Now that the formalism of all the possible arguments has been detailed, to find out what argu- ments are possible in the dialogue at any time, two protocols have been defined: theC-inqproto- col in Section3.4.1that finds all the possible defeasible facts, defeasible rules andB-arguments that can be communicated in the C-inq dialogue; and the C-pAct protocol in Section 3.4.2 that finds all the possibleC-arguments and critical questions that can be communicated in the C-pActdialogue. It is then left to the agents themselves to autonomously decide which (if any) of the possible moves to use.
3.4.1 Defining the Inquiry Protocol
An inquiry protocol to find the state of the world, is now formally defined in Definition 73 (below) through the Ξ function, using the protocol format detailed in [21]. This Ξ protocol function returns the set of possible moves denoted℘I(M)(from the set of all possible moves M) that are legal for each agent (fromN) that has joined the dialogue (Dt
r) of the typeC-Inq. This protocol will not allow any proposition to become a claim of a B-argument without supporting evidence. Supporting evidence takes the form of defeasible facts or a fully supported defeasible rule. A defeasible ruleλis fully supported when there is a defeasible derivation for the consequence of the rule that includes the rule and can be constructed from the union of all the commitment stores.
The protocol works by initially allowing each agentjto identify all of itsrelevant beliefsthat can be asserted in the current dialogue that are not already present in the commitment store (see
Ξa(Dt1, j)in the following definition). A belief can be either a defeasible rule or a defeasible fact (seeΞa(D1t, j)part (1)). A defeasible fact isrelevantto the current dialogue (seeΞa(D1t, j) part (2)) if: (2)(i) it is not already present in the commitment store and is present in agent j’s belief base (i.e.Σj); and (2)(ii,a) it is an element of the dialogue topic or (2)(ii,b) it is an element of a defeasible rule in the combined commitment store of all of the agents. A defeasible rule is relevantto the current dialogue (seeΞa(D1t, j)part (3)) if: (3)(i) it is not already present in the commitment store and is present in agent j’s belief base (i.e. Σj); and (3)(ii,a) its consequent
returns a defeasible fact that is an element of the dialogue topic or (3)(ii,b) its consequent returns a defeasible fact that is an element of another defeasible rule in theCoSt.
Next, the agent checks to see if any of its asserted beliefs (denotedφ) are now fully supported by a set of defeasible facts and defeasible rules (denotedΦ) in the commitment store, i.eΦ ⊆ CoSt(seeΞb(Dt1, j)in the following definition). If some asserted beliefs are found to be fully supported, these beliefs can be asserted asB-arguments in the formB=hΦ, φi, as long as they have not been asserted already (i.e.B∈/ CoSt). Each agentjcan only assert aB-argument with claimφif it asserted the belief that includedφ(i.e. φ∈ CoStti). This is to eliminate multiple assertions ofB-arguments. Lastly if the agents cannot assert anything new then the moves that are returned are the ‘close’ move to be used when the agents want to remain in the dialogue to hear what the other agents have to say, or the ‘leave’ move to be used if the agent wants to leave the dialogue completely.
Definition 73: TheC-inq protocolfor aC-inqdialogue is a functionΞ :D ×N 7→ ℘I(M). If j ∈ N is the given agent and Drt is the given dialogue where Ags(Drt) = N, CoSt =
S
∀k∈NCoSttk,Type(Dtr) =C-Inqand1≤t, thenΞ(Drt, j)is
Ξa(D1t, j)∪Ξb(Dt1, j)∪ {hj, close, dialogue(C-Inq,Topic(Dtr))i} ∪ {hj, leave, dialogue(C-Inq,Topic(Dtr))i}
where
Ξa(D1t, j) ={hj, assert,Ψi|
(1)Ψ6=∅whereΨis a set of beliefs, and (2)∀φ∈Ψwhereφis a defeasible fact:
(i)φ6∈CoSt, φ∈Σj , and either (ii,a)φ∈Topic(Dtr),
or (ii,b)∃λ∈CoSts.t.φ∈DefeasibleSection(λ)
(3)∀λ∈Ψwhereλis a defeasible rule: (i)λ6∈CoSt, λ∈Σj, and
either (ii,a)DefeasibleProp(λ)∈Topic(Dt r),
or (ii,b)∃λ0 ∈CoSts.t.DefeasibleProp(λ)∈DefeasibleSection(λ0) Ξb(Dt1, j) ={hj, assert,Υi|
(1)Υ6=∅,Υis a set ofB-arguments , and
(2)∀B ∈Υ:B=hΦ, φiwhere:Φ⊆Ψ∪CoSt,φ∈Ψ∪CoSttiandB∈/ CoSt
If every agent is using theC-inqprotocol then a C-inqdialogue, just like the inquiry dia- logue of [22], is guaranteed to terminate according to the following Theorem:
Theorem 3.1. If each agent in aC-inqdialogue is using theC-inqprotocol, then the dialogue is guaranteed to terminate (given each agent’s beliefs is of finite size).
