: |————| End-1 < Start-2|——————|

Meets: |————| End-1 = Start-2|——————|

Overlaps: |————| Start-1 < Start-2 < End-1|——————|

Costarts: |————| Start-1 = Start-2|——————|

During: |————| Start-2 < Start-1|——————| End-1 < End-2

Cofinishes: |————| End-1 = End-2|——————|

Equal


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Using the Interval RelationsUsing the Interval Relations

“The reign of George VI preceded that of Elizabeth II.”

(Precedes (ReignOf GeorgeVI) (ReignOf ElizabethII))

“The reign of Elvis overlapped with the 1950’s.”

(Starting-Point Fifties (Starting-Point AD1950))

(Ending-Point Fifties (Ending-Point AD1959))

(Overlaps Fifties (ReignOf Elvis))


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““A Week in January”A Week in January”

(and (Subclass-Of Week-In-January Convex-Time-Interval)

(=> (Week-In-January ?w)

(and (Duration ?w (The-Quantity Day 7))

(exists ?j

(and (Calendar-January ?j)

(or (During ?w ?j)

(Costarts ?w ?j)

(Cofinishes ?w ?j)))))))


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Class Hierarchy of Time OntologyClass Hierarchy of Time Ontology

Calendar-January

. . .. . .. . .

Time-Point Time-Interval

Convex-Time-Interval

Regular- Non-Convex -Time-Interval

Time-Interval-Left-Open

Time-Interval-Right-Open

Time-Interval-Right-Closed





Calendar-Month Calendar-Day

Non-Convex-Time-Interval

Calendar-Sunday

Calendar-Saturday

Calendar-Day-1

Calendar-Day-31

Calendar-December

Time-Quantity

Time-Interval-Left-Closed


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Infinity and DensityInfinity and Density

Infinite-PastInfinite-Past and Infinite-FutureInfinite-Future are time points

(and (Time-Point Infinite-Past)

(=> (Time-Point ?p) (not (Before ?p Infinite-Past))))

The time line is considered to be dense (=> (and (Time-Point ?i) (Time-Point ?j) (Before ?i ?j))

(exists ?k (and (Before ?i ?k) (Before ?k ?j))))


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Class Frame Time-PointClass Frame Time-PointTime-Point

Subclass-Of: Thing *Location-Of

Maximum-Cardinality: 1Value-Type: Time-Quantity

*Year-OfMaximum-Cardinality: 1Value-Type: Integer

*Month-OfMaximum-Cardinality: 1Value-Type: Calendar-Month-Type

*Day-OfMaximum-Cardinality: 1Value-Type: Calendar-Day-Type

*Week-Day-OfMaximum-Cardinality: 1Value-Type: Calendar-Week-Day-Type

*Hour-OfMaximum-Cardinality: 1Value-Type: Integer

*Minute-OfMaximum-Cardinality: 1Value-Type: Integer

*Second-OfMaximum-Cardinality: 1Value-Type: Integer

*Granularity-OfSlot-Cardinality: 1Value-Type: Time-

Granularity


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Time GranularityTime Granularity

Time cannot be measured with infinite accuracy

Need a notion of “abstracted” time points

E.g., 1984, May-1927, 12:50 p.m. February 14, 2000

Time intervals are not sufficient

Specify a granularity for a time point Provides for uncertainty that a point occurs somewhere

in a certain time interval

E.g., The time point 1984 at granularity “year” occurs

somewhere during the convex interval 1984


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Time GranularityTime GranularityTime-Granularity

Subclass-Of: Thing*Time-Unit-Of

Value-Type: Time-UnitMax-Cardinality: 1

Year-GranularityInstance-Of: Time-GranularityTime-Unit-Of: Year

Infinitely-Fine-GranularityInstance-Of: Time-GranularityTime-Unit-Of:

Slot-Cardinality: 0


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Equal-PointEqual-Point

(<=> (Equal-Point ?i ?j)

(or (and (Granularity-Of ?i Infinitely-Fine-Granularity)

(Granularity-Of ?j Infinitely-Fine-Granularity)

(= (Location-Of ?i) (Location-Of ?j)))

(and (Granularity-Of ?i ?gran)

(Granularity-Of ?j ?gran)

(= (LINLT (Quantity-Magnitude (Location-Of ?i)

(Time-Unit-Of ?gran)))

(LINLT (Quantity-Magnitude (Location-Of ?j)

(Time-Unit-Of ?gran)))))))

Two time points on two different levels of granularity cannot be said to be equal to each other


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Full Definition of BeforeFull Definition of Before(=> (and (Time-Point ?i) (Time-Point ?j))

(<=> (Before ?i ?j)

(or (and (Granularity-Of ?i Infinitely-Fine-Granularity)

(Granularity-Of ?j Infinitely-Fine-Granularity)

(< (Location-Of ?i) (Location-Of ?j)))

(and (Granularity-Of ?i Infinitely-Fine-Granularity) (Granularity-Of ?j ?gran-j)

(< (Location-Of ?i)

(The-Quantity (LINLT (Quantity-Magnitude (Location-Of ?j)

(Time-Unit-Of ?gran-j)))

(Time-Unit-Of ?gran-j)))) ... (and (Granularity-Of ?i ?gran-i) (Granularity-Of ?j ?gran-j)

(< (The-Quantity (SINLT (Quantity-Magnitude (Location-Of ?i)

(Time-Unit-Of ?gran-i))) (Time-Unit-Of ?gran-i))

(The-Quantity (LINLT (Quantity-Magnitude (Location-Of ?j)

(Time-Unit-Of ?gran-j))) (Time-Unit-Of ?gran-j)))))))


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Styles of Temporal RepresentationsStyles of Temporal Representations Timeless Quantification

Functions and relations have a time argument

E.g., (Married Joe Anne 1993)

> Situation calculus

Objects have time intervals associated with them

E.g., (contains (time-of (Marriage Joe Anne)) 1993)

Sentences “hold true” at timesE.g., (holds (Married Joe Anne) 1993)

Tense logicsE.g., (F (Married Joe Anne))

(F (and (not (Married Joe Anne))

(P (Married Joe Anne))))

Documents

2/12/01 Professor Richard Fikes Representing Time Computer Science Department Stanford University CS222 Winter 2001 Knowledge Systems Laboratory, Stanford