Events, Numbers, and Time

1
Events, Numbers, and Time
2
Zeroth order theory

• Time is what keeps everything from happening all
  at once.
• Space is what keeps everything from happening to
  you.
• Number is what tells you just how bad things are.

3
Qualitative Representations

• Symbolic reasoning about continuous properties
  requires quantization
• Qualitative representations of space and time
  need to be based on natural decompositions
• Where and when what is happening is different
• Task-specific constraints

4
Time Some typical choices

• Represent time in terms of intervals and instants
• Sometimes instants mapped to numbers, intervals
  to numerical intervals
• Ex t17.3 seconds, engine firing 223, 457
• Reasoning can include calculations of durations
• Often purely relational vocabulary used
• Ex The lecture was before lunch, but the
  projector broke during the lecture.
• Reasoning by transitivity The projector broke
  before lunch
• Properties of temporal algebras have been heavily
  studied

5
How to individuate time?

• Properties of physics
• Cultural conventions
• Occurrence of events
• Uniform behavior or events
• Quantization of time often tied to quantization
  of number and space
• The water heated up and then started to boil
• After a long flight, the passengers disembarked.

6
Temporal representations can havecomplex
structure
7
Fluents

• Time-dependent physical quantity (Newton)
• Functions (Leibniz)
• Implicit notation
• ? always
• ? sometimes
• Explicit notation
• ( tTime)
• ( tTime)

8
Issues in representing numbers

• Resolution
• Composability
• Graceful Extension
• Relevance

9
Resolution

• Fine versus coarse?
• How many distinctions can be made?
• Fixed versus variable?
• Can the number of distinctions made be varied to
  meet different needs?

10
Composability

• Compare
• How much information is available about relative
  magnitudes?
• Propagate
• Given some values, how can other values be
  computed?
• Combine
• What kinds of relationships can be expressed
  between values?

11
Graceful Extension

• If higher resolution information is needed, can
  it be added without invalidating old conclusions?

12
Relevance

• Which tasks is this notion of value suitable for?
• Which tasks are unsuitable for a given notion of
  value?

13
The Reals

• Most familiar
• Originally intended as a model of continuous
  things
• Some important properties
• Continuity
• Density

14
Continuity

• You cant get from A to B without going through C.

15
Density

• Between any two real numbers you can always find
  a third

16
Finite Fixed Algebras

• One of the first qualitative schemes proposed
• Example Height is one of
• very short, short, medium, tall, very tall
• Built-in order relation
• Example tall gt short

17
Problems with finite fixed algebras

• Compositionality
• short tall ???
• Hard to extend
• If height 6 foot
• ? ?height short
• ? ?height tall

18
Signs

• The first representation used in QR
• The weakest that can support continuity
• if A - then it must be before A 0 before
  A
• Can describe derivatives
• ?? ??increasing
• ?? 0 ??steady
• ?? - ??decreasing

19
Status Abstraction

• Even weaker than signs!
• A okay
• A faulted
• Useful for diagnosis
• Faultfinder (Abbott, 1988)
• Xerox PARC group (1990s)

20
Ordinals

• Describe value via relationships with other
  values
• A gt B A lt C Alt D
• Allows partial information
• in the above, dont know relation between C and D
• Like signs, supports continuity and derivatives

21
Nomenclature

• Quantity Space (range)
• Value defined by set of comparisons with other
  parameters
• Example T(w) gt T(freeze) T(w) lt Tboil(w)
  T(w) lt T(stove)
• T(freeze), Tboil(w) are examples of limit points
• Specialization Value Space
• Quantity space where imposed order is total
• Example The above, but also Tboil(w) lt T(stove)

22
Intervals

• Pin value down to a range (typically of reals)
• Example T(w) (0, 100) at room conditions
• Can be used to express tolerances
• Variable resolution
• Use wider or narrower intervals
• Whole field of mathematics devoted to
  calculations using intervals

23
Intervals can be tricky

• Consider
• AZ/(X-Y)
• Z1,2 X2,4 Y3,5
• A?

24
Fuzzy Logic

• Claim Endpoints of intervals are too crisp
• Represent value by distribution

25
Issues in Fuzzy Logic

• Elkins critique
• Violates basic logical intuitions
• Applications successful because of continuous
  parameters, not specific claims of fuzzy logic
• Some find it useful
• Leitchs work on FuSim
• Flakey the robot

26
Order of Magnitude

• Provides stratification of values
• Larger effects completely dominate
• Smaller effects can be ignored
• Example Ignore evaporation when boiling is
  occurring.
• Several interesting formalisms, still some
  shaking out to do
• Logical consistency
• When do small effects break through to the next
  level?
• Phase shifts

27

• Events in our lives happen in a sequence in time,
  but in their significance to ourselves they find
  their own order the continuous thread of
  revelation.
• Eudora Welty