The Evolution of Commitments during Reconfiguration
6.5.2.2 Case 2: an Action Failed
In the sequel, some actions fail during team action. Then, we show the evolution of collective commitment according to the reasons for failure given in the reconfiguration algorithm. Inevitably, the “old” collective commitment has to be dropped because the social commitments with respect to the failed actions from S-COMMG,P(ϕ) do not exist anymore. After an action failure, the situation is not a priori hopeless: the collective commitment may still evolve, leading to a good end. This evolution is done according to
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a conservative revision of the social plan P , resulting in a new plan P. The particular cases will differ with respect to the stage where the re-planning actually starts: at action allocation, at means-end analysis or even earlier, at task division. Thus, we split this situation into four cases of varying difficulty.
Case 2a: Reallocation possible
When other agents are prepared to realize the failed actions, that is, when action re- allocation is possible, a new planPis devised, starting from a new action allocation. In this way the results of the previous task division and means-end analysis are conserved, taking minimal costs: only a new action allocation is performed. Finally, a new collective commitment based onP is constructed. This is expressed by the property below. Property: reallocation possible
Suppose that there is an(i, α)∈ P such that failed(i, α) and objectiveG(α) and no failed α blocks ϕ, that is,¬necessary(α, ϕ) holds for all objectively failed actions. Then for the
current action sequenceτ and a new social plan P we have for all Kripke modelsM in which the teamwork axioms hold, and all worldsw:
C-INTG(ϕ)∧ division(ϕ, σ ) ∧ means(σ, τ) →
[confirm(succ(action-allocation(τ, P);construction(ϕ, G, P)))] S-COMMG,P(ϕ)
Proof SupposeM, w | C-INTG(ϕ)∧ division(ϕ, σ ) ∧ means(σ, τ). Now by the second property in Lemma 6.1, it suffices to show that if:
M, w | succ(action-allocation(τ, P);construction(ϕ, G, P)), then M, w | S-COMMG,P(ϕ); so suppose:
M, w | succ(action-allocation(τ, P);construction(ϕ, G, P)) It immediately follows by axiom FR1 that M, w | allocation(τ, P). Com-
bined with M, w | division(ϕ, σ ) ∧ means(σ, τ) this implies by axiom C0 that
M, w | constitute(ϕ, P). On the other hand, by the first property of Lemma 6.1 we
deriveM, w | succ(construction(ϕ, G, P)) from:
M, w | succ(action-allocation(τ, P);construction(ϕ, G, P)) Thus we have:
M, w | C-INTG(ϕ)∧ constitute(ϕ, P)∧ succ(construction(ϕ, G, P) and so by postulate CTR we conclude M, w | S-COMMG,P(ϕ), as desired.
The example Suppose thatl does not succeed in proving Lemma 1 and in fact believes
that it cannot as it misses some knowledge about elliptic curves, whicht does have. After t communicates that it will pitch in for l, COMM(l, t, provelemma1 ) (and thus the old
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collective commitment) is dropped, and a new social plan is devised, for example:
P =provelemma2, l provelemma1, t; provetheorem, t
checklemma1, c; checklemma2, c; checktheorem, c
Finally, a new strong collective commitment is constructed, containing the social com- mitment COMM(t, l, provelemma1 ).
Case 2b: some failed action blocks the goal
In this case some action α that was necessary for achieving the goal failed and cannot
be re-allocated, that is objectiveG(α) and necessary(α, ϕ) hold. This is the most serious
negative case, inevitably leading to system-failure.
Property: goal blocked
Suppose that there is an(i, α)∈ P such that failed(i, α) and objectiveG(α) and α blocks ϕ, that is, necessary(α, ϕ) holds for an objectively failed action. Then for all Kripke
models M in which the teamwork axioms hold, and all worlds w:
M, w | failed(i, α) ∧ objectiveG(α)∧ necessary(α, ϕ) → do(system-failure(ϕ))
This formalizes the postulate to be ensured by the system designer. Thus, if it is discovered that a failed action blocks ϕ, the system fails to achieve ϕ and stops. This implies that
neither a collective intention nor an evolved collective commitment towards ϕ will be
established. In the Appendix, an alternative account of this case is formalized in the language of branching time temporal logic, with a focus on formalizing the concept of blocking.
The example Suppose that, while checkingt’s proof of the theorem from the lemmas, c
discovers that not only the proof is wrong but also finds a counterexample to the theorem. Then nothing can be done to remedy the problem. This concludes the case.
Case 2c: New means-end analysis possible
In this case action reallocation is not possible because there are some objectively failed actions. This means that for every relevant social planP, allocation with respect to the current action sequence τ fails. Furthermore, in this case each objectively failed action
does not block the goal. In this situation, the old collective commitment is dropped but its evolution is still possible, if a new means-end analysis yields new actions realizing the failed subgoals, allowing for a new allocation of them. This is expressed by the following property.
