Ausencia y presencia del fuego

and interdependencies given above, will be called TEAMLOGind (the individual part of teamwork logic) in the sequel .

As to the side-effect problem, note that TeamLogind _{fortunately does not prove that an}

agent intends all the consequences it believes its intentions to have, that is the believed side-effects of its intentions.

Thus, TeamLogind  BEL(i, ϕ → ψ) → (INT(i, ϕ) → INT(i, ψ)).

There is a weaker version that does hold, though, namely if| ϕ → ψ (a quite strong assumption!), then by R2I, we have| INT(i, ϕ) → INT(i, ψ). Therefore, agents intend all logical consequences of their intentions. This is similar to the logical omniscience problem for logics of knowledge and belief discussed in Section 2.7.1. For a discussion of the ‘side-effect problem’ for intentions, see Bratman (1987), Cohen and Levesque (1990) and Rao and Georgeff (1991).

3.4

Collective Intention Constitutes a Group

Collective intention, as a specific joint mental attitude, is the central topic addressed in teamwork. We agree with Levesque et al. (1990) that:

Joint intention by a team does not consist merely of simultaneous and coordinated individual actions; to act together, a team must be aware of and care about the status of the group effort as a whole.

2_{These and similar correspondences can be automatically computed at http://www.fmi.uni-sofia.bg/}

Collective Intentions 37

In TeamLog, teams are created on the basis of collective intentions: a team is constituted as soon as a collective intention among the members is present and stays together as long as the collective intention persists. In this chapter we focus on defining several notions of collective intentions in Section 3.5 and Section 3.7, abstracting from the team formation process. We refer the interested reader to Chapter 8 and Castelfranchi et al. (1992), Dignum et al. (2001a, b), Jennings (1993) and Wooldridge and Jennings (1999).

In contrast to Cohen and Levesque (1990), we are interested in generic characteristics of intentions, resigning from classifying them further along different dimensions. The collective choice that is ‘hidden’ in group intention directly leads to a collective com- mitment. The essential characteristics of commitments follow the linguistic tradition that while intentions ultimately lead to actions, the immediate triggers of these actions are commitments. In fact, social (or bilateral) commitments are related to individual actions, while collective commitments are related to plan-based team actions.

As mentioned before, we agree with Bratman (1987) that in human practical reasoning, intentions are first class citizens, that are not reducible to beliefs and desires. They form a rather special consistent subset of an agent’s goals, that it wants to focus on for the time being. This way they create a screen of admissibility for the agent’s further, possibly long-term, deliberation. In this chapter we extend this view to the collective intention case. In TeamLog, collective intentions are not introduced as primitive modalities, with some restrictions on the semantic accessibility relations (as in, for example, Cavedon

et al . (1997)). We do give necessary and sufficient conditions for collective motivational

attitudes to be present, making teamwork easier to predict. Collective commitments are treated in Dunin-K ¸eplicz and Verbrugge (1996, 1999) and most extensively in Chapter 4. In the philosophical and MAS literature there is an ongoing discussion as to whether collective intentions may be reduced to individual ones plus common beliefs about them (see Castelfranchi (1995), Haddadi (1995) and Tuomela and Miller (1988)). Even though our definition in Section 3.5 seems to be reductive, it involves infinitely nested intentions and group epistemic operators, making them much deeper than a simple compound built out of individual intentions and common beliefs by propositional connectives only.

Despite the overall complexity of collective motivational notions, in our investiga- tion we tried to find minimal conditions characterizing them, and not to weigh down the definitions with all possible aspects applicable in specific situations. Such elements as conventions, abilities, opportunities, power relations and social structure (see Singh (1997), Tuomela (1995) and Wooldridge and Jennings (1999) for a thorough discussion) certainly are important. Therefore, we leave open the possibility of extending TeamLog by additional properties. For example, abilities and opportunities are important in dia- logues recognizing potential towards a specific goal. These dialogues will be discussed in Chapter 8 (see also Dignum et al. (2001b)). Power relations and social structure, on the other hand, are reflected in collective commitments (see Chapter 4).

