Temporal Constraints

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• Temporal Constraints
• Time and Time Again The Many Ways to Represent
  Time
• James F Allen

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• Overview
• Various representation techniques
• Based on Dating Schemes
• Constraint Propagation Approaches
• Duration based approaches
• Temporal Knowledge Representation Systems
• Conclusion

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• Representation Based on Dating Schemes
• Absolute dating
• When events are instantaneous and their complete
  linear order is known.
• A time stamp is associated with each event eg.
  secminhour
• Time comparisons are reduced to numerical
  calculations. Duration between events can be
  calculated.
• Domains
• Database transactions using a central clock
• Appointment scheduling

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• Pseudo Dates
• Assign a number to each event such that numerical
  ordering reflects the linear ordering.
• Used when absolute time is unknown, but the
  linear order is known.
• (e1, T1)gt event e1 occurs at time T1
• (e2, T2) gt event e2 occurs at time T2
• If e3 occurs b/w e1 and e2 then
• (e3, (T1T2)/2)
• Duration information is lost.

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• More complex representations
• Events that occur over a range but still remain
  instantaneous.
• An event is associated with a pair (e1,l1)
• This representation does not work with
  pseudo-times

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• Constraint Propagation Approach
• A graph-based representation.
• Arcs gt relations
• Nodes gt time points (events)
• Can represent binary relations (e1lte2)
• Cannot represent disjunctions involving more than
  two time points
• (e1lte2 OR e2lte3)
• When new information is added, it may constraint
  existing information.
• Constraints based on transitivity can be used to
  update graph incrementally.

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• Constraint Propagation Approach (cont.)
• Representation becomes complex when events take
  place over intervals. (non-instantaneous)
• Some relationships cannot be captured by
  point-based representations.
• E1(e1,l1) and E2(e2, l2)
• E1 before E2 gt (l1lte2)
• E1 overlaps E2 gt (e1 lt e2) (e2 lt l1) (l1 lt
  l2)

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• Constraint Propagation Approach (cont.)
• Allens constraint propagation algorithm
• Based on intervals, rather than points.

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• Constraint Propagation Approach (cont.)
• Ability to capture 213 constraints compared to
  181 expressible by point-based representation.
• A NP-complete algorithm exists that can handle
  such a representation.
• Allen developed a heuristic O(n3) algorithm. This
  algorithm is complete with respect to any
  situation that can be expressed as point-based
  constraints.
• The algorithm also handles more complex
  situations but without the guarantee of
  completeness.

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• Duration based representations
• This representation makes use of duration
  information.
• Makes use of an acyclic directed graph that has
  a well defined start and end node.
• Each node represents an event and the arcs
  indicate the associated duration.
• Each node indicates the earliest and latest
  starting time.
• This technique is very useful for scheduling
  applications, for which it was initially
  developed.

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Duration based representations (cont.)
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• Duration based representations (cont.)
• This technique is useful for domains where
  duration is known.
• One unknown duration can render this technique
  useless.
• Partial ordering of endpoints cannot be
  represented by this technique.

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• Duration based representations (cont.)
• Dean and McDermott developed a similar
  representation.
• Rather than using durations of events as a base,
  they represent all information as durations
  between time points.
• More qualitative information can be described by
  using infinity as a value for unknown duration
  values.

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• Temporal Logic
• For a temporal representation to be successful,
  it must be embedded within a more general
  representation that can encode general assertions
  about the world.
• Green(frog1,t1)
• frog1 is green at time t1
• frog1 is green over the time interval t1
• This technique does not work with all predicates
• sun rose over the interval t1 is not equivalent
  to saying sun rose at every interval in the
  interval t1.
• Such predicates take intervals as
  arguments----Rise(t1,t2)

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• Temporal Knowledge Representation Systems
• The RHET system developed at Rochester
  integrates a temporal reasoner to a general
  purpose reasoning system.
• It is a horn based AI representation language
  that has as a subcomponent the TIMELOGIC temporal
  reasoning system developed by Koomen and based on
  Allen's interval logic.
• RHET is a hybrid system, rather than using a
  single uniform proof technique, each predicate
  defined in RHET could potentially use its own
  specialized techniques for computing its
  truthhood.
• A(x,y) B(x) gt P(x,y)

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• Temporal Knowledge Representation Systems (cont.)
• All temporal relations are represented as
  TimeRel t1 r t2
• If all variables are bounded then the temporal
  database is used directly.
• TIMELOGIC determines set of all possible bindings
  and this is passed to RHET.
• Once all the variables are bounded then this
  information is passed to TIMELOGIC which
  evaluates the consequences of this binding using
  constraint propagation.

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• Conclusion
• Persistence problem
• If P is true over some time interval T1, what can
  be said about P being true after T1.
• Techniques designed to tackle this problem are
  crude and depend on assumptions.
• Current techniques on time representation work
  well in select domains.
• Not effective in domains where agent has limited
  knowledge of the world.