Action State Semantics for Practical Reasoning

In [8], a semantics for practical reasoning is presented based on a formalisation that handles the temporal aspects of practical reasoning, and enables the justification of ac- tions. The formalism presents a practical reasoning approach using states and actions with regard to temporal considerations. The formalism greater expressiveness allows the modelling problems where there are temporal windows of opportunity where coordina- tion is required, and where the likelihood of an event varies with time.

An approach to model practical reasoning through argumentation temporal aspects and the intrinsic worth of actions is presented in [8]. Time is important because actions change the state of the world and when agents argue about joint-actions they consider future states as well as past and present ones. States can be reached by several actions through actions justified by the values they promote/demote.

2.3.5.1 Action-based Alternating Transition Systems

To define formally the argumentation scheme for action proposals and the associated critical questions, Atkinson and Bench-Capon used a structure based on Alternating Transition Systems12named Action Based Alternating Transition Systems (AATS) [178]. Such a structure is needed to reason about actions and their effects in terms of transitions from one state to another. In this way the effects of actions can be made dependent on the action of other agents, and other events in the environment. In [178], these structures are used to describe coordination in multi-agent systems using social laws (introduced through the work of Moses and Tennenholtz [125]).

An AATS models joint-actions that may be performed by a set Ag of agents in a state and the effects of these actions. The systems modelled with an AATS may be

12

Alternating Transition Systems (ATS) were originally developed to underpin the Alternating-time Temporal Logic of [3].

in any of a finite set Q of possible states, with q = 0 ∈ Q as the initial state. Each agent i∈Ag is associated with a setAci of possible actions unique to agents. A joint actionjC (coalition C) for set of agents, is a tuple{α1, ..., αk} where for eachαj (where 1 ≤j ≤k) there is some i∈ C such thatαj ∈Aci. Given an elementj of JC and an agenti∈C,is action inj is denoted byji .

Specifically, an AATS model defines semantic structures useful to represent joint- actions for multiple agents, their preconditions and the states that will result from the transition. An AATS is an (n+7)-tuple of the form:

S=hQ, q0, Ag, Ac1, ..., Acn, ρ, τ,Φ, πi where:

• Qis a finite non-empty set of states;

• q0 ∈Q is theinitial state;

• Ag ={1, ..., n}is a finite non-empty set of agents;

• Aci is a finite, non-empty set of actions, for eachi∈Ag, whereAci∩Acj =∅for all i6=j∈Ag;

• ρ : AcAg → 2Q is an action precondition function, which for each action α ∈ AcAg defines the set of states ρ(α) from which α may be executed;

• τ : Q×JAg → Q is a partial system transition function, which defines the state τ(q, j) that would result by the performance of j from state q, note that, as this function is partial, not all joint actions are possible in all states (cf. the precondition function above);

• Φ is a finite, non-empty set of atomicpropositions; and

• π:Q→2Φis aninterpretation function, which gives the set of primitive propo- sitions satisfied in each state: if p ∈π(q), then this means that the propositional variablep is satisfied (equivalently, true) in stateq.

In [9], Atkinson and Bench-Capon extended this AATS model to enable the repre- sentation of a theory of practical reasoning related to arguments about action through which values were added to the system. The use of the term value follows [11] where values are qualitative social interests of agents. The extensions to theAATS model are:

• Avi, is a finite, non-empty set of valuesAvi ⊆V, for each i∈Ag.

• δ: Q×Q×AvAg → {+,−,=} is avaluation function which defines the status (promoted (+), demoted (-) or neutral (=)) of a value vu ∈ AvAg ascribed by the agent to the transition between two states: δ(qx, qy, vu) labels the transition betweenqx and qy with one of {+,−,=}with respect to the value vu ∈AvAg.