3.5

Definitions of Mutual and Collective Intentions

In this book, we focus on strictly cooperative teams, where ‘cooperative’ is meant in a stronger sense than the homonymous concept in game theory. See Bratman (1999) and Tuomela (1995) for good philosophical discussions on strong types of cooperation and collaboration needed in teamwork. This essence of cooperation makes the concept of collective intention rather powerful.

38 Teamwork in Multi-Agent Systems

So, what motivates a group of agents to combine their efforts to achieve a given goal ϕ? First, they all need to individually intend ϕ. This leads to the so-called general intention E-INTG(ϕ) in the group G (see Table 3.2 for the relevant formulas). In fact, this necessary condition is taken to fully characterize collective intention in Rao et al. (1992) (see Wooldridge and Jennings (1996) for a similar definition of collective goal). However, this is certainly not sufficient. Imagine that two agents want to reach the same goal but are in a competition, willing to achieve it exclusively. Therefore, to exclude cases of competition, all agents should intend all members to have the associated individual intention, as well as the intention that all members have the individual intention, and so on. This simply means that general intention should be iterated in order to express the reciprocity of this process: ‘everyone intends that everyone intends that everyone intends that. . . ϕ’. We will call the resulting attitude a mutual intention M-INTG(ϕ).

Table 3.2 Group formulas and their intended meanings.

E-INTG(ϕ) Every agent in groupG has the individual intention to achieve ϕ

M-INTG(ϕ) GroupG has the mutual intention to achieve ϕ

C-INTG(ϕ) GroupG has the collective intention to achieve ϕ

Even though mutual intention creates the motivational core of group intention, it would’t be enough, if the agents weren’t aware about their mutual attitudes. Thus, group members need to be aware of their reciprocal intentions. As discussed in the previous chapter, there are many ways of defining group awareness. Paradigmatically, in teamwork it is expressed by common belief C-BELG(M-INTG(ϕ)), which we choose for the time being. This way a loosely coupled group becomes a strictly cooperative team. Of course, team members remain autonomous in maintaining their other motivational attitudes and may even be in competition about other issues.

In order to formalize the above conditions, a general intention E-INTG(ϕ) (standing for ‘everyone intends’) is defined by the following axiom, corresponding to the semantic condition thatM, s | E-INTG(ϕ) iff for all i∈ G, M, s | INT(i, ϕ):

M1 E-INTG(ϕ)↔
i∈GINT(i, ϕ).

The mutual intention M-INTG(ϕ) is meant to be true if everyone in G intends ϕ, every- one inG intends that everyone in G intends ϕ, etc. As we do not have infinite formulas to

express this, let E-INT1_G(ϕ) be an abbreviation for E-INTG(ϕ), and let E-INTkG+1(ϕ) for k > 1 be an abbreviation of E-INTG(E-INTkG(ϕ)). Thus we haveM, s | M-INTG(ϕ) iff M, s | E-INTk

G(ϕ) for all k≥ 1.

Define worldt to be GI-reachable from world s iff (s, t)∈ ( 

i∈GIi)+, the transitive closure of the union of all individual accessibility relations. Formulated more informally, this means that there is a path of length ≥ 1 in the Kripke model from s to t along accessibility arrowsIi that are associated with members i of G.

Then the following property holds (see Section 2.4 and Fagin et al. (1995) for analogous properties for common belief and common knowledge, respectively):

Collective Intentions 39

Using this property, it can be shown that the following fixed-point axiom and rule can be soundly added to the union of KDn and M1:

M2 M-INTG(ϕ)↔ E-INTG(ϕ∧ M-INTG(ϕ))

RM1 Fromϕ→ E-INTG(ψ ∧ ϕ) infer ϕ → M-INTG(ψ) (Induction Rule)

The resulting system is called TeamLogmint _{(the part of teamwork logic for mutual}

intentions) and is sound and complete with respect to Kripke models where alln accessibil-

ity relations for intentions are serial. The completeness proof will be given in Section 3.8. Now we will show the soundness of Rule RM1 with respect to the given semantics. (The other axioms and rules of TeamLogmint are more intuitive, so we leave their soundness to the reader.)