Proof. Assume that allnagents in aC-inqdialogue game are using theC-inqprotocol but the dialogue never terminates. For this to occur, the agents must be constantly asserting arguments of defeasible rules or defeasible facts (because n leave moves will terminate the dialogue as
there will be no agents left andnclose moves will terminate the dialogue as no more agents have any more arguments to add).
Yet according to theC-inqprotocol condition (2)(i) and (3)(i), an argument for a defeasible fact or defeasible rule can only be asserted if it has not previously been asserted by any agent. As it is given that each agent’s beliefs is of finite size, then in the worst case, each agent will assert all of the defeasible rules and defeasible facts that it is aware of. Once this has occurred, the only moves the agents will have left is the close or leave move andnleave moves ornclose moves in a row will terminate the dialogue, which contradicts the assumption.
Thus, as every belief of an agent could theoretically be communicated, the tractability of theC-inqdialogues (like the inquiry dialogues in [22]) depends on the total number of possible beliefs of all the agents.
3.4.2 Extending the pAct Protocol
APersuasion for ActionProtocol for finding arguments for coalition formation is now formally defined in Definition 74 through theΠfunction, using the protocol format detailed in [21]. This protocol returns the set of possible moves denoted℘P(M)(from the set of all possible moves M) that are legal for each agent (fromN) that has joined the dialogue (Dt
r) of the typeC-pAct. The protocol works by initially allowing each agent ito assert or propose all its relevant C-arguments and critical questionsto the current dialogue that are not already present in the commitment store (seeΞa(D1t, i)in the following definition). AC-argument and critical ques- tion is relevant if it achieves the dialogue topic (given by Topic(Drt)) or attacks an argument present in the combined commitment store CoSt. The agent should use an assert move for arguments with completedξ tuples, or theproposemove for non-completedξtuples. Any ar- guments returned that are either aC-argument or one of the critical questions CQ5, CQ6, CQ7 or CQ11 allow the proposal or assertion of a new coalition in the ξ tuple, because this is an argument where the condition CoalAct(A0) = ξ0 (ξ = ξ0)for an argumentA0 ∈ CoStis not enforced. The other CQs use a previously proposed or asserted coalition to argue in favour or against it.
Lastly, if all the agents cannot propose or assert anything new, then the moves that are returned are either the ‘close’ move to be used if the agents want to remain in the dialogue to hear what the other agents have to say, or the ‘leave’ move used when the agent wants to leave the dialogue completely.
