AMPLIACION Y DISMINUCION DE LA SUMA AFIANZADA

Definition 3.4.3 (Satisfaction). Given the model M, the satisfaction of a CTLcc,α_formula

ϕin a social state s denoted by (M, s)|= ϕ is defined in Tables 4, 5, 6, 7, 8, and 9.

The formal semantics of the CTL formulae, a temporal fragment of CTLcc,α_{, is intro-}

duced in Chapter 2. In Table 4, the state formula W CC(i, j, ψ, ϕ) is satisfied in the model M at s iff the consequence ϕ holds in every state satisfying ψ and accessible from s via

∼i→j. The semantics of the strong commitment SCC(i, j, ψ, ϕ) is similar, but we add

Condition 1 to stress that the antecedent ψ and the consequent ϕ of commitments should be achieved at least in one state accessible from s, which is necessitated in business com- mitments in a strong manner. The proposed semantics is close to the semantics introduced

Table 4: The semantics of the weak and strong commitment formulae (M, s) |= W CC(i, j, ψ, ϕ) iff ∀s _{∈ S s.t. s ∼}

i→j s and (M, s) |= ψ, we have

(M, s_{) |= ϕ}

(M, s) |= SCC(i, j, ψ, ϕ) iff 1) ∃s∈ S s.t. s∼_i→j sand (M, s) |= ψ, and 2) (M, s) |= W CC(i, j, ψ, ϕ)

by Singh [81] for practical commitments. Singh’s semantics is defined by first comput- ing the set of states where the consequence holds and then testing whether these states are among the set of sets of states satisfying the antecedent, which in turn explicitly means this set of states should satisfy the consequence and antecedent of commitment together. Moreover, our semantics guarantee that any conflict between two or more strong commit- ments (enforceable commitments) will not exist as if one commitment has been activated, the other conflicting commitments will be not active in the same state (cf. R₁₁ in Section 3.6). Also, two conflicting commitments can be individually enforceable in different states in the future.

In Table 5, the state formula F uW (i, W CC(i, j, ψ, ϕ)) is satisfied in the model M at s iff s satisfies the consequence ϕ and the negation of the weak commitment W CC(i, j, ψ, ϕ) as well as there exists a state s _{at which i performs an action to fulfill her commitment}

holding at s _{and s is “seen” and reached from this state via ∼}

i→j and Rci. The idea

behind this semantics is to say that a weak commitment is fulfilled when we reach an Table 5: The semantics of the fulfillment action formulae

(M, s) |= F uW (i, W CC(i, j, ψ, ϕ)) iff ∃s _{∈ S s.t. s} _∼

i→j s and (li(s), Fulfilli,

l_i(s)) ∈ R_ciand (M, s) |= W CC(i, j, ψ, ϕ) and (M, s)|= ϕ ∧ ¬W CC(i, j, ψ, ϕ)

(M, s) |= F uS(i, SCC(i, j, ψ, ϕ)) iff ∃s ∈ S s.t. s ∼_i→j s and (l_i(s), Fulfill_i, l_i(s)) ∈ R_ciand (M, s) |= SCC(i, j, ψ, ϕ) and (M, s)|= ψ ∧ ¬SCC(i, j, ψ, ϕ)

accessible state from the weak commitment state by performing the fulfillment action in which the consequence holds and the weak commitment becomes no longer active. The semantics of the strong fulfillment F uS(i, SCC(i, j, ψ, ϕ)) is similar, but the focus is on checking the satisfiability of the antecedent ψ. This is because—from the semantics of

the strong conditional commitment—we guarantee that whenever ψ holds in an accessible state, then the consequence ϕ holds as well. By stressing that the active commitment should be terminated in the fulfillment state, we address the fulfillment paradox appeared in [15] (Proposition 2) and discussed in Chapter 1. Recall that this paradox results from the assumption: unconditional commitment should be active when it comes time to its fulfillment, formally: F u(C(i, j, ϕ)) → C(i, j, ϕ). Terminating (or deleting) commitment being fulfilled is stated explicitly in the operational semantics introduced in [106, 97, 81, 23]. Singh, for instance, uses the following postulate: ϕ → ¬C(i, j, ψ, ϕ) where C could be a practical or dialectical conditional commitment [81].

In Table 6, the formula CaS(i, SCC(i, j, ψ, ϕ)) (respectively, CaW (i, W CC(i, j, ψ, ϕ))) is satisfied in M at s iff there exists a state s _{at which i performs an action to cancel her}

strong (respectively, weak) commitment holding at s _{and s is seen and reached from the}

state s via ∼i→j and Rci. And the current state s satisfies the negation of both the an-

tecedent ψ (respectively, consequence ϕ) and strong commitment SCC(i, j, ψ, ϕ) (respec- tively, weak commitment W CC(i, j, ψ, ϕ)).

