A Mathematical notations and symbols

We list here both classic mathematical notations, e.g., $\RR$, and some symbols specific to our work, e.g., $s$.

$\RR$ : the “reals”, or set of all real numbers.

$t$ : a time step.

$\length{X}$ : depending on the context, either the cardinality of a set $X$, or the absolute value of a real $X$.

$\LAgts$ : the set of learning agents.

$\ActionSpace_l$ : the action space of a learning agent $l$, typically $\ActionSpace_l \subseteq \RR^d$.

$d$ : number of dimensions of the agent’s action space.

$\mathbf{a}$ : an action in the action space, i.e., a vector of $d$ dimensions.

$\ObsSpace_l$ : the observation space of a learning agent $l$, typically $\ObsSpace_l \subseteq \RR^g$.

$g$ : number of dimensions of the agent’s observation space.

$\mathbf{o}$ : observations in the observation space, i.e., a vector of $g$ dimensions.

$\gamma$ : Discount factor.

$\mathcal{U}$ : set of neurons in the State-(D)SOM, i.e., the neurons that learn the observation space.

$\mathcal{W}$ : set of neurons in the Action-(D)SOM, i.e., the neurons that learn the action space.

$\mathbf{U_i}$ : prototype vector associated to neuron $i$ of the State-(D)SOM.

$\mathbf{W_j}$ : prototype vector associated to neuron $j$ of the Action-(D)SOM.

$\pos_U(i)$ : position of neuron $i$ in the State-(D)SOM.

$\pos_W(j)$ : position of neuron $j$ in the Action-(D)SOM.

$\psi_U$ : neighborhood of the State-(D)SOM.

$\psi_W$ : neighborhood of the Action-(D)SOM.

$\Q$ : the Q-Function, or Q-Table, which stores the interest for every state-action pair.

$s$ : usually, a discrete state identifier.

$\JAgts$ : the set of judging agents.

$\BeliefSpace$ : the space of all possible beliefs about a current situation.

$\Beliefs$ : a set of beliefs about a current situation, with $\Beliefs \in \BeliefSpace$.

$\ValSpace$ : the set of possible moral valuations, $\ValSpace = \left\{ \moral, \immoral, \neutral \right\}$.

$\ME_j$ : the Moral Evaluation function of a judging agent $j$, $\ME_j : \BeliefSpace \times \RR \rightarrow \ValSpace$.

$\Judgment_j$ : the Judgment function of a judging agent $j$, $\Judgment : \LAgts \rightarrow \ValSpace^d$.

$\Feedback$ : the Feedback function.

$\Args$ or $AF_{[\Args]}$ : a set of arguments in an Argumentation Framework for Judging a Decision (AFJD).

$\Att$ or $AF_{[\Att]}$ : a binary attack relationship in an AFJD.

$\Fp$ or $AF_{[\Fp]}$ : the set of pros arguments in an AFJD.

$\Fc$ or $AF_{[\Fc]}$ : the set of cons arguments in an AFJD.

$\Jj$ : the Judgment function of a judging agent $j$, $\Jj : \mathcal{P}(AF_j) \rightarrow \RR$.

$\gagr$ : the Aggregation function, $\gagr : \RR^{\length{\JAgts}} \to \RR$.

$\grd$ : the grounded extension of an AFJD.

$\gPareto$ : the Pareto-dominance operator : $\mathbf{x} \gPareto \mathbf{y} \Leftrightarrow \left( \forall i : x_i \geq y_i \right) \text{and} \left( \exists j : x_j > y_j \right)$.

$\Profiles$ : the set of all exploration profiles.

$p$ an exploration profile $\in \Profiles$.

$\DiscreteStates$ : the set of discrete state identifiers.

$\StateFn$ : function that returns a state identifier from an observations vector, $\StateFn : \ObsSpace \rightarrow \DiscreteStates$.

$\DiscreteActions$ : the set of discrete action identifiers.

$\ActionFn$ : function that returns action parameters from a discrete action identifier, $\ActionFn : \DiscreteActions \rightarrow \ActionSpace_l$.

$\PFFn$ : function that returns a Pareto Front (PF) from a set of possible actions.

$\ThresholdSpace$ : the space of possible ethical thresholds.

$\Contexts$ : the set of all learned contexts.