Definition 74:TheC-pAct protocolfor aC-pActdialogue is a functionΠ :D×N 7→℘P(M). If agentj ∈N is the agent given andDrtis the dialogue given where: Ags(Dtr) =N,CoSt=
S
∀k∈NCoSttk,Type(Dtr) =C-pAct,Topic(Drt) =pand1≤t, thenΠ(Dt1, j)is
Πa(D1t, j)∪ {hj, close, dialogue(C-pAct,Topic(Dtr))i} ∪ {hj, leave, dialogue(C-pAct,Topic(Dtr))i}
where
Πa(D1t, j) ={hj, propose∨assert,Ψi|
(2)∀A∈Ψ:
(i)A6∈CoSt, and
either (ii,c) A =hqx, µ, ξ, qy, p, v,+iwhere A is aC-argument
or (ii,2) A =hµ, ξ, qyiwhere A is acq2-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0), EndState(A0) =q0y(qy 6=q0y),
or (ii,3) A =hµ, ξ, qy,∅iwhere A is acq3-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0),
EndState(A0) =q0y(qy ≈q0y, p /∈qy, p∈qy0)
or (ii, 4) A =hµ, ξ, qy, v,{=,−}iand where A is acq4-argument∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0), EndState(A0) =q0y(qy ≈q0y)
Value(A0) =v(v = v’),Polarity(A0) = +,
or (ii, 5) A =hµ, ξ, qyiwhere A is acq5-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ6=µ0)
EndState(A0) =q0y(qy ≈q0y)
or (ii, 6) A =hµ, ξ, qy, v,+iwhere A is acq6-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ6=µ0)
EndState(A0) =q0y(qy ≈q0y)
Value(A0) =v(v=6 v0),Polarity(A0) = +,
or (ii, 7) A =hµ, ξ, qy, v,+iwhere A is acq7-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ6=µ0)
EndState(A0) =q0y(qy 6=q0y)
Value(A0) =v(v = v’),Polarity(A0) = +,
or (ii, 8) A =hµ, ξ, qy, v,−iwhere A is acq8-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0), EndState(A0) =q0y(qy ≈q0y)
Value(A0) =v(v = v’),Polarity(A0) = +,
or (ii,9)A=hµ, ξ, qy, v,−iwhere A is acq9-argumentand∃A0∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0), EndState(A0) =q0y(qy ≈q0y)
Value(A0) =v(v=6 v0),Polarity(A0) = +,
or (ii,10)A=hµ, ξ, qy, v,+iwhere A is acq10-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0)
CoalAct(A0) =ξ0 (ξ=ξ0), EndState(A0) =q0y(qy ≈q0y)
Value(A0) =v(v=6 v0),Polarity(A0) = +,
or (ii, 11) A =hµ, ξ, qy, v,+iwhere A is acq11-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ6=µ0)
EndState(A0) =q0y(qy 6=q0y)
Value(A0) =v(v=6 v0),Polarity(A0) = +,
or (ii, 13) A =h¬µiwhere A is acq13-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0 (µ=µ0and∃ac∈µwhereac /∈Aci)
or (ii,14) A =h¬qyiwhere A is acq14-argumentand∃A0 ∈CoSts.t. EndState(A0) =q0y(qy ≈q0yand¬∃qz ∈Qiwhereqy ≈qz), or (ii, 15) A =h¬piwhere A is acq15-argumentand∃A0 ∈CoSts.t.
Goal(A0) =p0, (p=p0,p∈Φi and¬∃qz∈Qjwherep∈π(qz)) or (ii, 16) A =h¬viwhere A is acq16-argumentand∃A0∈CoSts.t.
Value(A0) =v0 (v=v0andv /∈Avi).
or (ii, 17) A =hµ,¬ξiwhere A is acq17-argumentand∃A0 ∈CoSts.t. Action(A0) =µ0,(µ=µ0and∃j∈ξwhereζi(j,AgAction(j, ξ)) =⊥)
elseΠ(Dt1, j) =∅.
If every agent is using theC-pActprotocol then aC-pActdialogue, just like the persuasion dialogue of [21], is guaranteed to terminate according to the following Theorem:
Theorem 3.2. If each agent in aC-pActdialogue is using theC-pActprotocol, then the dia- logue is guaranteed to terminate (given each agent’s VATS is of finite size).
Proof. Assume that allnagents in aC-pActdialogue game are using theC-pActprotocol but the dialogue never terminates. For this to occur, the agents must be constantly proposing or asserting arguments for coalitions to form (becausenleave moves will terminate the dialogue as there will be no agents left andnclose moves will terminate the dialogue as no more agents have any more arguments to add).
Yet according to theC-pActprotocol condition (2)(i), an argument for a coalition to form can only be proposed or asserted if it has not previously been proposed or asserted by any agent. As it is given that each agent’s VATS is of finite size, then in the worst case, each agent will propose or assert every possible argument it can make out of its VATS. Once this has occurred, the only moves the agents will have left is the close or leave move andnleave moves ornclose moves in a row will terminate the dialogue, thus contradicting the assumption.
Thus, as every possible argument in a VATS of an agent could theoretically be communi- cated, the tractability of theC-pActdialogue game (like the persuasion dialogue game in [21]) depends on the total number of possible arguments that can be generated from all of the agent’s VATS.