Dynamically generated commitment protocols in open systems

Abstract

Agent interaction is a fundamental part of any multiagent system. Such interactions are usually regulated by protocols, which are typically defined at design-time. However, in many situations a protocol may not exist or the available protocols may not fit the needs of the agents. In order to deal with such situations agents should be able to generate protocols at runtime. In this paper we develop a three-phase framework to enable agents to create a commitment protocol dynamically. In the first phase one of the agents generates candidate commitment protocols, by considering its goals, its abilities and its knowledge about the other agents’ services. We propose two algorithms that ensure that each generated protocol allows the agent to reach its goals if the protocol is enacted. The second phase is ranking of the generated protocols in terms of their expected utility in order to select the one that best suits the agent. The third phase is the negotiation of the protocol between agents that will enact the protocol so that the agents can agree on a protocol that will be used for enactment. We demonstrate the applicability of our approach using a case study.

Appendix: formal proofs

This appendix provides the formal proofs for the theorems in Section 3.3.

Theorem 3

(Soundness) Algorithm 2 is sound (see Definition 9).

Proof

we need to show that the algorithm, when given a set of goals, generates commitments that are sufficient to ensure support for these goals. The algorithm closely follows the structure of Definition 8, and we prove the desired result by induction over the structure of the algorithm’s first argument.

• Lines 1–2 of the algorithm correspond to the base case: if there are no propositions to be supported, then the agent’s goals ($\top$) are trivially supported by the empty protocol.

• Lines 3–5 correspond to the second case: the condition $d_1 \wedge d_2$ is supported if both $d_1$ and $d_2$ are supported, (and more generally, $d_1 \wedge \cdots \wedge d_n$ is supported if all of the $d_i$ are supported). By the induction hypothesis, the recursive calls to GoalBased yield sets of protocols $P_1$ and $P_2$ such that for $d_1$ we have that for any $p^1_i \in P_1 : {x, p^1_i} \Vdash {d_1}$, and respectively for $d_2$ we have that for any $p^2_j \in P_2 : {x, p^2_j} \Vdash {d_2}$. Consider now $p_{ij} = p^1_i \cup p^2_j$ (recall that protocols are just sets of commitments, so they can be merged using set union). We have that ${x, p_{ij}} \Vdash {d_1}$ and ${x, p_{ij}} \Vdash {d_2}$ and therefore, by the second case of Definition 8, that ${x, p_{ij}} \Vdash {d_1 \wedge d_2}$ as desired. This holds for any selection of $p^1_i$ and $p^2_j$ which is exactly what the merge in line 5 does.

• Lines 13–20 correspond to the first part of the last case of Definition 8 (agent using its own ability): The loop in line 13 finds all cases where the first part of the condition in Definition 8 is satisfied. Line 14 creates a fresh goal queue, and lines 15–17 decompose $d$ into propositions. The recursive call in line 18 generates protocols $P'$ such that (by inductive hypothesis) for all $p \in P' : {x, p} \Vdash {d}$. Given this, and that (as per line 13) $A_{x}({d}, {r'}) \in \mathcal {A}$ where $r' \Rightarrow r$, by Definition 8 we conclude that for all $p \in P' : {x, p} \Vdash {r}$ as desired.

• Lines 21–34 correspond to the second part of the last case of Definition 8: The loops in lines 21–22 find all cases where $S_{x}({y}, {d}, {r'}) \in \mathcal {B}$ (where $r' \Rightarrow r$) and $I_{x}({y}, {w}, {r}) \in \mathcal {B}$. The next few lines create a fresh goal queue, and decompose $d$ into propositions, inserting them into the goal queue, and then (line 27) also inserting $w$ into the goal queue. The recursive call in line 28 thus, by induction hypothesis, generates a set of protocols $P$ such that for any $p \in P : {x, p} \Vdash {d \wedge w}$. We then define $P' = \{ p \cup \{C({y}, {x}, {d \wedge w}, {r})\} \; | \; p \in P \}$. By Definition 8 we then conclude that for any $p' \in P'$, we have that ${x, p'} \Vdash {r}$ as desired.$\square$

Theorem 4

(Completeness) Algorithm 2 is complete (see Definition 11).

Proof

Algorithm 2 follows Definition 10, and the proof, by induction over the structure of the first argument to the algorithm, follows the algorithm’s structure.

• The first case of Definition 10 corresponds to lines 1–2: if there are no conditions to be supported, then a single empty protocol is returned. This meets the completeness requirement: the only minimal commitment set in this case is the empty set, and this is exactly what the algorithm returns.

• The second case corresponds to lines 3–5: given $d_i \wedge d_j$ the algorithm is called recursively to obtain support for each of the conjuncts. By the induction hypothesis we have that $P_i$ and $P_j$ (respectively the set of protocols that support $d_i$ and $d_j$) contain all (and only) minimal commitment sets. The merge in line 5 selects $p \in P_i$ and $p' \in P_j$ and merges them ($p \cup p'$) in line with Definition 10. Hence the algorithm, which considers all possible combinations of $p$ and $p'$, generates exactly all possible minimal protocols that support $d_i \wedge d_j$.

• The first part of the last case corresponds to lines 13–20: by induction hypothesis the recursive call generates all minimal commitment sets that support $d$, and, by Definition 10 these are exactly the ones that support $r$ in this case, which is what the algorithm returns.

• The final part of the last case corresponds to lines 21–34: by the induction hypothesis the recursive call generates all minimal commitment sets that support $d \wedge w$. The algorithm then considers these commitment sets, and adds the extra commitment $C({y}, {x}, {d\wedge w}, {r})$ to each one, yielding exactly the complete collection of minimal commitment sets for $r$. $\square$