3.1 Belief graph

A BG is a set of propositional atoms for valid transitions between mental states. The mental states within a BG represent configurations of factors that contribute to specific mental states. These factors, known as mental fluents, are determined by psychological theories and encompass a range of possible values, thus defining the potential states within a BG. These mental fluents are defined as follows:

A mental state space $S$ is defined as the set of all possible combinations of mental fluents.

Definition 2 (Mental state space) Let $C = \{c_1, \ldots , c_n\}$ be a set of psychological classes, and let $V = \{V_{c_1}, \ldots , V_{c_n}\}$ be a set where each $V_{c_i} (1 \leq i \leq n)$ is the set of possible values for the class $c_i$ . The mental state space $S$ is defined as:

\begin{equation*} S = \bigcup _{ v_1 \in V_{c_1}, v_2 \in V_{c_2},\ldots , v_n \in V_{c_n}} \{ f(c_1, v_1), f(c_2, v_2), \ldots , f(c_n, v_n) \} \end{equation*}

where each set $\{ f(c_1, v_1), f(c_2, v_2), \ldots , f(c_n, v_n) \}$ denotes a unique combination of mental fluents, called a mental state.

We now proceed by defining a BG with states $S$ and directed edges $E \subseteq S \times S$ . These edges between states capture principles of mental change suggested by psychological theories, providing rules for allowed mental state transitions.

A BG needs to be characterized for each specific application since different approaches to mental influence may be applicable. Consequently, to support controlled system behavior and principle-based assessment, it is essential to establish reasoning principles of “monotonicity” governing valid transitions between mental states. These principles can be motivated from various sources, such as psychological theories or insights from human experts. In the setting of reasoning about transitions between emotions, there are different principles and theories that have contrasting views on emotional change. For instance, principles of hedonic emotion regulation (Zaki Reference Zaki2020) aim to increase positive emotions and decrease negative emotions. Another set of principles regards utilitarian emotion regulation (Tamir et al. Reference Tamir, Chiu and Gross2007), which aim to increase emotions that provide utility, such as control. Hence, the BG is agnostic to the choice of psychological theories. We show how this flexibility enables the framework to model, and compare, psychological theories through the principle-based assessment of generated trajectories.

3.2 The ${\mathcal{C}}_{\boldsymbol{MT}}$ action language

${\mathcal{C}}_{MT}$ consists of a set of symbols representing actions and fluents, forming the alphabet of the action language. Given that the purpose of an action language is to provide a higher-level framework that makes modeling dynamic systems more natural and modular, several aspects of the language serve as syntactic sugar to simplify the modeling of mental states. The primary addition, beyond a specialized alphabet, is the incorporation of forbids to cause rules that given a set of mental fluents in the current state forbids a set of mental fluents to hold in the next state. By capturing elements of both the external environment and the human mind, the language facilitates the description and analysis of how environmental events influence human mental states and behavior, and through the new forbidding constraints, actions as well as indirect effects on mental fluents can be constrained. We require this addition to the language in order to model principles of mental change.

Within ${\mathcal{C}}_{MT}$ , a domain description defines static and dynamic causal laws for actions. These laws precisely express the expected influences exerted on mental fluents, either as direct effects of actions or as indirect causal effects. The laws governing mental change operate by modulating the variables of a given mental state while adhering to the constraints outlined by the BG.

Definition 5 ( ${\mathcal{C}}_{MT}$ domain description language)The ${\mathcal{C}}_{MT}$ domain description language $D^{MT}(\textbf {A}, \textbf {F})$ consists of static and dynamic causal laws of the following form:

where $a \in \textbf {A}$ , $a^h \in \textbf {A}^H$ , and $a_i \in \textbf {A}$ ( $0 \leq i \leq n$ ) and $f_j \in \textbf {F}$ , $(0 \leq j \leq n)$ and $g_j \in \textbf {F}$ , $(0 \leq j \leq m)$ , and $f_1^h, \ldots , f_n^h \in \textbf {F}^H$ , $g_1^h, \ldots , g_m^h \in \textbf {F}^H$ are mental fluents of the form $f(c,v)$ , where $c$ is a psychological class and $v$ is a psychological value.

We next provide the formal specification of the action language components in terms of states, rules, and transition conditions. Once this foundation is in place, we proceed to their characterization under answer set semantics, where the operational meaning of the laws is captured by logic programs and their answer sets.

BGs are expressed in terms of ${\mathcal{C}}_{MT}$ logic programs, that is finite sets of dynamic and causal laws on mental states. This characterization allows us to ensure controlled mental change by restricting the states and state-transitions. The mental state, within the context of the domain description, represents an interpretation of the current state of the system.

In this definition, a state is determined by the satisfaction of static causal laws within the domain description $D^{MT}(\textbf {A}, \textbf {F})$ . The first condition ensures that if the prerequisite mental fluents for an action are true, then the consequent mental fluents will also be true in the state. The second condition specifies that if certain mental fluents influence a particular mental fluent, then the influenced mental fluent must be true when all the influencing mental fluents are true.

Let us define the laws of the domain description more precisely.

Intuitively, the rules describe how actions may or may not occur, and how fluents are connected across and within states. Rule (1) states that certain fluents can inhibit an action, making it inactive in the current state. Rule (2) captures when fluents trigger an action, meaning it must occur if its triggers are satisfied and it is not inhibited. Rule (3) specifies when fluents allow an action, meaning the action is permitted but not enforced. Rules (4) and (5) extend this to human actions: some mental fluents can facilitate a human action, while others can contravene it. Rule (6) introduces dynamic causal laws, describing how actions under certain conditions cause fluents to hold in the next state. Rule (7) introduces static causal laws, which specify that if some fluents hold in a state, then others must co-hold in the same state. Rule (8) adds dynamic influence for mental fluents: when an action occurs under certain conditions, it brings about specified mental fluents in the next state. Rule (9) specifies static influence for mental fluents: when its conditions hold, the corresponding mental fluents must also hold in the same state. Finally, Rule (10) introduces forbids to cause, which regulates mental change by specifying that if certain mental fluents hold in the current state, then some mental fluents are not allowed to appear in the next state.

The output of the action language is in terms of trajectories. A mental-state trajectory consists of a sequence of valid transitions, represented as

$\langle s_0, A_1, s_1, A_2, \ldots , A_n, s_n \rangle$ , with alternating sets of mind-altering actions $A \subseteq \mathbf{A}$ and mental-states $s \in S$ , following the constraints of the BG.

The set $ B$ in condition (9) represents a subset of actions restricted by noconcurrency constraints in $ D^{MT}(\textbf {A}, \textbf {F})$ , ensuring that actions in $ B$ cannot execute simultaneously. This prevents conflicts where multiple actions attempt to modify the same fluent at the same time. While actions affecting distinct fluents can occur concurrently, those altering the same fluent are mutually exclusive, maintaining consistency in fluent updates and ensuring well-defined state transitions.

Definition 9 (Action Observation Language) The action observation language of ${\mathcal{C}}_{MT}$ (similar to ${\mathcal{C}}_{TAID}$ ) consists of expressions of the following form:

\begin{equation*} (f \,\, \mathbf{ at } \,\,t ) \qquad (a\,\, \mathbf{occurs\_at}\,\, t )\qquad (8) \end{equation*}

where $f \in \textbf {F}$ , $a$ is an action and $t \in \mathbb{N}$ is a point in time.

The action observation language allows us to specify observations concerning the current state of mental fluents and the execution of actions that influence mental states. By combining observations with the causal laws of the domain description, we can generate plans, explanations, and predictions regarding the behavior of the system. This integration of observations and the domain description is referred to as an action theory.

The action theory forms the basis for constructing trajectory models, trajectories where all observations are satisfied, providing a structured representation of the system’s dynamics over time. Trajectory models enable us to analyze and reason about the evolution of mental states and actions, allowing for a deeper understanding how the mental state domain operates and how it responds to different observations and changes.

We can observe that actions in a trajectory model can be actions executed by a rational software agent to influence mental fluents, or actions estimated to be executed by a human agent.

An important consequence of the $\mathbf{forbids \,to \,cause}$ rule is that it can render an action theory inconsistent. Since such a rule blocks any transition to a state containing its forbidden fluents, a trajectory that reaches such a state is invalid. If all candidate trajectories are thus blocked, then no trajectory model exists and the action theory is inconsistent. This regulative effect is intentional, ensuring that unwanted mental fluents cannot appear in valid states.

The central computational task is to determine, for a given ${\mathcal{C}}_{MT}$ action theory $(D,O)$ , whether there exists a trajectory model of $(D,O)$ . This baseline feasibility check underpins subsequent reasoning tasks such as planning, explanation, and query answering.

We need a mechanism to write queries about state dynamics. The Action Query Language provides a means to inquire about specific sequences of actions and their impact on fluents and states. This is achieved by specifying subsets of the action set and their corresponding occurrences in time.

By formulating queries in the action language, causal relationships between actions and their effects can be investigated, contributing to a deeper understanding of system dynamics, informed decision-making and controlled methods for automated planning.

Given a ${\mathcal{C}}_{MT}$ action theory $(D,O)$ and a query $Q$ , the decision problem regards whether $Q$ holds in all (or some) trajectory models of $(D,O)$ .