Table 6: The semantics of the cancelation action formulae

(M, s) |= CaS(i, SCC(i, j, ψ, ϕ)) iff ∃s ∈ S s.t. s ∼_i→j s and (l_i(s), Cancel_i, l_i(s)) ∈ R_ciand (M, s) |= SCC(i, j, ψ, ϕ) and (M, s)|= ¬ψ ∧ ¬SCC(i, j, ψ, ϕ) (M, s) |= CaW (i, W CC(i, j, ψ, ϕ)) iff ∃s _{∈ S s.t. s} _∼

i→j s and (li(s), Canceli,

l_i(s)) ∈ R_ciand (M, s) |= W CC(i, j, ψ, ϕ) and (M, s)|= ¬ϕ ∧ ¬W CC(i, j, ψ, ϕ)

In Table 7, the formula ReS(i, SCC(i, j, ψ, ϕ)) (respectively, ReW (i, W CC(i, j, ψ, ϕ))) is satisfied in M at s iff there exists a state s _{at which j performs an action to release}

the agent i from her strong (respectively, weak) commitment holding at s _{and s is seen}

and reached from the state s _{via ∼}

i→j and Rcj. And the current state s satisfies the

negation of both the antecedent ψ (respectively, consequence ϕ) and strong commitment SCC(i, j, ψ, ϕ) (respectively, weak commitment W CC(i, j, ψ, ϕ)). The only difference between the semantics of cancelation and release action formulae is the intelligent agent that performs these actions and the label of the local transition.

In Table 8, the formula DeS(i, k, SCC(i, j, ψ, ϕ)) (respectively, DeW (i, k, W CC(i, j, ψ, ϕ))) is satisfied in M at s iff there exists a state s _{at which i performs an action to del-}

egate her strong (respectively, weak) commitment holding at s _{to another agent k and s}

is seen and reached from the state s _{via ∼}

i→j and Rci. And the current state s satis-

Table 7: The semantics of the release action formulae

(M, s) |= ReS(j, SCC(i, j, ψ, ϕ)) iff ∃s ∈ S s.t. s ∼_i→j s and (l_j(s), Release_j, l_j(s)) ∈ R_cj and (M, s) |= SCC(i, j, ψ, ϕ) and (M, s)|= ¬ψ ∧ ¬SCC(i, j, ψ, ϕ) (M, s) |= ReW (j, W CC(i, j, ψ, ϕ)) iff ∃s _{∈ S s.t. s} _∼

i→j s and (lj(s), Releasej,

l_j(s)) ∈ R_cj and (M, s) |= W CC(i, j, ψ, ϕ) and (M, s)|= ¬ψ ∧ ¬W CC(i, j, ψ, ϕ)

SCC(i, j, ψ, ϕ)(respectively, weak commitment W CC(i, j, ψ, ϕ)) as well as the new com- mitment created by k.

Table 8: The semantics of the delegation action formulae

(M, s) |= DeS(i, k, SCC(i, j, ψ, ϕ)) iff ∃s ∈ S s.t. s ∼_i→j s and (l_i(s), Delegate_i, l_i(s)) ∈ R_ci and (M, s) |= SCC(i, j, ψ, ϕ) and (M, s) |= ¬ψ ∧ ¬SCC(i, j, ψ, ϕ)∧

SCC(k, j, ψ, ϕ) (M, s) |= DeW (i, k, W CC(i, j, ψ, ϕ)) iff ∃s _{∈ S s.t. s} _∼

i→j s and (li(s), Delegatei,

l_i(s)) ∈ R_ci and (M, s) |= W CC(i, j, ψ, ϕ) and (M, s)|= ¬ϕ ∧ ¬W CC(i, j, ψ, ϕ) ∧W CC(k, j, ψ, ϕ)

In Table 9, the formula AsS(j, k, SCC(i, j, ψ, ϕ)) (respectively, AsW (j, k, W CC(i, j, ψ, ϕ))) is satisfied in M at s iff there exists a state s_{at which j performs an action to assign}

her strong (respectively, weak) commitment holding at s _{to another agent k and s is seen}

and reached from the state s_via∼

i→j and Rcj. And the current state s satisfies the negation

of the antecedent ψ (respectively, consequence ϕ) and strong commitment SCC(i, j, ψ, ϕ) (respectively, weak commitment W CC(i, j, ψ, ϕ)) as well as the new commitment cre- ated by k. The differences between the semantics of delegation and assignment action

Table 9: The semantics of the assignment action formulae

(M, s) |= AsS(j, k, SCC(i, j, ψ, ϕ)) iff ∃s ∈ S s.t. s ∼_i→j s and (l_j(s), Assign_j, l_j(s)) ∈ R_cj and (M, s) |= SCC(i, j, ψ, ϕ) and (M, s)|= ¬ψ ∧ ¬SCC(i, j, ψ, ϕ)∧ SCC(i, k, ψ, ϕ)

(M, s) |= AsW (j, k, W CC(i, j, ψ, ϕ)) iff ∃s _{∈ S s.t. s} _∼

i→j s and (lj(s), Assignj,

l_j(s)) ∈ R_cj and (M, s) |= W CC(i, j, ψ, ϕ) and (M, s)|= ¬ϕ ∧ ¬W CC(i, j, ψ, ϕ)∧ W CC(i, k, ψ, ϕ)

put, the idea of the proposed semantic rules is that a commitment is fulfilled (respectively cancelled, released, delegated and assigned) when we reach an accessible state from the commitment state by performing the fulfillment (respectively cancelation, release, delega- tion and assignment) action in which the antecedent holds (respectively not holds) and the commitment becomes no longer active as well as in the case of delegation and assignment action formulae, a new commitment is created.