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.