3.3 Action language in answer set semantics

In this section, we provide operational semantics for ${\mathcal{C}}_{MT}$ by translation into answer set programs. The goal is to construct an encoding such that the encoding has exactly one answer set for every trajectory model of a ${\mathcal{C}}_{MT}$ theory. Intuitively, each answer set represents a trajectory model: for a trajectory $\langle s_0,A_1,s_1,\ldots ,A_n,s_n\rangle$ , the atoms holds $(f,t)$ indicate that fluent $f$ holds in state $s_t$ , and the atoms holds(occurs(a),t) indicate that action $a$ occurs between $s_t$ and $s_{t+1}$ . Conversely, each answer set uniquely determines a trajectory by the fluents and actions it contains, thereby establishing a one-to-one correspondence between answer sets and trajectory models and reducing the central reasoning problems of ${\mathcal{C}}_{MT}$ to ASP computation.

To make the presentation self-contained, we include translations from (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008) for the ${\mathcal{C}}_{TAID}$ language (expressions 1–7 in Definition 5), and then extend them with the new constructs of ${\mathcal{C}}_{MT}$ . We organize the translations into distinct subsections, addressing the Action Description Language, the Action Observation Language, and the Action Query Language.

3.3.1 Encoding of the action description language

We consider the encoding of $\mathcal{C}_{TAID}$ action description language, which we extend with components of $\mathcal{C}_{MT}$ .

Define symbols for a fluent f $\in \textbf {F}^{\textbf {E}}$ and an action a $\in \textbf {A}^{\textbf {E}}$ (extending (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)).

Define symbols for a fluent e $\in \textbf {F}^{\textbf {H}}$ and an action u $\in \textbf {A}^{\textbf {H}}$ (extending (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)).

Define a range of time points $0 \leq t \leq t\_max$ , $t \in \mathbb{N}$ (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)).

Contradiction constraint (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A fluent $f \in \textbf {F}$ and its negation $\neg f$ cannot hold simultaneously at the same time step $T \in \mathbb{N}$ . In ASP, this is enforced as a constraint, ensuring that $\texttt {holds}(f,T)$ and $\texttt {holds(neg(f),T)}$ cannot both be true. The predicates $\texttt {fluent}(f)$ and $\texttt {time}(T)$ declare fluents and time steps.

Inertial fluents (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An inertial fluent persists across time steps unless modified by an action or a static causal law. In ASP, this is encoded by ensuring that if $\texttt {holds}(f,T)$ is true and not overridden, then $\texttt {holds}(f,T+1)$ holds by default. The predicates $\texttt {fluent}(f)$ and $\texttt {time}(T)$ track fluents over time.

Non-inertial fluents (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A non-inertial fluent resets to a default value unless explicitly updated. In ASP, this is encoded by inferring $\texttt {holds}(f,T)$ whenever $\texttt {default}(f)$ holds and $\texttt {holds(neg(f),T)}$ is not inferred. The predicates $\texttt {fluent}(f)$ , $\texttt {default}(f)$ , and $\texttt {time}(T)$ define non-inertial fluents and their default behavior.

Dynamic causal law (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A dynamic causal law $(a \,\, \mathbf{causes}$ $f$ $\mathbf{if}$ $g_1, \ldots , g_n)$ states that if an action $a \in \textbf {A}$ occurs at time $T \in \mathbb{N}$ and fluents $g_1, \ldots , g_n \in \textbf {F}$ hold, then a fluent $f \in \textbf {F}$ must hold at time $T+1$ . In ASP, one rule is generated for each dynamic causal law, ensuring that if $\texttt {holds(occurs}(a,T)\texttt {)}$ and $\texttt {holds}(g_1,T), \ldots , \texttt {holds}(g_n,T)$ hold, then $\texttt {holds}(f,T+1)$ is inferred. The predicates $\texttt {fluent}(f)$ and $\texttt {fluent}(g_1), \ldots , \texttt {fluent}(g_n)$ declare fluents, $\texttt {action}(a)$ declares the action, and $\texttt {time}(T)$ ensures valid time progression.

Static causal law (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A static causal law $(f \,\, \mathbf{if} \,\, g_1, \ldots , g_n)$ states that a fluent $f \in \textbf {F}$ holds at time $T \in \mathbb{N}$ if fluents $g_1, \ldots , g_n \in \textbf {F}$ hold at the same time. In ASP, one rule is generated for each static causal law, ensuring that if $\texttt {holds}(g_1,T), \ldots , \texttt {holds}(g_n,T)$ hold, then $\texttt {holds}(f,T)$ is inferred. The predicates $\texttt {fluent}(f)$ and $\texttt {fluent}(g_1), \ldots , \texttt {fluent}(g_n)$ declare fluents, and $\texttt {time}(T)$ ensures valid time progression.

Inhibition rule (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An inhibition rule $(f_1, \ldots , f_n \textbf { inhibits } a)$ states that an action $a \in \textbf {A}$ is prevented from occurring at time $T \in \mathbb{N}$ if fluents $f_1, \ldots , f_n \in \textbf {F}$ hold. In ASP, one rule is generated for each inhibition rule, ensuring that if $\texttt {holds}(f_1,T), \ldots , \texttt {holds}(f_n,T)$ hold, then $\texttt {holds(ab(occurs}(a),T)\texttt {)}$ is inferred, marking the action as inhibited. The predicates $\texttt {fluent}(f_1),$ $\ldots ,$ $\texttt {fluent}(f_n)$ declare fluents, $\texttt {action}(a)$ declares the action, and $\texttt {time}(T)$ ensures valid time progression.

As in $\mathcal{C}_{TAID}$ , unlike standard ASP planning encodings, we do not use a choice rule for action generation, such as

{ holds(occurs(A), T): action(A) } = 1 :- T = 1.t_max. Instead, actions are determined by logical conditions: they occur if they are explicitly triggered, allowed, and not inhibited. This ensures that only valid and necessary actions are selected, avoiding unnecessary non-determinism.

Triggering rule (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A triggering rule $(f_1, \ldots , f_n \textbf { triggers } a)$ states that an action $a \in \textbf {A}$ occurs at time $T \in \mathbb{N}$ if fluents $f_1, \ldots , f_n \in \textbf {F}$ hold and no inhibition rule is active. In ASP, one rule is generated for each triggering rule, ensuring that if $\texttt {holds}(f_1,T), \ldots , \texttt {holds}(f_n,T)$ hold and $\texttt {not holds(ab(occurs}(a,T)\texttt {))}$ , then $\texttt {holds(occurs}(a,T)\texttt {)}$ is inferred. The predicates $\texttt {fluent}(f_1), \ldots , \texttt {fluent}(f_n)$ declare fluents, $\texttt {action}(a)$ declares the action, and $\texttt {time}(T)$ ensures valid time progression.

Allowance rule (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An allowance rule $(f_1, \ldots , f_n \textbf { allows } a)$ states that an action $a \in \textbf {A}$ is permitted to occur at time $T \in \mathbb{N}$ if fluents $f_1, \ldots , f_n \in \textbf {F}$ hold and no inhibition rule is active. In ASP, one rule is generated for each allowance rule, ensuring that if $\texttt {holds}(f_1,T), \ldots , \texttt {holds}(f_n,T)$ hold and $\texttt {not holds(ab(occurs}(a,T)\texttt {))}$ , then $\texttt {holds(allow(occurs}(a),T)\texttt {)}$ is inferred. The predicates $\texttt {fluent}(f_1), \ldots , \texttt {fluent}(f_n)$ declare fluents, $\texttt {action}(a)$ declares the action, and $\texttt {time}(T)$ ensures valid time progression.

Ensure exogenous actions can always occur (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): This rule ensures that an action $a \in \textbf {A}$ is always permitted to occur at any time step $T \in \mathbb{N}$ , regardless of specific preconditions. In ASP, this is encoded by inferring $\texttt {holds(allow(occurs}(a),T)\texttt {)}$ for every declared action $\texttt {action}(a)$ and time point $\texttt {time}(T)$ . This guarantees that exogenous actions remain available throughout the execution.

No-concurrency constraint (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): The actions $a_1, \ldots , a_n \in \textbf { A}$ cannot occur simultaneously at time $T \in \mathbb{N}$ . In ASP, this is encoded as a constraint that ensures selecting two or more actions at $T$ leads to inconsistency. The rule applies over declared actions $\texttt {action}(a_1), \ldots , \texttt {action}(a_n)$ and time points $\texttt {time}(T)$ .

3.3.2 Encoding of the action observation language

We consider the encoding of $\mathcal{C}_{TAID}$ action observation language, which we extend with components of $\mathcal{C}_{MT}$ . The action observation language allows for the specification of observations about fluents and actions over time. Observations provide constraints on the initial state and subsequent state transitions, ensuring that execution traces align with known facts.

Initial state fluent observations (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A fluent $f \in \textbf {F}$ that holds in the initial state at time $T = 0$ is directly asserted. In ASP, this is represented by $\texttt {holds}(f,0)$ , ensuring that observed fluents are set at the start of execution, forming the basis for reasoning about subsequent state transitions.

Fluent observations for other states (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): If a fluent $f \in \textbf {F}$ is observed to hold at time $T \in \mathbb{N}$ , it must be enforced. In ASP, this is encoded as a constraint ensuring that $\texttt {holds}(f,T)$ must be true whenever $f$ is observed. The rule applies over declared fluents $\texttt {fluent}(f)$ and time points $\texttt {time}(T)$ .

Generate possible completions of the initial state (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): A fluent $f \in \textbf {F}$ in the initial state at $T = 0$ must either hold or its negation $\neg f$ must hold, but not both. In ASP, this is encoded using a default completion principle, ensuring that $\texttt {holds}(f,0)$ is inferred unless $\texttt {holds(neg(f),0)}$ is explicitly stated, and vice versa. The rule applies over declared fluents $\texttt {fluent}(f)$ at time step $\texttt {time}(0)$ .

