Quantum affective processes for multidimensional decision-making

In modeling the human affective system and applying lessons learned to human–robot interaction, the challenge is to handle ambiguous emotional states of an agency (whether human or artificial), probabilistic decisions, and freedom of choice in affective and behavioral patterns. Moreover, many cognitive processes seem to run in parallel whereas seriality is the standard in conventional computation. Representation of contextual aspects of behavior and processes and of self-directed neuroplasticity are still wanted and so we attempt a quantum-computational construction of robot affect, which theoretically should be able to account for indefinite and ambiguous states as well as parallelism. Our Quantum Coppélia (Q-Coppélia) is a translation into quantum logics of the fuzzy-based Silicon Coppélia system, which simulates the progression of a robot’s attitude towards its user. We show the entire circuitry of the Q-Coppélia framework, aiming at contemporary descriptions of (neuro)psychological processes. Arguably, our work provides a system for simulating and handling affective interactions among various agencies from an understanding of the relations between quantum algorithms and the fundamental nature of psychology.

: Simulated fractional (relative) difference between the weighted mean (or inner product) determined by the classical fuzzy-based Silicon Coppélia system and quantum logics. The scenario in Eq. (S11) is taken in this simulation. The quantum circuit consists of successive single-control C-R y rotations, as depicted in Eq. (3) and Fig. 3h. Each condition is made up of one distinct weight factor |φ i = cos φi 2 |0 + sin φi 2 |1 , where φ i ∈ [0, π] and i φ i = π. The angle of rotation α i π for each condition is arbitrary with α i ∈ [0, π] and i α i = 1. 50,000 combinations of φ i and α i were randomly generated to evaluate the resultant target state |ψ . The probability of observing state |1 is compared to its classical counterpart, which is given by i sin 2 αiπ 2 sin 2 φi 2 . A positive relative difference implies a greater value obtained by quantum logic. A single-conditioned C-R y rotation gives exact results. The distributions for C-R y rotations under 2-4 control qubits are similar, with a high proportion showing little deviation.
S-1 S-3 S1 Silicon Coppélia and Terminology S1.1 Encoding During encoding the features of a human or virtual agency, each feature is numbered k in a set F , for their level of ethics (good E + or bad E − ), aesthetics (beautiful S + or ugly S − ), and epistemics (realistic P + or unrealistic P − ). During the encoding, moreover, the robot evaluates how far the user has certain affordances (aids F + or obstacles F − ), action possibilities that make the other agency functional as a tool to achieve robot goals or not (e.g., to recharge its battery, to keep it from falling, to serve its user).
Features receive different feature weights w (k) to represent subjective judgments of any kind (weights are unconstrained). Each appraisal variable also has its appraisal weight for the feature d (l = µ ν ∈ D) as an indicator of its dominance in the appraisal process. Empirical research found that features pertaining to ethics and affordances are better predictors of use intentions and engagement than epistemic aspects (e.g., the agency is real or virtual) or aesthetics, i.e., d The encoding process generates the perceived weight of the appraisal variables (S1) Table S1 summarizes the parameters used in the encoding process.

S1.2 Comparison
In the comparison phase, the features are judged for relevance to robot goals j in a set of goals G (relevant (R (k) ) or irrelevant (R †(k) )) and valence to robot goals (positive or negative outcome expectancies (V (k) or V †(k) )) using the membership functions. Relevance determines the intensity of the affective response, whereas valence determines its direction. The variables involved in the comparison phase are summarized in Table S2. An agency's features (e.g., its intentions) encoded as positive (e.g., "good") may afford the facilitation of a desired robot goal, for example, to have maintenance every once in a while. This instigates positive outcome expectancies. When the features are found to be highly relevant to the robot goals (for either positive or negative outcome expectancies), a high level of relevance results.
S-5 Table S2: Variables involved in the relevance and valence evaluation as well as their related action tendencies in the Silicon Coppélia system, and their corresponding kets in the Quantum Coppélia system. Name Si-Coppélia notation Description Qubit notation Set of choices of Agreement to any action for goal j for feature k Specifically, relevance and irrelevance are evaluated from a set of (1) goal-and-action-based rules and (2) appraisal-variables-based rules. For the former, an action i (k) for feature k may be chosen from a set of choice of action A (k) where i (k) can be mapped to some action tendency in a set T via t : A (k) → T (see Satisfaction in Section S1.3.3 for details). On the other hand, a goal j from a set G , desired or undesired, defines what the agency wants to achieve or avoid. The goal is important whether it is desired or undesired. The action i (k) has an effect on (affects) goal j if it either facilitates or inhibits the goal.
Two statements concerning a particular feature k can be set up: ζ 1 : any i (k) ∈ A (k) affects goal j, and ζ 2 : goal j is important. The agency may agree or disagree to ζ 1 and ζ 2 . Agree and disagree act as a bidimensional unipolar scale as one can simultaneously agree to the statements in some aspects and S-7 disagree in other aspects. The three statements result in the following rule: or, abbreviated in terms of membership function, R or, abbreviated in terms of membership function, . The two sets of rules are similar for irrelevance evaluation. The negation of the conditions leads to irrelevance. Nonetheless, for agree and disagree as a pair of bidimensional unipolar scales (one can simultaneously agree and disagree to one entity given multiple perspectives), we consider S-8 the negation of agree to be disagree. Therefore, or, abbreviated in terms of membership function, or, abbreviated in terms of membership function,

