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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Temporal Constraint Propagation (Preemptive Case)

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Temporal constraint propagation preemptive case

Temporal Constraint Propagation(Preemptive Case)


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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