Assume that | ϕ → E-INTG(ψ∧ ϕ), meaning that ϕ → E-INTG(ψ∧ ϕ) holds in all worlds of all Kripke models. We need to show that | ϕ → M-INTG(ψ). So take any Kripke model M = (W, {Bi :i∈ A}, {Gi :i∈ A}, {Ii :i∈ A}, Val) with A = {1, . . . , n}, and any world s ∈ W with M, s | ϕ. Now assume that t is

GI-reachable from s in k steps along the path w0, . . . wk with w0= s and wk = t, by k≥ 1 relations of the form Ij(j∈ {1, . . . , n}). We need to show that M, t | ψ, for which we can show step by step that ψ∧ ϕ holds in all worlds wi, i≥ 1, on the path from s to t. For the first step, for example sIjw1, we can use the fact that

M, s | ϕ → E-INTG(ψ∧ ϕ), and thus M, s | E-INTG(ψ∧ ϕ), to conclude that M, w1| ψ ∧ ϕ. Repeating this reasoning on the path to t, we conclude that for

all i with 1≤ i ≤ k, M, wi | ψ ∧ ϕ, in particular M, t | ψ. We conclude that M, s | ϕ → M-INTG(ψ), as desired.

Finally, the collective intention is defined by the following axiom: M3 C-INTG(ϕ)↔ M-INTG(ϕ)∧ C-BELG(M-INTG(ϕ))

The definition would be even stronger if common knowledge were applied in M3. However, because common knowledge is almost impossible to establish in multi-agent systems due to the unreliability of communication media (see Chapter 2), we do not pursue this strengthening further.

Definition 3.5 The resulting system, which we call TeamLog, is the union of T eamLogmint(for mutual intentions), KD45C_n (for common beliefs) and axiom M3.

3.5.1 Some Examples

Let us give an informal example of the establishment of a collective intention. Two violinists,a and b, have studied together and have toyed with the idea of giving a concert

together someday. Later this has become more concrete: they both intend to perform the two solo parts of the Bach Double Concerto, expressed in INT(a, ϕ) and INT(b, ϕ),

where ϕ stands for ‘a and b perform the solo parts of the Bach Double Concerto’.

After communicating with each other, they start practising together. Clearly, a mutual intention M-INT_{a,b}(ϕ) as defined in M2 is now in place, involving nested intentions like

INT(a, INT(b, INT(a, ϕ))) and so on. The communication established a common belief

C-BELG(M-INTG(ϕ)) about their mutual intention with G= {a, b}, according to M3. As sometimes happens in life, when people are ready, an opportunity appears: Carnegie Hall plans a concert for Christmas Eve, including the Bach Double Concerto. Now they

40 Teamwork in Multi-Agent Systems

refine their collective intention to a more concrete C-INTG(ψ) (where ψ stands for ‘a andb perform the solo parts of the Bach Double Concerto at the Christmas Eve concert

in Carnegie Hall’). Luckily, our two violinists are chosen from among a list of candidates to be the soloists, and both sign the appropriate contract. Because they do this together, common knowledge, not merely common belief of their mutual intention is present:

M-INTG(ψ)∧ C-KNOWG(M-INTG(ψ))

One important difference between common knowledge and common belief is that com- mon knowledge can be justified if needed and a commonly signed contract provides a perfect basis for this. Clearly, the two violinists have developed a very strong variant of collective intention due to their common knowledge of the mutual intention.

3.5.2 Collective Intentions Allow Collective Introspection

The following lemma about positive introspection for collective intentions follows easily from the definition of collective intention, using Lemma 2.1, as we will show below.