S1.2.2 Valence
Valence evaluation relies primarily on the available beliefs (b ij < 0) the goal, the ambition to the goal (a j ) (desired (a j > 0) or undesired (a j < 0)), and the agreement (g (k) ij ) of the agency to the former two statements (agree (g ij < 0)). Depending on the combination of facilitates/inhibits, desired/undesired and S-9 agree/disagree, the positive and negative valence are evaluated as follows: if ζ 1 : feature k is aid obstacle then feature k has positive, sgn b or, abbreviated in terms of membership function, is the value of negative valence. µ V and µ V † are the corresponding membership functions.
The measures in the encode phase -mediated by relevance and valence in the comparison phase and moderated by similarity between the features of robot and user (similar or dissimilar) -determine the robot's responses Fig. 1. [1] However, empirical research showed that relevance and valence are more important to engagement than similarity. [1] For the sake of simplicity, we modeled relevance and valence alone.

S1.3 Response
In the response phase, the robot generates rational cognitions about use intentions and runs affective processes for engagement. Cognition and affect are then collectively considered to estimate the expected satisfaction. Decision-making is done based on the expected satisfaction upon various response strategies (positive and negative approach, change, avoid, etc.) S-10

S1.3.1 Use intentions
In the response phase, the robot calculates a value for the so-called use intentions, the willingness to employ the user or another agency as a means to achieve robot goals. The variables involved in the evaluation of use intentions are summarized in Table S3. Utility as an intermediate variable to indicate the usefulness of the feature to the agency with respect to its goals is calculated first. The expected utility is given by The mean expected utility over goals for an action i (k) can then be calculated, subject to a weight for (S11) Then, the indicative utility (u where the 2 × 10 matrix B ui is the weight matrix for use intentions, containing the weights for r (k) ui , the component vector for use intention in R 10 : where ⊗ denotes the Kronecker product and s and s † are the predefined similarity and dissimilarity.
Abuse of notation is used to represent the construction of r

S1.3.2 Engagement as involvement and distance
The robot establishes engagement with an agency by calculating the levels of involvement with and distance towards, for instance, its user. Involvement and distance are two tendencies that occur in parallel and may compensate for one another. This trade-off in the engagement process is accompanied by specific emotions, which may differ in each particular case. The variables involved in the comparison phase are summarized in Table S4. Similar to use intentions, involvement-distance is modeled as a complex composition of various factors, each carrying a particular weight. The involvement (E where the 2 × 26 matrix B id is the weight matrix for involvement-distance, containing the weights for id , the component vector in R 26 : where the convention of vectorization is the same as that in Eq. (S13). The involvement-distance trade-off parameter is then calculated as the weighted mean of the fuzzy-or operator and the compensation between involvement with distance and the compensation factor for the two (β idt ): Compensation factor

S1.3.3 Satisfaction
Use intentions, involvement, and distance together determine the overall satisfaction of the robot with its user or another agency. The level of satisfaction determines the robot's decision of whether and how it is going to interact with the user (or avoid -stop and turn to another). The variables involved in the calculation are summarized in Table S5. The decision-making process is carried out by calculating the expected satisfaction of interacting (again) with a user, which is the weighted mean of the involvementdistance trade-off (IDT) and the average of the indicative and counter-indicative use intentions with the respective weights σ idt and σ ui : (S18) Eq. (S18) reveals that the robot makes a decision on the more rationally generated use intentions in unison with the more affectively generated involvement-distance trade-off. The feature k max that promises the highest expected satisfaction during the interaction is selected, i.e., treating S (k) as a function of k, 4. avoid the user: t i = a.
The 4 types of action tendencies are depicted in the report of Silicon Coppélia. [2] In general, a feature may not offer all the 4 action tendencies. Additional action tendencies are also possible (e.g., do nothing).
The selection of the action tendency is done by calculating the respective expected satisfaction upon the action: or, in matrix representation, (S21) The action with the highest expected satisfaction will be chosen as the final response, i.e., treating S (kmax) i as a function of i (kmax) ,