Exogenous action observations (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An exogenous action $a \in \textbf {A}$ that occurs at time $T \in \mathbb{N}$ is recorded as a fact. This corresponds to an external execution of $a$ , independent of triggering, allowance, or inhibition conditions. In ASP, this is represented by asserting $\texttt {holds(occurs}(a),T\texttt {)}$ , ensuring that the action is registered as having taken place. The rule applies over declared actions $\texttt {action}(a)$ and time points $\texttt {time}(T)$ .

Observed non-occurrence of actions (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): If an action $a \in \textbf {A}$ does not occur at time $T \in \mathbb{N}$ , this must be explicitly recorded. This corresponds to cases where no triggering or allowance rule is active, or an inhibition rule prevents execution. In ASP, this is enforced as a constraint, ensuring that $\texttt {holds(neg(occurs}(a),T\texttt {))}$ is inferred when the action does not occur. The rule applies over declared actions $\texttt {action}(a)$ and time points $\texttt {time}(T)$ .

Action execution conditions (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An action $a \in \textbf {A}$ occurs at time $T \in \mathbb{N}$ if at least one allowance rule $(f_1, \ldots , f_n \mathbf{allows}\,a)$ is active, no inhibition rule $(f_1, \ldots , f_n \mathbf{inhibits}\,a)$ applies, and no external constraint negates its occurrence. In ASP, this is encoded by ensuring that $\texttt {holds(occurs}(a),T\texttt {)}$ is inferred under these conditions. Additionally, if an action does not occur, its negation $\texttt {holds(neg(occurs}(a),T\texttt {))}$ is inferred. The rule applies over declared actions $\texttt {action}(a)$ and time points $\texttt {time}(T)$ , where $T \lt t_{\max }$ .

3.3.3 Encoding of the action query language

We consider the encoding of $\mathcal{C}_{TAID}$ action query language, which we extend with components of $\mathcal{C}_{MT}$ .

Goal achievement constraint (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): The goal must be achieved in every valid plan. This corresponds to ensuring that at least one fluent configuration satisfying the goal holds. In ASP, this is enforced as a constraint ensuring that $\texttt {achieved}$ is inferred.

Initial goal satisfaction (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): The goal is considered achieved if it already holds at the initial time step $T=0$ .

Persistence of goal achievement (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): If the goal is achieved at time $T+1$ but was not achieved at $T$ , then it remains achieved for all subsequent time steps. The predicates $\texttt {time}(T)$ ensure valid time progression.

Fluent-based goal satisfaction (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): The goal is achieved at time $T$ if a set of fluents $f_1, \ldots , f_n \in \textbf {F}$ required for goal satisfaction hold. This corresponds to a static causal law of the form $(f_1, \ldots , f_n \mathbf{if} \,\, \text{goal_conditions})$ . In ASP, this is encoded by inferring $\texttt {achieved}(T)$ when $\texttt {holds}(f_1,T), \ldots , \texttt {holds}(f_n,T)$ hold.

Final goal satisfaction (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): The goal must hold at the maximum time step $t_{\max }$ . This ensures that the trajectory satisfies the required final state.

Execution of allowed actions (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): An action $a \in \textbf {A}$ occurs at time $T \in \mathbb{N}$ if it is allowed by an active allowance rule $(f_1, \ldots , f_n \mathbf{allows}\,a)$ , is not inhibited by an inhibition rule $(f_1, \ldots , f_n \mathbf{inhibits} a)$ , and the goal has not yet been achieved. In ASP, this ensures that $\texttt {holds(occurs}(a),T\texttt {)}$ is inferred only when these conditions hold. The predicates $\texttt {fluent}(f_1), \ldots , \texttt {fluent}(f_n)$ declare fluents, $\texttt {action}(a)$ declares actions, and $\texttt {time}(T)$ defines time steps.

Explicit non-occurrence of actions (as in (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008)): If an action $a \in \textbf {A}$ does not occur at time $T \in \mathbb{N}$ , this must be explicitly recorded. This corresponds to enforcing that an action not chosen in the trajectory is negated.

3.3.4 Encoding of the mental state language extension

We now proceed by presenting the translations for static and dynamic causal laws for mental state specifications introduced in $\mathcal{C}_{MT}$ .

Dynamic causal law for mental fluents (introduced in $\mathcal{C}_{MT}$ ): A dynamic causal law $(a \,\, \mathbf{influences} \,\, f_1^h, \ldots , f_n^h \,\, \mathbf{if} \,\, g_1, \ldots , g_m)$ specifies that an action $a \in \textbf {A}$ causes mental fluents $f_1^h, \ldots , f_n^h \in \textbf {F}^H$ to hold at $T+1$ if fluents $g_1, \ldots , g_m \in \textbf {F}$ hold at $T \in \mathbb{N}$ . The ASP encoding generates a rule for each mental fluent $f_i^h \in \textbf {F}^H$ ( $1 \leq i \leq n$ ), of the form:

Static causal law for mental fluents (introduced in $\mathcal{C}_{MT}$ ): A static causal law $(g_1, \ldots , g_m \,\, \mathbf{influences} \,\, f_1^h,\ldots ,f_n^h)$ states that if fluents $g_1, \ldots , g_m \in \textbf { F}$ hold at time $T$ , then mental fluents $f_1^h,\ldots ,f_n^h \in \textbf {F}^H$ also hold at $T \in \mathbb{N}$ . The ASP encoding generates a rule for each mental fluent $f_i^h \in \textbf {F}^H$ ( $1 \leq i \leq n$ ), of the form:

The following translations regard contravening and facilitating rules (introduced in $\mathcal{C}_{MT}$ ). Unlike the "inhibition" and "allowance" rules outlined in $\mathcal{C}_{TAID}$ (Dworschak et al. Reference Dworschak, Grell, Nikiforova, Schaub and Selbig2008), which impact actions in the current time step, "contravenes" and "facilitates" relate particularly about a human action $a^h \in \textbf {A}^H$ and mental fluents $f^h \in \textbf {F}^H$ . As we have previously discussed; appraisal theories have suggested that emotional states have distinctive “action tendencies” (Roseman et al.Reference Roseman, Wiest and Swartz1994). Capturing these semantics, while seemingly redundant due to the general rules in the action language, supports knowledge representation.

Facilitation rule (introduced in $\mathcal{C}_{\boldsymbol{MT}}$ ): A facilitation rule $(g_1^h,\ldots ,g_m^h$ $\mathbf{facilitates} \,\, a^h)$ states that if mental fluents $g_1^h, \ldots , g_m^h \in \textbf {F}^H$ hold at time $T \in \mathbb{N}$ , and the action $a^h \in \textbf {A}^H$ is not inhibited, then $a^h$ occurs. The ASP encoding ensures that $\texttt {holds(occurs}(a^h),T\texttt {)}$ is inferred if $\texttt {not holds(ab(occurs}(a^h),T)\texttt {)}$ holds.

Contravening rule (introduced in $\mathcal{C}_{\boldsymbol{MT}}$ ): A contravening rule $(g_1^h,\ldots ,g_m^h$ $\mathbf{contravenes} \,\, a^h)$ states that if mental fluents $g_1^h, \ldots , g_m^h \in \textbf {F}^H$ hold at time $T \in \mathbb{N}$ , then the human action $a^h \in \textbf {A}^H$ is inhibited from occurring. The ASP encoding infers $\texttt {holds(ab(occurs}(a^h),T\texttt {))}$ under these conditions.

Forbidding rule (introduced in $\mathcal{C}_{\boldsymbol{MT}}$ ): A forbidding rule $(g_1^h,\ldots ,g_m^h \ \textbf{forbids to} $ $\mathbf{cause}$ $f_1^h, \ldots , f_n^h)$ enforces constraints on mental state transitions in the BG. If mental fluents $g_1^h, \ldots , g_m^h \in \textbf {F}^H$ hold at time $T \in \mathbb{N}$ , then fluents $f_1^h, \ldots , f_n^h \in \textbf {F}^H$ are forbidden to hold at $T+1$ . This ensures that the belief graph’s constraints on state transitions are respected. The ASP encoding introduces an integrity constraint to prohibit transitions that violate these restrictions.

In the upcoming section, we will explore a case study that applies a specialization of $\mathcal{C}_{MT}$ for emotional reasoning. The case study models emotion states and dynamics based on well-established psychological theories. By examining this specialized application, we aim to demonstrate the effectiveness and versatility of the $\mathcal{C}_{MT}$ framework for modeling different kinds of mental state dynamics.

4.1 Emotion theories: AE, HER and UER

The Appraisal theory of emotion (AE) (Roseman Reference Roseman1996) proposes that emotions are caused by an appraisal of a situation in terms of 1) being consistent or inconsistent with needs, importance of the situation, 2) being consistent or inconsistent with goals, the attainability/potential to achieve goals, 3) who/what is accountable/caused the situation, which can be the environment, others, or oneself, and 4) as being easy or difficult to control. According to AE, the difference between goal consistency/attainability and need consistency/importance determines negative, stable and positive emotions. More intense negative emotions (e.g. Anger or Fear) arise when the need consistency is greater than the goal consistency, while less intense negative emotions can arise when both the need consistency and goal consistency are low. On the other hand, positive emotions (e.g. Joy or Liking) arise when the goal consistency is greater than the need consistency, or when both are high. By ranking consistency values as Low $\lt$ Undecided $\lt$ High and by looking at the difference between need and goal consistency, positive and negative emotions can be distinguished.

