Representing Time

1
Representing Time
Professor Richard Fikes
CS222 Winter 2001
Computer Science Department Stanford University
Knowledge Systems Laboratory, Stanford University
2/12/01
2
About 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

3
Points and Intervals

• Time point
• Position on a temporal coordinate system
• E.g., 222 p.m., February 2, 2000
• Primitive object
• Analogous to a real number
• Also represented at varying granularities
• E.g., March 14, 1994
• Time interval
• Set of two or more time points
• E.g., the 16th century
• 1050 to 1100 a.m. on May 30, 1993
• noon to 100 p.m. every Tuesday in
  2000
• Primitive object
• Convex 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

4
Class Hierarchy of Time Ontology
5
Class Frame Time-Point

• Time-Point
• Subclass-Of Thing
• Location-Of
• Maximum-Cardinality 1
• Value-Type Time-Quantity
• Year-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Month-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Month-Type
• Day-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Day-Type
• Week-Day-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Week-Day-Type

• Hour-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Minute-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Second-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Granularity-Of
• Slot-Cardinality 1
• Value-Type Time-Granularity

6
Location of Time Points

• Function Location-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-1-1900
• Instance-Of Time-Point
• Location-Of Time-Instant
• Time-Instant
• Instance-Of Time-Quantity
• (gt (Time-Unit ?u) (Quantity-Magnitude
  Time-Instant ?u 0))

7
Year-Of, Month-Of, Day-Of, etc.

• Derivable from Location-Of
• Value of Year-Of, Hour-Of, Minute-Of, Second-Of
  is an integer
• (gt (Time-Point ?p)
• ( (Year-Of ?p)
• ( (LINLT (Quantity-Magnitude
  (Location-Of ?p) Year-Unit)) 1900)))
• (gt (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-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-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-Of is a Calendar-Week-Day-Type

8
Binary Relations on Time-Points

• Before, After, Equal-Point
• Defined in terms of Location-Of
• (gt (and (Time-Point ?i) (Time-Point ?j))
• (ltgt (Before ?i ?j)
• ... (lt (Location-Of ?i)
  (Location-Of ?j)) ... )
• (gt (Physical-Quantity ?q1) (Physical-Dimension
  ?q1 ?d)
• (Physical-Quantity ?q2) (Physical-Dimension
  ?q2 ?d)
• (Unit-Of-Measure ?u) (Unit-Dimension ?u ?d)
• (ltgt (lt ?q1 ?q2) (lt (Quantity-Magnitude ?q1
  ?u)
• 
  (Quantity-Magnitude ?q2 ?u))))
• Point-In-Interval
• Primitive relation

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Class Frame Time-Interval

• Time-Interval
• Instance-Of Class
• Starting-Point
• Value-Type Time-Point
• Slot-Cardinality 1
• Ending-Point
• Value-Type Time-Point
• Slot-Cardinality 1
• --------------------------------------------------
  --------------------------
• (gt (Time-Interval ?i)
• (and (Before (Starting-Point ?i)
  (Ending-Point ?i))
• (gt (Time-Point ?p)
• (and (gt (Before ?p
  (Starting-Point ?i))
• (not
  (Point-In-Interval ?p ?i)))
• (gt (Before
  (Ending-Point ?i) ?p)
• (not
  (Point-In-Interval ?p ?i)))))))

10
Functions on Time-Interval

• Starting-Point and Ending-Point
• (gt (and (Time-Point ?s) (Time-Interval ?i))
• (ltgt (Starting-Point ?i ?s)
• (and (not (exists ?j (and
  (Time-Point ?j)
• 
  (Before ?j ?s)
• 
  (Point-In-Interval ?j ?i))))
• (gt (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)))))))
• Duration
• Convex time interval
• Time quantity whose magnitude is the difference
  between the location of the intervals 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-Interval

• James Allen defined a complete set of relations
  on time intervals
• Precedes End-1 lt Start-2
• Meets End-1 Start-2
• Overlaps Start-1 lt Start-2
  lt End-1
• Costarts Start-1 Start-2
• During Start-2 lt Start-1
• End-1 lt End-2
• Cofinishes End-1 End-2
• Equal

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Using 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 1950s.
• (Starting-Point Fifties (Starting-Point AD1950))
• (Ending-Point Fifties (Ending-Point AD1959))
• (Overlaps Fifties (ReignOf Elvis))

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A Week in January

• (and (Subclass-Of Week-In-January
  Convex-Time-Interval)
• (gt (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 Ontology
15
Infinity and Density

• Infinite-Past and Infinite-Future are time points
• (and (Time-Point Infinite-Past)
• (gt (Time-Point ?p) (not (Before ?p
  Infinite-Past))))
• The time line is considered to be dense
• (gt (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-Point

• Time-Point
• Subclass-Of Thing
• Location-Of
• Maximum-Cardinality 1
• Value-Type Time-Quantity
• Year-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Month-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Month-Type
• Day-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Day-Type
• Week-Day-Of
• Maximum-Cardinality 1
• Value-Type Calendar-Week-Day-Type

• Hour-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Minute-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Second-Of
• Maximum-Cardinality 1
• Value-Type Integer
• Granularity-Of
• Slot-Cardinality 1
• Value-Type Time-Granularity

17
Time Granularity

• Time cannot be measured with infinite accuracy
• Need a notion of abstracted time points
• E.g., 1984, May-1927, 1250 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 Granularity

• Time-Granularity
• Subclass-Of Thing
• Time-Unit-Of
• Value-Type Time-Unit
• Max-Cardinality 1
• Year-Granularity
• Instance-Of Time-Granularity
• Time-Unit-Of Year
• Infinitely-Fine-Granularity
• Instance-Of Time-Granularity
• Time-Unit-Of
• Slot-Cardinality 0

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Equal-Point

• (ltgt (Equal-Point ?i ?j)
• (or (and (Granularity-Of ?i
  Infinitely-Fine-Granularity)
• (Granularity-Of ?j Infinitely-Fine-Granu
  larity)
• ( (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

20
Full Definition of Before

• (gt (and (Time-Point ?i) (Time-Point ?j))
• (ltgt (Before ?i ?j)
• (or (and (Granularity-Of ?i
  Infinitely-Fine-Granularity)
• (Granularity-Of ?j
  Infinitely-Fine-Granularity)
• (lt (Location-Of ?i)
  (Location-Of ?j)))
• (and (Granularity-Of ?i
  Infinitely-Fine-Granularity) (Granularity-Of ?j
  ?gran-j)
• (lt (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)
• (lt (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 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 times
• E.g., (holds (Married Joe Anne) 1993)
• Tense logics
• E.g., (F (Married Joe Anne))
• (F (and (not (Married Joe Anne))
• (P (Married Joe Anne))))