1
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2
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- 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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3
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- Time point
- Position on a temporal coordinate system
- E.g., “2:22 p.m., February 2, 2000”
- Primitive object
- Analogous to a real number
- Also represented at varying granularities
- Time interval
- Set of two or more time points
- E.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 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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4
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5
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- Time-Point
- Instance-Of: Class
- *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
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6
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- 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
- (=> (Time-Unit ?u) (Quantity-Magnitude Time-Instant ?u
0))
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7
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- Derivable from Location-Of
- Value of Year-Of, Hour-Of, Minute-Of, & Second-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-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
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8
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- Before, After, Equal-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-Interval
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9
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- Time-Interval
- Instance-Of: Class
- *Starting-Point:
- Value-Type: Time-Point
- Slot-Cardinality: 1
- *Ending-Point:
- Value-Type: Time-Point
- Slot-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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10
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- Starting-Point and Ending-Point
- Defining axiom for Starting-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))))
-
(forall ?p (=> (Time-Point ?p)
-
(not (exists ?k
(and (Time-Point ?k)
-
(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 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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11
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- 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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12
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- “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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13
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- (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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14
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15
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- Infinite-Past and Infinite-Future are time points
- Defining axiom:
- (and (Time-Point Infinite-Past)
- (=> (Time-Point ?p) (not (Before ?p Infinite-Past))))
- The time line is considered to be dense
- Defining axiom:
- (=> (and (Time-Point ?i) (Time-Point ?j) (Before ?i ?j))
- (exists ?k
(and (Before ?i ?k) (Before ?k ?j))))
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16
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- Time-Point
- Instance-Of: Class
- *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
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17
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- 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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18
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- Time-Granularity
- Instance-Of: Class
- *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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19
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- (<=> (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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20
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- (=> (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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21
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- Qualification by time
- Functions and relations have a time argument
- E.g., (Married Joe Anne 1993)
[I.e., “Joe and Anne were married during 1993.”]
- Situation calculus
- Objects have time intervals associated with them
- E.g., Joe-Anne-Marriage
- Documentation: “A marriage of Joe and Anne during
1993”
- Instance-Of: Marriage
- Husband: Joe
- Wife: Anne
- Time-Of: 1993
- Sentences “hold true” at times
- E.g., (holds (Married Joe
Anne) 1993) [I.e., “Joe and Anne were married
during 1993.”]
- Tense logics
- E.g., (F (Married Joe
Anne)) [I.e., “Joe and Anne will
be married at some future time.”]
- (F (and (not
(Married Joe Anne))
-
(P (Married Joe Anne))))
- [I.e., “At some future time Joe and Anne will have been married and
will not be married.”]
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