Hedonic emotion regulation (HER) (Zaki Reference Zaki2020) is a theory for regulating emotions, guided by the goals to increase positive emotion and decrease negative emotion. According to HER, both of these emotion regulation goals are associated with improved well-being, where decreasing of negative emotion has been most effective (Ortner et al. Reference Ortner, Corno, Fung and Rapinda2018). The principles of HER can be applied to reason about emotional change. For instance, the relation between goal attainability and goal importance has been empirically explored, showing that goal attainability, rather than goal importance, was positively linked to well-being (Bühler et al. Reference Bühler, Weidmann, Nikitin and Grob2019). Another empirical study analyzed self-responsibility and emotions, and showed that accountability does not by itself affect whether the emotion is negative or positive. Nevertheless, when something else than when an individual (self/other) is perceived as accountable, less negative emotions appear (Passyn and Sujan Reference Passyn and Sujan2006). Another empirical study examined the relationship between control potential and emotions (Roseman Reference Roseman1996). The study found that the perception of high or low control potential influenced the experience of accommodating emotions or contending emotions, respectively. However, the study did not find a direct effect of control potential on the experience of positive or negative emotions.

By analyzing the four dimensions of emotion as proposed by AE (Need consistency, Goal consistency, Accountability, and Control potential), we can derive meaningful interpretations about the relationship between HER and AE in the context of hedonic emotional change. These interpretations are summarized in Table 1.

Table 1.

Hedonic emotion regulation as basis for principles of change

UER (Tamir et al. Reference Tamir, Chiu and Gross2007) is a theory emotion regulation that, in contrast to HER, is guided by the goals to experience emotions in the short-term for a long-term utilitarian purpose. This may involve accepting a temporary negative emotion to increase their capability of dealing with a situation. For instance, valuing long-term goals (i.e. focusing on need, importance and motivation) (Dweck Reference Dweck2017) over immediate pleasure makes individuals more likely to engage in UER. Furthermore, a study on goal-setting (Schunk Reference Schunk2001) found that when individuals perceive a discrepancy between their present performance and a desired goal, it can lead to dissatisfaction. This dissatisfaction, in turn, can serve as a motivator for increased effort and striving towards the goal. Another study, exploring accountability and emotion (Autry and Langenbach Reference Autry and Langenbach1985), suggests that self-accountability for a situation has utilitarian gains by having effects on the individual’s “locus of control.” Moreover, a study exploring control and emotion (Tamir et al. Reference Tamir, Mitchell and Gross2008) found that increasing control potential, such as through an increased level of anger, can have utilitarian gains. Another study on the emotion of fear (Tamir and Ford Reference Tamir and Ford2009) proposes the concept of “fear as function for goal pursuit,” highlighting its role in motivating goal-directed behavior. Yet another study (Parrott Reference Parrott2002) suggests that “negative” emotions can modify an individual’s “readiness to think and act, and their potential functional utility.”

By analyzing the four dimensions of emotion as proposed by AE (Need consistency, Goal consistency, Accountability, and Control potential), we can derive meaningful interpretations about the relationship between UER and AE in the context of utilitarian emotional change. These interpretations are summarized in Table 2.

Following the interpretations of HER and UER, we can define transition constraints in an emotion specialized BG. This will allow certain sets of trajectories depending on which emotional-change theory we model. In both cases, we use the state space defined by AE.

In the following section, we present a formalization of these theories in terms of $\mathcal{C}_{MT}$ . We first define an emotion state in terms of the dimensions of AE. We then define a set of transition constraints that restrict particular configurations (states) of AE appraisals to arise. Two sets of transition constraints are defined, constituting two different sub-graphs of valid transitions in the AE state-space, a so-called hedonic emotion state and an utilitarian emotion state, based on HER and UER principles, respectively.

4.2 Formalizing emotional reasoning

Components of AE and HER (resp. UER) are formalized as a specialized BG to reason about emotion states and hedonic (resp. utilitarian) emotional change to reduce unintended emotional side-effects. The constraints of the specialized BG serve as safety restrictions for emotion-influencing actions, applicable for different types of emotional reasoning settings.

Recall that AE defines emotions as a composition of an individual’s appraisal of a situation, in terms of need consistency (importance), goal consistency (attainability), accountability (who/what) and control potential. By following this definition of emotional causes, we can define emotion fluent, a changeable emotion variable.

Definition 16 (Emotion fluent) An emotion fluent is a mental fluent $f(c, v)$ , a ground atom of arity 2, where $c \in C$ , and $v \in V_c$ , such that $C = \{ne, go, ac, co\}$ is a set of constants denoting psychological classes and $V = \{V_{ne}, V_{go}, V_{ac}, V_{co}\}$ a set of total ordered sets of constants denoting psychological values for each class of $C$ , where:

Table 2.

Utilitarian emotion regulation as basis for principles of change

$V_{ne} = \{high, low, undecided\}$

$V_{go} = \{high, low, undecided\}$

$V_{ac} = \{self, other, environment, undecided\}$

$V_{co} = \{high, low, undecided\}$ .

where $ne$ denotes need consistency, $go$ denotes goal consistency, $ac$ denotes accountability and $co$ denotes control potential.

By defining a set of emotions following AE in this way, and by utilizing different principles of emotion regulation, we can specify preferred (e.g. hedonic or utilitarian) transitions between emotion states. In the following subsection, we specify a BG to reason about emotional transitions.

Following the AE-theory, 16 emotion states are specified, corresponding to each basic emotion: {Anger, Dislike, Disgust, Sadness, Hope, Frustration, Fear, Distress, Joy, Liking, Pride, Surprise, Relief, Regret, Shame, Guilt}. It is important to note that various other emotion states can be defined (in total 108) by different combinations of the four AE dimensions, and it is not necessary, nor always preferred, to assign a specific label, such as “Joy,” to each of them. While labels aid in expressing these states in a human readable way, they do not inherently contribute to the functionality of a system’s reasoning. It is worth noting that emotional expressions vary across individuals, cultures, languages, and other factors. Therefore, it is crucial for the system to analyze emotion states using a multidimensional format. We can model the states and transitions as an EG that represents valid emotional change given a recognized emotion state configuration. In Figure 3, examples of emotion states are presented.

Fig 3.

Emotion states following the appraisal theory of emotion (Roseman Reference Roseman1996). Each emotion state is here expressed by an intuitive “emotion” label on top, and an appraisal configuration consisting of a set of variable: value pairs below. The variables are: ne = need consistency, go = goal consistency, ac = accountable, co = control potential. The values are: l = low, h = high, u = undecided, o = other, s = self, e = environment.

The specialized belief graph for emotion is captured by a given program specified by the semantics of the action language $\mathcal{C}_{MT}$ , with constraints in terms of HER or UER, serving as restrictions for hedonic contra utilitarian emotional change, presented in the following section.

4.3 Action language emotion specification

The $\mathcal{C}_{MT}$ alphabet is specialized with an emotion related vocabulary and causal laws to specify fluents and actions according to AE-theory.

The emotional reasoning semantics is characterized by the constraints of a specialized BG for emotion, specified through a set of static causal laws and mental state constraints. In this way, we can restrict states and state-transitions to comply with principles of emotional change.

For any particular application, we need to define a BG that, based on application specific interaction goals and relevant theories, avoids unintended mental states. This specifies an BG with a subset of transitions (in the fully connected graph) that is considered valid. We define a hedonic theory specification that follows principles of HER, aiming to increase positive emotion and decrease negative emotion (see more details of HER in Section 4.1). In an abstraction of HER, the hedonic emotion specification is defined to not allow entering a negative emotion state and to preserve a positive balance in emotion.

Recall Figure 1, which illustrates the BG representing possible trajectories over environmental and mental states. Transitions (edges) in this graph are defined by a set of dynamic causal laws of the form $(a\,\mathbf{causes}\,f_1,\ldots ,f_n\,\mathbf{if}\,g_1, \ldots , g_m)$ , capturing how actions produce observable changes. Overlaid on these are forbidding rules of the form $(g_1^h,\ldots ,g_m^h\,\mathbf{forbids \,to \,cause}\,f_1^h, \ldots , f_n^h)$ , which specify mental state transitions that are disallowed based on psychological theory. The red arrows in the graph correspond to environmental transitions that are excluded because they would cause such forbidden mental transitions. In this way, the Belief Graph integrates both causal dynamics and psychological constraints to filter out inadmissible action trajectories.

The hedonic theory specification can thus be formally expressed as follows.

We further define an utilitarian emotion theory specification that follows principles of UER, aiming to prioritize particular emotional dimensions (e.g. need_consistency) that are associated with increased utilitarian gain (see more details of UER in Section 4.1). In an abstraction of UER, the utilitarian emotion theory specification is defined to allow entering a negative emotion state in the short-term if it may result in a long-term utilitarian gain.

These sets of $\mathbf{forbids \,to \,cause}$ rules are encoded as integrity constraints. Given a logic program $P_{EG}$ (based on the translation in Section 3.3.1), we attach sets of integrity constraints representing HER (Listing 1) or UER (Listing 2) based principles. These integrity constraints are independent of the main logic program $P_{EG}$ . By extending the program with different sets of integrity constraints, based on different emotion-regulation principles, we can analyze and compare their respective trajectories.

Listing 1.

Integrity Constraints for Hedonic Emotion Regulation.

Listing 2.

Integrity Constraints for Utilitarian Emotion Regulation.

