12/2/98 Prof. Richard Fikes Representing Time Computer Science Department Stanford University CS222 Fall 1998

2 Knowledge Systems Laboratory, Stanford University Senses of Time - 1 u A physical dimension (the Time-Dimension) u Time plenum ä Large temporal space in which all events are located E.g., “time line” “temporally possible worlds” u Time intervals ä Pieces of time E.g., “during the 1994 Winter Olympics” “the 16th century” “10:50 to 11:00 a.m. on May 30, 1993”

3 Knowledge Systems Laboratory, Stanford University Senses of Time - 2 u Durations E.g., “a century” “25 minutes” “as long as it takes for the kettle to boil” u Time points A time interval of 0 duration u Position on a temporal coordinate system E.g., “March 14, 1994” “3:45 p.m.”

4 Knowledge Systems Laboratory, Stanford University Views of Intervals and Points are u View 1: Points are intervals ä Time is discrete ä Points are single clock ticks ä Points are called “moments” ä Points have no subintervals › No internal separable time points ä Points do not overlap or contain one another

5 Knowledge Systems Laboratory, Stanford University Views of Intervals and Points u View 2 - Point continuum ä Point is a primitive object ä An interval is a set of points ä Intervals are either open or closed ä A closed interval can consists of a single point u View 3 - Glass continuum ä Interval is a primitive object ä The point where intervals meet is not contained in either interval ä No distinction between open and closed intervals ä An interval cannot consist of a single point

6 Knowledge Systems Laboratory, Stanford University Styles of Temporal Representations u Timeless Quantification ä Functions and relations have a time argument E.g., (Married Joe Anne 1993) › Situation calculus ä Objects have time intervals associated with them E.g., (contains (time-of (Marriage Joe Anne)) 1993) u Sentences “hold true” at times E.g., (holds (Married Joe Anne) 1993) u Tense logics E.g., (F (Married Joe Anne)) (F (and (not (Married Joe Anne)) (P (Married Joe Anne)

7 Knowledge Systems Laboratory, Stanford University Relations on Time Intervals

8 Knowledge Systems Laboratory, Stanford University Using the Interval Relations u “The reign of Elizabeth II followed that of George VI.” ä (After (ReignOf ElizabethII) (ReignOf GeorgeVI)) u “The reign of Elvis overlapped with the 1950’s.” ä (Overlaps Fifties (ReignOf Elvis)) ä (= (Start Fifties) (Start AD1950)) ä (= (End Fifties) (End AD1959))

9 Knowledge Systems Laboratory, Stanford University Time Abstractions u Time points can be abstracted Time-Point *Year-Of: *Month-Of: *Day-Of:... u Intervals can have abstract start and end times E.g., [1984 May-1993]

10 Knowledge Systems Laboratory, Stanford University Example Axiom For Abstract Points