Property: new means-end analysis possible
Suppose that there is an(i, α)∈ P such that failed(i, α) and objectiveG(α) and no failed α blocks ϕ, that is,¬necessary(α, ϕ) holds for all objectively failed actions. Then for the
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τ and P excluding the objectively failed actions such that the following holds for all Kripke modelsM in which the teamwork axioms hold, and all worlds w:
C-INTG(ϕ)∧ division(ϕ, σ ) →
[confirm(failed(action-allocation(τ, P)))] [confirm(succ(means-end-analysis(σ, τ);
action-allocation(τ, P);construction(ϕ, G, P)))] S-COMMG,P(ϕ)
Proof Suppose M, w | C-INTG(ϕ)∧ division(ϕ, σ ). Now by the second property in Lemma 6.1, it suffices to show that if:
M, w | succ(means-end-analysis(σ, τ);action-allocation(τ, P);
construction(ϕ, G, P));
then M, w | S-COMMG,P(ϕ); so suppose:
M, w | succ(means-end-analysis(σ, τ);action-allocation(τ, P);
construction(ϕ, G, P)).
It immediately follows by axiom FR2 that M, w | means(σ, τ)∧ allocation(τ, P).
Combined with M, w | division(ϕ, σ ) this implies by axiom C0 that
M, w | constitute(ϕ, P).
On the other hand, by the first property of Lemma 6.1 we derive:
M, w | succ(construction(ϕ, G, P))
and, exactly as in case 2a, we derive M, w | S-COMMG,P(ϕ) by CTR.
The example As in case 2a, suppose thatl does not succeed in proving Lemma 1, but now t and c do not believe they can prove it, either. The team does a new means-end analysis
based on the old subgoal sequence, and comes up with some other lemmas (say 3, 4 and 5) that together hopefully imply the theorem. This gives rise to a new action sequence
τ= provelemma3 , provelemma4 , provelemma5 , checklemma3 , checklemma4 , check- lemma5 , provetheorem, checktheorem . They allocate the actions in a similar way as
before, creating a new social plan P, for example:
P= provelemma3, l; provelemma4, l;
provelemma5, l provetheorem, t; checklemma3, c; checklemma4, c; checklemma5, c; checktheorem, c.
Finally, by public communication they establish new social commitments leading to a new strong collective commitment.
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Case 2d: New task division possible
When no objectively failed action blocks the goal but neither action reallocation, nor a new means-end analysis is possible for the failed actions, this means that for the current
τ , action allocation fails to deliver any social plan P and then means-end analysis with respect to the current σ fails to deliver any action sequence τ not containing the objec- tively failed actions. Even in this difficult case, the evolution of collective commitment is still possible. This happens when task division for the goalϕ is successfully executed,
resulting in a new goal sequenceσ. Then, this sequence is a subject of a new round of means-end analysis, establishing a new action sequence τ. Next follows action alloca- tion, to create a new social planP on the basis ofτ. The following property describes the result.
Property: new task division possible
Suppose there is an(i, α)∈ P such that failed(i, α) and for all failed α, ¬necessary(α, ϕ).
Then for the current goal sequence σ and action sequence τ , and for every social plan P and action sequenceτ, there are σ,τ andP such that:
C-INTG(ϕ)→
[confirm(failed(action-allocation(τ, P)))] [confirm(failed(means-end-analysis(σ, τ)))]
[confirm(succ(division(ϕ, σ);means-end-analysis(σ, τ); action-allocation(τ, P);construction(ϕ, G, P)))] S-COMMG,P(ϕ)
Proof Suppose M, w | C-INTG(ϕ). By the second property in Lemma 6.1, it suffices to show that if:
M, w | succ(division(ϕ, σ);means-end-analysis(σ, τ); action-allocation(τ, P);construction(ϕ, G, P)); then M, w | S-COMMG,P(ϕ); so suppose:
M, w | succ(division(ϕ, σ);means-end-analysis(σ, τ);
action-allocation(τ, P);construction(ϕ, G, P)).
It immediately follows by axiom FR3 that:
M, w | division(ϕ, σ)∧ means(σ, τ)∧ allocation(τ, P)
This implies by axiom C0 thatM, w | constitute(ϕ, P).
On the other hand, by the first property of Lemma 6.1 we derive:
M, w | succ(construction(ϕ, G, P))
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The example Suppose that the theorem has been divided into lemmas several times and each time it was impossible to prove some essential lemma. Then the team concludes that they are not able to prove the theorem by formulating and proving suitable lemmas. Then they may come up with a completely different task division, for example σ= σ3, σ4
whereσ3= “a theorem analogous to T has been found in a different area of mathematics”
andσ4= “a suitable translation between the two contexts has been defined”. On means-
end analysis and action allocation result in a social plan P very different fromP .
In case 2d, if task division is not successful, the story of the current team is completed and a return to team formation is made in order to establish a new team attempting to achieve ϕ. In this way, the evolution of the collective commitment is completed
as well.