5.1 Analysis of translation to answer set semantics

In order to define the semantics of $\mathcal{C}_{MT}$ , we characterize trajectory models in terms of answer sets. This is formalized by the following theorem:

Theorem 1 Let $(D^{MT},O_{\text{initial}})$ be an action theory such that $O_{\text{initial}}$ specifies fluent observations in the initial state. Let $Q$ be a query, and define:

\begin{equation*} A_Q = \{(a\,\mathrm{occurs\_at}\,t_k) \mid a \in A_{k+1},\,\, 0 \leq k \lt t_{\max } \}. \end{equation*}

Let ${\mathcal P}_{MT} := \Pi (D^{MT}, O_{\text{initial}} \cup A_Q)$ be the $\mathcal{C}_{MT}$ logic program obtained via the translation operator $\Pi$ .

1. 1. If there is a trajectory model $\langle s_0, A_1, s_1, \ldots , A_{t_{\max }}, s_{t_{\max }} \rangle$ of $\mathcal{C}_{MT}(D^{MT}, O_{\text{initial}} \cup A_Q)$ , then there is an answer set $\mathcal A$ of ${\mathcal P}_{MT}$ such that for all time points $0 \leq k \leq t_{\max }$ :

   1. (a) $holds(f,k) \in {\mathcal A}$ iff $f \in s_k$ ,

   2. (b) $holds(neg(f),k) \in {\mathcal A}$ iff $f \notin s_k$ ,

   3. (c) $holds(occurs(a),k) \in {\mathcal A}$ iff $a \in A_{k+1}$ ,

   4. (d) $holds(neg(occurs(a)),k) \in {\mathcal A}$ iff $a \notin A_{k+1}$ .

2. 2. If there is an answer set $\mathcal A$ of ${\mathcal P}_{MT}$ and for each $0 \leq k \leq t_{\max }$ ,

   1. (a) $s_k = \{f \mid holds(f,k) \in {\mathcal A}\} \cup \{\neg f \mid holds(neg(f),k) \in {\mathcal A}\}$ ,

   2. (b) $A_{k+1} = \{a \mid holds(occurs(a),k) \in {\mathcal A}\}$ ,

   then there is a trajectory model $\langle s_0, A_1, s_1, \ldots , A_{t_{\max }}, s_{t_{\max }} \rangle$ of $\mathcal{C}_{MT}(D^{MT}, O_{\text{initial}} \cup A_Q)$ .

Proof We prove Theorem 1 in two directions using induction over the maximal time point $t_{\max }$ of the trajectory. The proof applies the Splitting Set Theorem (Lifschitz and Turner Reference Lifschitz and Turner1994) to justify that answer sets can be constructed incrementally over time.

Part 1: From trajectory to answer set.

Let $\langle s_0, A_1, s_1, \ldots , A_{t_{\max }}, s_{t_{\max }} \rangle$ be a trajectory model of $\mathcal{C}_{MT}(D^{MT}, O_{\text{initial}} \cup A_Q)$ . We define a set $X$ of atoms such that:

\begin{align*} X := &\, \{ holds(f, k) \mid f \in s_k,\,\, 0 \leq k \leq t_{\max } \} \\[2pt] & \cup \{ holds(neg(f), k) \mid f \notin s_k,\,\, 0 \leq k \leq t_{\max } \} \\[2pt] & \cup \{ holds(occurs(a), k) \mid a \in A_{k+1},\,\, 0 \leq k \lt t_{\max } \} \\[2pt] & \cup \{ holds(neg(occurs(a)), k) \mid a \notin A_{k+1},\,\, 0 \leq k \lt t_{\max } \}. \end{align*}

Let $P_k$ be the set of ASP rules in $\mathcal{P}_{MT}$ whose time arguments are $\leq k$ . Then $\langle P_0, P_1, \ldots , P_{t_{\max }} \rangle$ forms a splitting sequence of $\mathcal{P}_{MT}$ .

We prove by induction on $k$ that $X_k := X \cap \text{Atoms}(P_k)$ is an answer set of $P_k$ .

Base Case ( $k = 0$ ). By construction of $X_0$ , for every $f \in s_0$ , we have $holds(f, 0) \in X_0$ , and for every $f \notin s_0$ , we have $holds(neg(f), 0) \in X_0$ . These atoms match the translation of initial observations in $O_{\text{initial}}$ . Hence, all rules in $P_0$ are satisfied, and $X_0$ is an answer set of $P_0$ .

Induction Step. Assume $X_k$ is an answer set of $P_k$ . Consider $P_{k+1}$ . The new rules introduced in $P_{k+1}$ ∖ $P_k$ have head atoms with time $k+1$ and body atoms with times $\leq k+1$ .

By the construction of $X$ , and since $f \in s_{k+1}$ can only hold by one of the rules of Definition 8, we distinguish the following cases:

• • Case 1: $f \in s_{k+1}$ due to inertia: $holds(f,k) \in X_k$ , $neg(holds(neg(f),k+1)) \in X_{k+1}$ .

• • Case 2: $f \in s_{k+1}$ due to a dynamic causal law. There exists $a \in A_k$ and preconditions $g_1,\ldots ,g_m \in s_k$ such that $holds(occurs(a),k), holds(g_i,k) \in X_k$ . The rule head $holds(f,k+1)$ is in $X_{k+1}$ .

• • Case 3: $f \in s_{k+1}$ due to a static causal law. Preconditions $g_1,\ldots ,g_m \in s_{k+1}$ imply $holds(g_i,k+1) \in X_{k+1}$ . Hence $holds(f,k+1) \in X_{k+1}$ .

• • Case 4: $f \in s_{k+1}$ by default (non-inertial fluent), that is no rule for $\neg f$ is applicable. Then $holds(neg(f),k+1) \notin X$ , so $holds(f,k+1)$ is derived by the default rule.

• • Case 5: $f \in s_{k+1}$ is mental: derived by static causal law as in Case 3.

• • Case 6: $f^h \in s_{k+1}$ due to dynamic causal law. Same as Case 2 but with $f^h$ , and $holds(f^h,k+1) \in X$ .

• • Case 7: $f^h \in s_{k+1}$ due to static causal law. As in Case 3.

• • Case 8: $f^h \notin s_{k+1}$ due to a forbidding constraint $(g_1^h, \ldots , g_m^h\,\mathbf{forbids \,to \,cause} f^h) \in D^{MT}$ with $g_1^h, \ldots , g_m^h \in s_k$ . The translated integrity constraint

  \begin{equation*} :-\ holds(f^h, k+1),\ holds(g_1^h, k),\ \ldots ,\ holds(g_m^h, k) \end{equation*}
  must be satisfied in $X$ , so $holds(f^h, k+1) \notin X$ .

Thus, all new rule bodies are satisfied, and all forbidden transitions are avoided. By the Splitting Set Theorem (Lifschitz and Turner Reference Lifschitz and Turner1994), $X_{k+1}$ is an answer set of $P_{k+1}$ . Therefore, $X$ is an answer set of $\mathcal{P}_{MT}$ .

Part 2: From answer set to trajectory.

Assume there is an answer set $\mathcal{A}$ of the logic program $\mathcal{P}_{MT}$ . We construct a trajectory model

\begin{equation*} \langle s_0, A_1, s_1, \ldots , A_{t_{\max }}, s_{t_{\max }} \rangle \end{equation*}

of $\mathcal{C}_{MT}(D^{MT}, O_{\text{initial}} \cup A_Q)$ as follows.

State and Action Set Construction. For every $0 \leq k \leq t_{\max }$ , define:

\begin{align*} s_k := \{f \mid holds(f,k) \in \mathcal{A} \} \cup \{\neg f \mid holds(neg(f),k) \in \mathcal{A}\}, \\[-30pt] \nonumber \end{align*}

\begin{align*} A_{k+1} := \{a \mid holds(occurs(a),k) \in \mathcal{A} \}. \end{align*}

We will prove by induction on $t_{\max }$ that the constructed sequence satisfies all trajectory conditions of Definition 8.

Base Case ( $t_{\max } = 0$ ): Then the trajectory reduces to $\langle s_0 \rangle$ . The initial observations $O_{\text{initial}}$ specify fluents and mental fluents that are true or false at time $0$ .

By translation of $O_{\text{initial}}$ into facts of the form $holds(f,0)$ and $holds(neg(f),0)$ in $\mathcal{P}_{MT}$ , and since $\mathcal{A}$ is an answer set, the state $s_0$ includes exactly the correct fluents and mental fluents. Hence, $s_0$ satisfies the initial conditions.

Induction Step: Assume that for $t_{\max } = r$ , the sequence

\begin{equation*} \langle s_0, A_1, s_1, \ldots , A_r, s_r \rangle \end{equation*}

is a trajectory model of $\mathcal{C}_{MT}$ . We now extend it to

\begin{equation*} \langle s_0, A_1, s_1, \ldots , A_r, s_r, A_{r+1}, s_{r+1} \rangle \end{equation*}

and prove that the extension satisfies the conditions.

Let $X = \mathcal{A}$ . Since an atom of the form $holds(f,k) \in X$ can only be present if it is derived by some rule in $\mathcal{P}_{MT}$ , we consider each possible case for $f \in s_{r+1}$ :

• • Case 1 (Dynamic Law for $f$ ): If $holds(f, r+1) \in X$ is derived from a dynamic causal law, then there exists $a \in A_r$ such that

  \begin{equation*} (a\,\mathbf{influences}\,f\,\mathbf{if}\,g_1, \ldots , g_m) \in D^{MT} \end{equation*}
  with $holds(g_1, r), \ldots , holds(g_m, r) \in X$ , hence $g_1, \ldots , g_m \in s_r$ and $f \in s_{r+1}$ .

