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Temporal Constraint Propagation (Preemptive Case)

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Temporal Constraint Propagation (Preemptive Case). Outline. New variables Definition Implementation Relations between the variables Temporal constraints. New variables (definition). set(A) = {t such that A executes at time t} W A (t) = 1 when t Î set(A), 0 otherwise

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Presentation Transcript
outline
Outline
  • New variables
    • Definition
    • Implementation
  • Relations between the variables
  • Temporal constraints
new variables definition
New variables (definition)
  • set(A) = {t such that A executes at time t}
  • WA(t) = 1 when tÎset(A), 0 otherwise
  • start(A) = mintÎset(A)(t)
  • end(A) = maxtÎset(A)(t + 1)
  • duration(A) = |set(A)|
  • span(A) = end(A) - start(A)
new variables implementation
New variables (implementation)
  • Three possible implementations for set(A)
    • Explicit set variable set(A)
    • Explicit Boolean variables WA(t)
    • Dynamic list of intervals Ii(A) = [si(A), ei(A)) with
      • W(Ii(A)) = 1 if "tÎ[si(A), ei(A)), tÎset(A)
      • W(Ii(A)) = 0 if "tÎ[si(A), ei(A)), tÏset(A)
      • W(Ii(A)) = unknown otherwise
  • Explicit or implicit integer variables for start(A), end(A), duration(A), and span(A)
relations between the variables
Relations between the variables
  • end(A) = start(A) + span(A)
  • duration(A) £ span(A)
  • duration(A) = |set(A)|

cardinality constraint

specific implementation for a list of intervals

relations between the variables1
Relations between the variables
  • start(A) = mintÎset(A)(t)

start(A)Îset(A)

[t = startmin(A) = startmax(A)] implies [tÎset(A)]

[t = startmin(A)Ïset(A)] implies [t < start(A)]

[t = startmax(A)Ïset(A)] implies [start(A) < t]

"tÎset(A), start(A) £ t

[tÎset(A)] implies [start(A) £ t]

[t < startmin(A)] implies [tÏset(A)]

relations between the variables2
Relations between the variables
  • end(A) = maxtÎset(A)(t + 1)

(end(A) - 1)Îset(A)

[t = endmin(A) = endmax(A)] implies [(t - 1)Îset(A)]

[t = (endmin(A) - 1)Ïset(A)] implies [(t + 1) < end(A)]

[t = (endmax(A) - 1)Ïset(A)] implies [end(A) < (t + 1)]

"tÎset(A), t < end(A)

[tÎset(A)] implies [t < end(A)]

[endmax(A) £ t] implies [tÏset(A)]

relation between the variables
Relation between the variables
  • pos(A) = {t such that WA(t) can be 1}
  • |{t\'Îpos(A) such that t\' < t}| < durationmin(A)

implies [t < end(A)]

  • |{t\'Îpos(A) such that t £ t\'}| < durationmin(A)

implies [start(A) < t]

temporal constraints
Temporal constraints
  • Constraints between start and end variables

Similar to the non-preemptive case when start(A) and end(A) are explicit

  • Other constraints

"tÎset(A), tÎset(B) (inclusion)

"tÎset(A), tÏset(B) (exclusion)

"tÏset(A), tÎset(B) (coverage)

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