El procedimiento de análisis, validez y confiabilidad

Given a solution to the transition-based constraint formulation, for each state variable svwe create a partial order schedule,POS(sv), which is a directed graph, as follows:

• For eachachieve-relevant transition pair Tsv and Tsv0 , if achieve(Tsv,Tsv0 ) = 1, then

addTsvandTsv0 as nodes inPOS(sv), and add a directed edge fromTsvtoTsv0 .

• For eachcan-followtransition pair Tsv andTsv0 , if follow(Tsv,Tsv0 ) = 1, then addTsv

andT_sv0 as nodes inPOS(sv), and add a directed edge fromTsvtoTsv0 .

We only add a node for a transition in POS(sv)above if it doesn’t exist before. Note that each edge represents a precedence constraint between transitions. Given a state variable sv, thePOS(sv)created by the solution has the following properties:

1. For each transitionTsvon the state variablesv, whereinplan(Tsv) =true, there exists

a corresponding node forTsvinPOS(sv). This is the case because we add an achieve-

§3.5 Solution To the Constraint Model 63

This means that we need to show that ifachieve(Tsv,Tsv0 ) =1, then

inplan(Tsv) =inplan(Tsv0 ) =true

Ifachieve(Tsv,Tsv0 ) = 1, then Constraint6ensures that inplan(Tsv0 ) = true. Note

thatT_sv0 can be either an EFFECT transition or a PREVAIL transition.

IfT_sv0 is an EFFECT transition, then Constraint7ensures thatinplan(Tsv) =true.

If T_sv0 is a PREVAIL transition, then Constraint 8 makes sure that there exists an EF- FECT transition T_sv00 such that follow(T_sv0 ,T_sv00) = 1,. Since achieve(Tsv,Tsv0 ) = 1

andfollow(T_sv0 ,T_sv00) = 1, andT_sv0 is a PREVAIL transition, Constraint 9derives that

achieve(Tsv,Tsv00) = 1. SinceTsv andTsv00 both are EFFECT transition, it implies that

T_sv00 is included in the plan (Constraint6), andTsvis included in the plan (Constraint7),

i.e.inplan(Tsv) =true.

In addition to this, Constraint19, includes the dummy actions in the plan, and Constraint

5ensures that all dummy start and end transitions are included in the plan.

2. The dummy start transition on sv, T_svstart, has no incoming edge. This is due to the fact that each pair < Tsv,Tsvstart > is not achieve-relevant. Similarly, since each pair < Tend

sv ,Tsv >is notachieve-relevant, the dummy end transitionTsvendhas no outgoing

edge.

3. All EFFECT transitions inPOS(sv), except for the dummy start transitionT_svstart, have exactly one incoming edge from one other EFFECT transition inPOS(sv)that achieves its pre-condition. This is the case because for each EFFECT transition Tsv that is in-

cluded in POS(sv), Constraint 6 implies that there exists an EFFECT transition that achieves its pre-condition. Similarly, all EFFECT transitions, except for the dummy end transition T_svend, have exactly one outgoing edge to another EFFECT transition in POS(sv), due to Constraint7. Since each action and its transitions can occur at most once, Constraint6and Constraint7together imply that inPOS(sv)all EFFECT transi- tions are sequenced, and since no transition occur more than once this also implies that the sequence is acyclic.

4. Each PREVAIL transition inPOS(sv)appears between two consecutive EFFECT tran- sitions inPOS(sv). This means that each PREVAIL transition has exactly one incoming edge from an EFFECT transitionTsvand exactly one out going edge to another EFFECT

transitionT_sv0 , where Tsv 6= Tsv0 , and there exists an edge fromTsv toTsv0 . This is due

to the fact that each transition inPOS(sv)must have their pre-condition satisfied (Con- straint6) and if the EFFECT transitionTsv achieves the pre-condition of the PREVAIL

Figure 3.5: Example ofPOSfor a state variable

transitionT_svP, and the EFFECT transitionT_sv0 followsT_svP, then Constraint9ensures that Tsvmust achieve the pre-condition of the EFFECT transitionTsv0 .

For each state variablesv, POS(sv)represents a set of ordering relations between the active transitions on the state variablesv, such that the EFFECT transitions are sequenced, where T_svstartappears in the first position andT_svendis at the last position, and each PREVAIL transition appears in between two EFFECT transitions, i.e. is ordered with the EFFECT transitions that appear immediate before and after it. Figure 3.5 describes an example of such POS(sv), whereStartandEndrepresents the dummy start and end transitions onsv,T1,T2,T3, andT6

are EFFECT transitions and T4 andT5 are PREVAIL transitions that appear in between two

EFFECT transitionsT3andT6.

For each state variable sv, the POS(sv)created by the solution to the constraint model, eachexecutionof thePOS(sv)achieves the conditions of avalid scheduleon a state variable, as described in Definition5in the previous chapter (page33). This is because eachexecution

ofPOS(sv)

• Ensures the correctevolutionof the state variable sv, because all EFFECT transitions are sequenced.

• Achieves the pre-conditions of all EFFECT transitions, and satisfies the overall condi- tions of the PREVAIL transitions. Each included PREVAIL transition appears in be- tween two EFFECT transitions, meaning that the PREVAIL transition starts execution

§3.5 Solution To the Constraint Model 65

after the EFFECT transition that achieves its required state and finishes before the EF- FECT transition that changes the state to another.

• Starts the evolution of the state variable sv from the initial state init(sv). The first transition inPOS(sv)isT_svstartthat achieves the initial state. The EFFECT transitionTsv

that executes immediately afterTstart

sv must havepre(Tsv) = post(Tsvstart) = init(sv).

That means every execution ofPOS(sv)starts from the stateinit(sv).

• Achieves the goal stategoal(sv), ifsvis a goal state variable. InPOS(sv)the last tran- sition is T_svend. Since all transitions’ pre-conditions are satisfied, the second to last EF- FECT transitionTsvin the sequence, must have post(Tsv) = pre(Tsvend) = goal(sv).

This means that each execution ofPOS(sv), wheresvis a goal state variable, ends with the stategoal(sv).

This means that eachexecutionofPOS(sv)represents a valid schedule on the state variable sv. In other words eachPOS(sv)represents a set ofvalid schedulesonsv.