• • Case 2 (Static Law for $f$ ): If $holds(f, r+1) \in X$ is derived from a static causal law, then

  \begin{equation*} (g_1, \ldots , g_m\,\mathbf{influences}\,f) \in D^{MT} \end{equation*}
  with $holds(g_1, r+1), \ldots , holds(g_m, r+1) \in X$ , hence $g_1, \ldots , g_m \in s_{r+1}$ and $f \in s_{r+1}$ .

• • Case 3 (Inertial Fluent $f$ ): If $f$ is inertial, and $holds(f, r) \in X$ but $holds(neg(f), r+1) \notin X$ , then $f$ persists by inertia, and $f \in s_{r+1}$ .

• • Case 4 (Default Non-Inertial $f$ ): If $f$ is non-inertial, and no law causes $neg(f)$ , and $default(f)$ applies, then $holds(f, r+1) \in X$ and $f \in s_{r+1}$ .

• • Case 5 (Action Occurrence): For every $a \in A_{r+1}$ , we have $holds(occurs(a), r) \in X$ . This satisfies the action occurrence condition.

• • Case 6 (Dynamic Mental Fluent): If $holds(f^h, r+1) \in X$ is derived from a dynamic mental causal law, then there exists $a \in A_r$ such that

  \begin{equation*} (a\,\mathbf{influences}\,f^h\,\mathbf{if}\,g_1, \ldots , g_m) \in D^{MT} \end{equation*}
  with $g_1, \ldots , g_m \in s_r$ . Hence $f^h \in s_{r+1}$ .

• • Case 7 (Static Mental Fluent): If $holds(f^h, r+1) \in X$ is derived from a static mental causal law, then

  \begin{equation*} (g_1, \ldots , g_m\,\mathbf{influences}\,f^h) \in D^{MT} \end{equation*}
  with $g_1, \ldots , g_m \in s_{r+1}$ . Hence $f^h \in s_{r+1}$ .

• • Case 8 (Forbidding Constraint): Suppose for contradiction that $f^h \in s_{r+1}$ and $f^h \in F(s_r)$ . Then there exists a forbidding rule

  \begin{equation*} (g_1^h, \ldots , g_m^h\,\mathbf{forbids \,to \,cause}\,f^h) \in D^{MT} \end{equation*}
  with $g_1^h, \ldots , g_m^h \in s_r$ . By translation, the integrity constraint
  \begin{equation*} :-\,holds(f^h, r+1),\,holds(g_1^h, r), \ldots , holds(g_m^h, r) \end{equation*}
  is present in $\mathcal{P}_{MT}$ . Since $\mathcal{A}$ is an answer set, this constraint must be satisfied, and $holds(f^h, r+1) \notin \mathcal{A}$ , hence $f^h \notin s_{r+1}$ .

The logic program $\mathcal{P}_{MT}$ is stratified by time: rules whose head involves $time = r+1$ do not appear in the body of rules at time $t \leq r$ . Thus, the ASP Splitting Set Theorem (Lifschitz and Turner Reference Lifschitz and Turner1994) applies. The bottom part (for $t \leq r$ ) is satisfied by induction hypothesis. The top part (for $t = r+1$ ) builds on $s_r$ , $A_{r+1}$ and produces $s_{r+1}$ as shown above.

By induction on $t_{\max }$ , the constructed sequence satisfies all conditions of Definition 8. Hence, $\langle s_0, A_1, s_1, \ldots , A_{t_{\max }}, s_{t_{\max }} \rangle$ is a valid trajectory model of $\mathcal{C}_{MT}(D^{MT}, O_{\text{initial}} \cup A_Q)$ .

5.2 Safety analysis: Emotional change

In this subsection, we aim to prove that trajectories generated by $\mathcal{C}_{MT}$ preserve certain safety properties that holds for all states along a trajectory, using the invariance principle (Hansen and Schmidt Reference Hansen and Schmidt2003), by considering particular $\mathcal{C}_{MT}$ specifications. As a particular case, we analyze principles for emotional change based on HER and UER, respectively. For particular emotion theories in $\mathcal{C}_{MT}$ , we define different MT Invariant properties.

Proposition 1 Let $ D^{MT}(\mathbf{A}, \mathbf{F})$ be a $\mathcal{C}_{MT}$ domain description, and let $ C$ be a set of psychological classes with associated value domains $ V_c \subseteq V$ for each $ c \in C$ . Let $ \mathbf{F}^H \subseteq \mathbf{F}$ denote the set of mental fluents. Let $ V$ be a totally ordered value domain. Let $ \mathcal{R} = \{ R_{=}, R_{\not =}, R_{\lt }, R_{\leq }, R_{\gt }, R_{\geq } \}$ , where each $ R_* \subseteq V \times V$ is defined by its standard semantic interpretation:

\begin{align*} R_{=}(v, w) \text{holds true } &\iff v = w \\[3pt] R_{\not =}(v, w) \text{ holds true } &\iff v \not = w \\[3pt] R_{\lt }(v, w) \text{ holds true } &\iff v \lt w \\[3pt] R_{\leq }(v, w) \text{ holds true } &\iff v \leq w \\[3pt] R_{\gt }(v, w) \text{ holds true } &\iff v \gt w \\[3pt] R_{\geq }(v, w) \text{ holds true } &\iff v \geq w \end{align*}

for all $ v, w \in V$ , where $ V$ is a totally ordered set.

If $ \!\! f_1(c_1, v_1) \land \ldots \land f_m(c_m, v_m) \,\,\mathbf{forbids \,to \,cause}\,\, f_1'(c_1', v_{c_1'}') \land \ldots \land f_n'(c_n', v_{c_n'}') \in D^{MT},$ then $D^{MT} \models R_1(v_{c_1'}', w_1') \land \ldots \land R_k(v_{c_k'}', w_k'),$ for some $ k \geq 1$ , where for all $ 1 \leq j \leq k$ : $ R_j \subseteq V \times V$ is a binary relation from the set $ \mathcal{R}$ , $ w_j' \in \{v_{c_1'}', \ldots , v_{c_n'}'\} \cup V$ , and $ c_1, \ldots , c_m, c_1', \ldots , c_n' \in C$ .

Proof Let the rule $f_1(c_1, v_1) \land \ldots \land f_m(c_m, v_m) \,\,\mathbf{forbids \,to \,cause}\,\, f_1'(c_1', v_1') \land \ldots \land f_n'(c_n', v_n') \in D^{MT}.$

For each $ 1 \leq j \leq n$ , let $ w_j' \in \{v_1', \ldots , v_n'\} \cup V$ , and define $ R_j \in \mathcal{R}$ such that the fluent $ f_j'(c_j', v_j')$ being forbidden implies that $ R_j(v_j', w_j')$ holds.

The following cases cover all binary comparisons over a totally ordered set:

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' = w_j'$ , then $ R_j = R_{\not =}$ .

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' \not = w_j'$ , then $ R_j = R_{=}$ .

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' \gt w_j'$ , then $ R_j = R_{\leq }$ .

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' \lt w_j'$ , then $ R_j = R_{\geq }$ .

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' \leq w_j'$ , then $ R_j = R_{\gt }$ .

• • If $ f_j'(c_j', v_j')$ is forbidden when $ v_j' \geq w_j'$ , then $ R_j = R_{\lt }$ .

Hence, for some $ k \geq 1$ , we obtain the conjunction of relational constraints:

\begin{equation*} D^{MT} \models R_1(v_1', w_1') \land \ldots \land R_k(v_k', w_k'), \end{equation*}

where each $ R_j \in \mathcal{R}$ , and each $ w_j' \in \{v_1', \ldots , v_n'\} \cup V$ .

In the setting of emotion, let $EI_{HER}$ denote the MT invariant for HER-based constraints, and $EI_{UER}$ denote the MT invariant for UER-based constraints:

Definition 20 (Emotional Invariant: Hedonic) Let $ D^{MT}_{AE}$ be a domain description with a hedonic theory specification (Definition 17).

The hedonic emotional invariant, denoted $ EI_{HER}$ , holds for a trajectory $ M = \langle s_0, A_1, s_1, \ldots , A_n, s_n \rangle$ , $\forall i \in \{0, \ldots , n{-}1\}$ , where $f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}') \subseteq s_{i+1}$ ,

\begin{equation*} \textbf {if } D^{MT}_{AE} \models f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}') \textbf { then } D^{MT}_{AE} \models R_{\leq }(v_{\text{ne}}', v_{\text{go}}') \end{equation*}

The intuition behind $EI_{HER}$ is that a positive balance between need consistency and goal consistency should be kept in any transition. Refer to Table 1 for HER-based principles of change.

Definition 21 (Emotional Invariant: Utilitarian) Let $ D^{MT}_{AE}$ be a domain description with a utilitarian theory specification (Definition 18).

The utilitarian emotional invariant, denoted $ EI_{UER}$ , holds for a trajectory $ M = \langle s_0, A_1, s_1, \ldots , A_n, s_n \rangle$ , $\forall i \in \{0, \ldots , n{-}1\}$ , where $f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}'), f(\text{ac}, v_{\text{ac}}'), f(\text{co}, v_{\text{co}}') \subseteq s_{i+1}$ ,

\begin{align*} \textbf {if } D^{MT}_{AE} \models f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}'), f(\text{ac}, v_{\text{ac}}'), f(\text{co}, v_{\text{co}}') \end{align*}

\begin{equation*} \textbf { then } D^{MT}_{AE} \models \left ( \begin{aligned} &R_{=}(v_{\text{ne}}', \text{high}) \land R_{\leq }(v_{\text{go}}', v_{\text{ne}}') \land R_{=}(v_{\text{co}}', \text{high}) \land {} \\ & R_{\neq }(v_{\text{ac}}', \text{other}) \land R_{\neq }(v_{\text{ac}}', \text{undecided}) \end{aligned} \right ) \end{equation*}

The intuition behind $EI_{UER}$ is to maintain a utilitarian state in every transition. According to UER-based principles of change, summarized in Table 2, this is achieved by ensuring 1) that goal consistency does not exceed need consistency; 2) the situation must be accountable either to the self or the environment; and 3) the individual must perceive high control potential.