2.3.5.2 Formalising Argumentation Schemes and Critical Questions

In [8, 9], the AATS is used to reformulate the problem establishing the relevant propo- sitions and values stated in the argumentation scheme for action proposals AS1. The argumentation scheme AS1 in terms of the extended AATS presented in table 2.3. The formalism can be used to “produce arguments” for action to be performed on its own merits and deal with temporal aspects. The critical questions were also formalised in these terms as follows [8]:

• CQ1: Are the believed circumstances true? [q0 6=qx andq0∈/ ρ(αi)]

• CQ2: Assuming the circumstances, does the action have the stated consequences? [τ(qx, jn) is not qy]

• CQ3: Assuming the circumstances and that the action has the stated consequences, will the action bring about the desired goal? [pa∈/ π(qy)]

• CQ4: Does the goal realise the value stated? [δ(qx, qy, vu) is not +]

• CQ05: Are there alternative ways of realising the same consequences? [Agent i∈Agcan participate in joint actionjm ∈JAg, wherejn6=jm, such thatτ(qx, jm) isqy]

• CQ06: Are there alternative ways of realising the same goal? [Agent i∈ Ag can participate in joint actionjm∈JAg, wherejn6=jm, such thatτ(qx, jm) isqy such thatpa∈π(qy) and pa∈/π(qx) or pa∈/ π(qy) andpa ∈π(qx)]

• CQ07: Are there alternative ways of promoting the same value? [Agent i ∈ Ag can participate in joint actionjm∈JAg, wherejn6=jm, such thatτ(qx, jm) isqz, such thatδ(qx, qz, vu) is +]

• CQ08: Does doing the action have a side effect which demotes the value? [In the initial state qx ∈ Q, if agent i ∈ Ag participates in joint action jn ∈ JAg, then τ(qx, jn) isqy, such thatpb∈π(qy), wherepa6=pb, such that δ(qx, qy, vu)is−]

• CQ09: Does doing the action have a side effect which demotes some other value? [In the initial state qx ∈ Q, if agent i∈Ag participates in joint actionjn ∈ JAg, thenτ(qx, jn) isqy, such that δ(qx, qy, vw) is −, where vu6=vw]

• CQ10: Does doing the action promote some other value? [In the initial state qx ∈ Q, if agent i∈Ag participates in joint action jn∈JAg, then τ(qx, jn) is qy, such thatδ(qx, qy, vw) is +, wherevu 6=vw]

• CQ11: Does doing the action preclude some other action which would promote some other value? [In the initial stateqx∈Q, if agenti∈Ag participates in joint actionjn∈JAg, thenτ(qx, jn) isqy and δ(qx, qy, vu) is + there is some other joint action jm ∈JAg, where jn 6=jm, such that τ(qx, jm) is qz, such that δ(qx, qz, vw) is +, wherevu 6=vw]

• CQ12: Are the circumstances as described possible? [qx ∈/Q]

• CQ13: Is the action possible? [jn∈/ JAG]

• CQ14: Are the consequences as described possible? [τ(qx, jn)∈/ Q]

• CQ15: Can the desired goal be realised? [pa∈/ π(q) for any q∈Q]

• CQ16: Is the value indeed a legitimate value? [vu ∈/ V]

Table 2.3: AS2: Formal AATS representation of the AS1 Argumentation Scheme.

In the initial stateq0 =qx ∈Q,

Agenti∈Ag should participate in joint actionjn∈JAg wherej_ni =αi

such thatτ(qx, jn) is qy,

such thatpa∈π(qy) and pa∈∈/ π(qx), orpa∈/ π(qy) andpa∈π(qx)

such that for somevu ∈Avi ,δ(qx, qy, vu) is +.

Figure 2.8 presents a simple transition system in AATS terms. The transitions are labelled with the values promoted and/or demoted by the execution of the actions. Each state corresponds to a set of possible worlds. A goal is represented as a formula the holds in a state. If the initial state is q0 and the goal holds in stateq2 there are three ways to achieve the goal in this transition system: executing actionsj1 followed by actionj2 to promote valuesv1 and v3; executing action j3 promoting valuev1 and executing action j4 promoting value v2. Using the argumentation scheme AS1 we can instantiate it to present an argument for actionj3 from stateq0 based on the promotion ofv1 to achieve state q0 and achieve the goal. If v1 is not a desired value then an alternative choice is to execute action j4 and promote value v2 to achieve the goal.

q0 j1 q1 q2 j2 V1+ V2+ V3+ j3 j4 V1+

Figure 2.8: AATS Action State Semantics. Boxes represent states qn and circles

represent joint-actionsjn labelled by the promotion of some valuevn.