Notes

Outline

Representing Time

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”

Points and Intervals 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 E.g., “March 14, 1994” 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

Class Hierarchy of Time Ontology

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

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

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

Binary Relations on Time-Points 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 Primitive relation

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 ---------------------------------------------------------------------------- (=> (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)))))))

Functions on Time-Interval Starting-Point and Ending-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))))))) 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

Relations 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

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 1950’s.” (Starting-Point Fifties (Starting-Point AD1950)) (Ending-Point Fifties (Ending-Point AD1959)) (Overlaps Fifties (ReignOf Elvis))

“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)))))))

Class Hierarchy of Time Ontology

Infinity and Density Infinite-Past and Infinite-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))))

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

Time 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

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

Equal-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

Full 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)))))))

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))))