5.2.1 Safety analysis: Hedonic emotional change

In the following theorem, we show that trajectories generated by HER-based constraints preserves the emotional invariant, $EI_{HER}$ , such that if the initial state is a hedonic emotion state, then the following states will be hedonic emotion states.

Theorem 2 (Hedonic emotional change) Let $(D^{MT}_{AE}, O_{initial})$ be an action theory where $D^{MT}_{AE}$ includes the hedonic emotion theory specification (Definition 17). Let $O_{initial}$ be the fluent observations of the initial state, and let $Q$ be a query according to Definition 14. Let $A_Q = \{(a\,\mathit{occurs\_at} \,t_i)\,|\,a \in A_i, 1 \leq i \leq m\}$ .

If there exists a trajectory model $M = \langle s_0, A_1, s_1, A_2, \ldots , A_n, s_m \rangle$ of $\mathcal{C}_{MT}(D^{MT}_{AE}, O_{initial} \cup A_Q)$ , where $A_i \subseteq \mathbf{A}$ for $0 \leq i \leq m$ , and $s_0$ satisfies the emotional invariant $EI_{HER}$ (Definition 20), then all transitions $s_t$ to $s_{t+1}$ in $M$ preserve $EI_{HER}$ , such that:

\begin{equation*} \text{if } f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}') \in s_{t+1} \text{ then } R_{\leq }(v_{\text{ne}}', v_{\text{go}}') \end{equation*}

Proof We prove the theorem by induction over the steps of the trajectory $M$ .

Base Case. Let $s_0$ be the initial state. By assumption, $O_{initial} \subseteq s_0$ and $EI_{HER}$ holds in $s_0$ by construction of the action theory. Hence, the base case is satisfied.

Inductive Step. Let $s_t \rightarrow s_{t+1}$ be an arbitrary transition in $M$ , and assume $EI_{HER}$ holds in $s_t$ . We must show that $EI_{HER}$ also holds in $s_{t+1}$ , that is if $f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}') \in s_{t+1}$ , then $R_{\leq }(v_{\text{ne}}', v_{\text{go}}')$ .

Assume, for contradiction, that $v_{\text{ne}}' \gt v_{\text{go}}'$ . Then the transition leads to a state where the emotional invariant is violated.

By the semantics of $\mathcal{C}_{MT}$ , a transition $s_t \rightarrow s_{t+1}$ is valid only if $F(s_t) \cap s_{t+1} = \emptyset$ , where $F(s_t)$ is the set of fluents forbidden to be caused in $s_{t+1}$ by the active forbids to cause rules in $s_t$ .

We now do case analysis over the possible values of $v_{\text{ne}}'$ .

• • Case 1: $f(\text{ne}, \text{high}) \in s_t$ . Then the following rules from Definition 17 are active:

  \begin{align*} &f(\text{ne}, \text{high}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{low}) \\[2pt] &f(\text{ne}, \text{high}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{undecided}) \\[2pt] &f(\text{ne}, \text{high}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{high}) \end{align*}
  Thus, any $f(\text{go}, v_{\text{go}}') \in s_{t+1}$ is forbidden. Contradiction.

• • Case 2: $f(\text{ne}, \text{undecided}) \in s_t$ . Then the following rules are active:

  \begin{align*} &f(\text{ne}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{low}) \\[2pt] &f(\text{ne}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{undecided}) \\[2pt] &f(\text{ne}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{go}, \text{high}) \end{align*}
  Again, no $f(\text{go}, v_{\text{go}}')$ can be valid in $s_{t+1}$ . Contradiction.

• • Case 3: $f(\text{ne}, \text{low}) \in s_t$ . In this case, no forbids to cause rule is active that forbids any value of $f(\text{go}, v_{\text{go}}')$ . Thus, any $v_{\text{go}}'$ is permitted, and since $\text{low} \leq v_{\text{go}}'$ for all values in $V = \langle \text{low} \lt \text{undecided} \lt \text{high} \rangle$ , the invariant holds.

In all cases, either the invariant is preserved by definition (Case 3), or a contradiction arises from the violation of active forbids to cause rules (Cases 1–2). Therefore, the assumption that $v_{\text{ne}}' \gt v_{\text{go}}'$ leads to contradiction, and we conclude:

\begin{equation*} f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}') \in s_{t+1} \Rightarrow R_{\leq }(v_{\text{ne}}', v_{\text{go}}') \end{equation*}

Hence, $EI_{HER}$ is preserved across all transitions in the trajectory $M$ .

5.2.2 Safety analysis: Utilitarian emotional change

In order to prove invariance for UER, we can follow a similar approach as the proof for hedonic emotion regulation, using the given constraints for UER. Let us define the theorem and present the proof.

Theorem 3 (Utilitarian emotional change) Let $(D^{MT}_{AE}, O_{initial})$ be an action theory where $D^{MT}_{AE}$ includes the utilitarian emotion theory specification (Definition 18 ), and let $O_{initial}$ be the fluent observations of the initial state such that $EI_{UER}$ holds in $s_0$ . Let $Q$ be a query according to Definition 14 , and let $A_Q = \{(a\,\mathit{occurs\_at}\,t_i)\,|\,a \in A_i, 1 \leq i \leq m\}$ .

If there exists a trajectory model $M = \langle s_0, A_1, s_1, A_2, \ldots , A_n, s_m \rangle$ of $\mathcal{C}_{MT}(D^{MT}_{AE}, O_{initial} \cup A_Q)$ , where $A_i \subseteq \mathbf{A}$ for $0 \leq i \leq m$ , then all transitions $s_t$ to $s_{t+1}$ in $M$ preserve the emotional invariant $EI_{UER}$ defined in Definition 21 , such that:

$\text{if } f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}'), f(\text{ac}, v_{\text{ac}}'), f(\text{co}, v_{\text{co}}') \in s_{t+1}$

$\text{ then } R_{=}(v_{\text{ne}}', \text{high}) \land R_{\leq }(v_{\text{go}}', v_{\text{ne}}') \land R_{=}(v_{\text{co}}', \text{high}) \land R_{\not =}(v_{\text{ac}}', \text{other}) \land R_{\not =}(v_{\text{ac}}', \text{undecided})$

Proof We prove the theorem by induction over the steps of the trajectory $M$ .

Base Case. Let $s_0$ be the initial state. From the assumptions of the theorem, $EI_{UER}$ holds in $s_0$ . Hence, the base case is satisfied.

Inductive Step. Let $s_t \rightarrow s_{t+1}$ be any transition in $M$ . Assume $EI_{UER}$ holds in $s_t$ . We must show that it holds in $s_{t+1}$ .

Suppose:

\begin{equation*} f(\text{ne}, v_{\text{ne}}'), f(\text{go}, v_{\text{go}}'), f(\text{ac}, v_{\text{ac}}'), f(\text{co}, v_{\text{co}}') \in s_{t+1} \end{equation*}

Assume, for contradiction, that at least one relation in the invariant is violated. We now perform case analysis.

• • Case 1: $R_{=}(v_{\text{ne}}', \text{high})$ does not hold. Then $v_{\text{ne}}' \neq \text{high}$ . Let $v_{\text{ne}}' \in \{\text{low}, \text{undecided}\}$ . Then by Definition 18, any of the following may be active in $s_t$ :

  \begin{align*} &f(\text{ne}, \text{low}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{ne}, \text{undecided}), f(\text{ne}, \text{high}) \\ &f(\text{ne}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{ne}, \text{low}), f(\text{ne}, \text{high}) \end{align*}
  Thus, $f(\text{ne}, v_{\text{ne}}') \in F(s_t)$ , contradicting $f(\text{ne}, v_{\text{ne}}') \in s_{t+1}$ .

• • Case 2: $R_{\leq }(v_{\text{go}}', v_{\text{ne}}')$ does not hold. Then $v_{\text{go}}' \gt v_{\text{ne}}'$ . Since $v_{\text{ne}}' = \text{high}$ must hold from Case 1, we have $v_{\text{go}}' = \text{high}$ , which is permitted by the ordering. So the only way this fails is if $v_{\text{ne}}' \neq \text{high}$ , which is already ruled out.

• • Case 3: $R_{=}(v_{\text{co}}', \text{high})$ does not hold. That is, $v_{\text{co}}' \in \{\text{low}, \text{undecided}\}$ . Then from Definition 18, the following rules are active:

  \begin{align*} &f(\text{co}, \text{low}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{co}, \text{high}) \\ &f(\text{co}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{co}, \text{high}) \end{align*}
  So if $f(\text{co}, v_{\text{co}}')$ with $v_{\text{co}}' \neq \text{high}$ appears in $s_{t+1}$ , it contradicts $F(s_t) \cap s_{t+1} = \emptyset$ .

• • Case 4: $R_{\not =}(v_{\text{ac}}', \text{other})$ or $R_{\not =}(v_{\text{ac}}', \text{undecided})$ fails. That is, $v_{\text{ac}}' \in \{\text{other}, \text{undecided}\}$ . From Definition 18:

  \begin{align*} &f(\text{ac}, \text{undecided}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{ac}, \text{self}), f(\text{ac}, \text{environment}) \\ &f(\text{ac}, \text{other}) \,\,\mathbf{forbids \,to \,cause}\,\, f(\text{ac}, \text{self}), f(\text{ac}, \text{environment}) \end{align*}
  Hence, transitions leading to $v_{\text{ac}}' \in \{\text{other}, \text{undecided}\}$ are forbidden from relevant preconditions in $s_t$ .

In each case, violation of the invariant implies the presence of a fluent in $s_{t+1}$ that contradicts an active forbids to cause rule from $s_t$ , hence violating $F(s_t) \cap s_{t+1} = \emptyset$ .

Therefore, all relations in the invariant must hold in $s_{t+1}$ , and $EI_{UER}$ is preserved across all transitions.

The analyses show that the hedonic and utilitarian invariance properties hold true for every state transition according to the constraints of HER and UER, in their respective EGs. Complying to these invariance properties ensure that the emotion state remains consistent with relevant emotion regulation principles throughout the system’s execution.

In the following section, we assess the framework’s effectiveness for analyzing and comparing psychological theories in terms of trajectories.

6.1 Emotional reachability analysis

Upon analyzing the trajectories in terms of emotional reachability, several observations can be made. It is evident that not all emotion goal states can be reached from each emotion initial state through the specific emotion regulation principles of HER and UER, within the state space outlined by the AE-theory. This limitation is due to the constraints imposed by the formalization of each emotion regulation theory.

Firstly, analyzing the trajectories based on the constraints of HER (see Figure 4, marked in blue), we can observe that configurations labeled as Hope, Joy, Relief, Liking, Pride, and Guilt are reachable from all initial configurations. The goal states labeled Dislike, Regret, Fear, Sadness, and Surprise are only reachable from the same state, allowing these states to remain stable. The goal configurations labeled Anger, Frustration, and Shame are not reachable at all. These observations align with previous empirical findings regarding HER (Zaki Reference Zaki2020), aiming to reduce negative emotions and enhance positive emotions. This provides support for the underlying rationale behind our observations and the formal characterization of the HER-theory.

Secondly, analyzing the trajectories based on the constraints of UER (see Figure 4, marked in red), we can observe that the configuration labeled Frustration is reachable from all initial configurations, encompassing each of the 16 AE-based emotions. Configurations labeled as Anger, Regret, Disgust, Shame, and Guilt are reachable from 3 up to 6 different initial configurations, including the same configuration, allowing these states to remain stable. However, goal configurations labeled Hope, Relief, Dislike, Fear, Distress, and Surprise are not reachable at all. The reasoning behind these observations, such as the inclusion of the goal of Frustration and the exclusion of the goal of Joy, bears similarity to prior empirical research on UER (Schunk Reference Schunk2001; Parrott Reference Parrott2002; Tamir et al. Reference Tamir, Chiu and Gross2007; Tamir and Ford Reference Tamir and Ford2009; Dweck Reference Dweck2017), indicating that individuals may opt for activities that elicit negative emotions when anticipating a challenging or threatening task, providing rationale for the formal characterization of the UER-theory.

Fig 4.

Emotional reachability: Counting generated trajectories (HER and UER based) between each initial configuration and each goal configuration, considering the 16 emotions of AE-theory. The vertical axis represents the number of initial configurations with reachability to a specific goal configuration. The horizontal axis represents goal configurations, which are labeled by emotion to provide intuition.

By examining trajectories that adhere to HER-based constraints (see Figure 5) and trajectories that adhere to UER-based constraints (see Figure 6), we can gain a more comprehensive understanding of the system’s behavior. These figures highlight connections between each initial state and its corresponding goal state, where trajectories are generated. Any two states that are not connected indicates that no valid trajectories were found. Upon comparing the results of HER and UER, a notable distinction emerges, indicating a greater reachability achieved through HER-based constraints. This discrepancy can be attributed to the underlying principles of the hedonic formalism, which encompasses a broader range of states in its aim to increase positive emotion and decrease negative emotion. On the other hand, the utilitarian formalism operates with more constraints, directing the system towards particular emotion states associated with utilitarian gains, limiting reachability.

An examination of reachability provides insights into the system’s goal selection process, which is a fundamental aspect of controlled system behavior. Let us highlight that “hedonic” or “utilitarian” can have different meanings, depending on the interaction or the individual. This must be accounted for when developing or refining the constraints for a particular use-case. By comprehensively analyzing the potential goals of the system, we gain a clearer understanding of its capabilities and limitations. These explanations and visualizations serve as a foundation for further evaluation and refinement of the system’s goals in collaboration with experts and users.

Fig 5.

Reachability: HER.

Fig 6.

Reachability: UER.

6.2 Emotional priority analysis

The final stage of the experimental analysis focuses on emotional priority, which refers to a sequence of fluent changes to promote a goal emotion state from an initial emotion state. Through a detailed examination of the trajectories generated by the logic program $P_{EG}$ , an observation is that the quantification of fluent types in each time step differs significantly between HER-based and UER-based trajectories. Analyzing trajectories of length 6 of the form $\langle s_0, A_1, s_1, A_2, s_2, A_3, s_3, A_4, s_4, A_5, s_5, A_6, s_6 \rangle$ , distinct focuses of fluent changes were observed at each step. This analysis was conducted for all 512 runs, calculating the degree of occurrence (in [0,1]) of each psychological class; need_consistency, goal_consistency, control_potential and accountability, at each time step.

In the context of hedonic emotion regulation (HER), an analysis was conducted on all HER-based trajectories, following the 256 test cases to determine the priorities of different influences at each step (see Figure 7 and Table 6). Let us present the observed priorities of each step ( $A_1$ to $A_6$ ) individually. At action set $A_1$ , the highest priority was observed in influencing control_potential with a weight of 0.8. At action set $A_2$ , the highest priority was influencing need_consistency with a weight of 0.5. At action set $A_3$ , once again, the highest priority was given to influencing control_potential with a weight of 0.6. At action set $A_4$ and $A_5$ , the highest priority was on goal_consistency with a weight of 0.6 in $A_4$ and 0.5 in $A_5$ . A priority on need_consistency with a weight of 0.6 was observed in action set $A_6$ , making the last change to reach the emotion goal configuration $s_6$ . This observed trend in the trajectories can be intuitively explained by the objective of hedonic emotion regulation, which aims to increase positive emotion and decrease negative emotion (Tamir et al. Reference Tamir, Mitchell and Gross2008). Recall that according to our interpretation of the AE-theory, the balance between need_consistency and goal_consistency determines the experience of positive and negative emotions (such that need_consistency $\leq$ goal_consistency means positive emotion), while control_potential and accountability regulate the intensity of the emotion by managing the feeling of control (Passyn and Sujan Reference Passyn and Sujan2006), and redirecting the focus on who/what is accountable (Roseman Reference Roseman1996) for a situation. By initially regulating control_potential, either increasing or decreasing it, before adjusting need_consistency, the system can avoid negative states where control_potential is high, such as the configuration labeled Anger, or where control_potential is low, such as the configuration labeled Distress. Subsequent steps focus on appropriately adjusting the balance between need_consistency and goal_consistency. Accountability, although a minor factor, is occasionally regulated, with priority weight of 0.2 and 0.3, in steps $A_1$ and $A_5$ , respectively. This can be explained by accountability not playing a significant role in the balance between positive and negative emotions (Roseman Reference Roseman1996).

Table 6.

Emotional priority: HER

In the context of utilitarian emotion regulation, an analysis was conducted on all UER-based trajectories, following the 256 test cases to determine the priorities of different influences at each step (see Figure 8 and Table 7). Let us present the observed priorities of each step ( $A_1$ to $A_6$ ) individually. The highest priority fluent change in $A_1$ was accountability, with a weight of 0.6. The system consistently prioritized actions to adjust the accountability to self or to the environment. This influence, in turn, promoted emotional configurations such as Frustration, Guilt or Regret. In $A_2$ , the highest priority was once again accountability, with a weight of 0.6. At $A_3$ , the highest priority was need_consistency, most often by increasing it, and accountability, both with weights of 0.4. At $A_4$ , one again accountability had highest priority, with a weight of 0.5. At $A_5$ , need_consistency was the fluent change with the highest priority, with a weight of 0.5. Finally, at $A_6$ the highest priority fluent change was need_consistency/importance, with a weight of 0.4. Goal_consistency/attainability was mostly unaffected. This intuitively reflects that utilitarian gains, such as self responsibility (Autry and Langenbach Reference Autry and Langenbach1985), and high need_consistency (Dweck Reference Dweck2017) and high control_potential (Tamir et al. Reference Tamir, Mitchell and Gross2008), prevailed over a priority to reach positive emotions (need_consistency $\leq$ goal_consistency), a significant difference from the HER-based trajectories.

Table 7.

Emotional priority: UER

Fig 7.

Emotional priority: HER.

This analysis has provided insights into the different behaviors emerging from the formalizations of hedonic versus UER approaches (both within the state-space defined by AE-theory). The emotional reachability analysis revealed that not all emotion goal-states can be reached from every initial state. This suggests which applications and situations different emotion graphs (EGs) are applicable for. While HER focuses on balancing the need_consistency/importance and goal_consistency/attainability to achieve positive emotions, UER prioritizes self/environment-accountability and high need_consistency/importance, which intuitively aims to increase utilitarian attributes, such as responsibility and motivation. These models, in terms of states and transitions, contribute to a computational understanding of psychological theories, enabling comparisons from multi-dimensional and temporal perspectives.