\section{Trust Management Framework}
\label{sec:trust_mng_frmwrk}

In this section, we present our proposed trust management framework. The framework is designed to be flexible, adaptable, and not restricted to any specific industry or context. Users can tailor the framework to meet their specific needs and requirements.

\subsection{Data Acquisition and Processing}
\label{sec:data_proc}

The trust management framework takes the results of the \ac{stl}-checker and uses them as input. The input consists of values representing the domain specified by the user's output from the \ac{stl}-checker. This step involves gathering and processing the evidence data required for assessing trust, including error handling and data validation.

Each evidence instance corresponds to the classification of indirect observations and carries information about the satisfaction (+1), uncertainty, or violation (-1) of predefined \ac{stl} properties. 

\subsection{Data Aggregation}
\label{sec:data_proc}

After acquiring and processing the data, the trust management framework aggregates instances of evidence over specific time intervals. The interval window \( \mathcal{I} \), represents a particular time frame within which observations are collected and processed. This interval window is user defined and can be customized to fit the needs of the observed system, guaranteeing flexibility to capture its dynamics and adapt to different operational situations. These observations are converted into a \(k\)-value evidence vector that holds the number of occurrences of each classification label. 

In our running example, we form multinomial opinions on a ternary domain \(\mathbb{Y} = \{+1, 0, -1 \}\), which is a mapping of discrete values that indicate the ``satisfaction (+1)'', ``uncertainty (0)'', or ``violation (-1)'' of predefined \ac{stl} properties. 

For a given interval window \( \mathcal{I} = 60 \hspace{0.5em} minutes\), \{6, 2, 2\} would mean six occurrences of +1, two occurrences of 0, and two occurrences of -1 in the 60-minute window. 

\subsection{Trust Snapshot Opinion (\( \omega_{Y}^{S} \))}
\label{sec:trust_snapshot}

The \textit{trust snapshot opinion} (\( \omega_{Y}^{S} \)) is a concise assessment of system trustworthiness formed by creating a multinomial opinion over the \(k\)-value evidence vector using Eq.~\eqref{eq:evd_op}. The non-informative prior weight is set to \( W = 2 \), and the base rate distribution to \( \textbf{a}_{Y}(y) = \frac{1}{k} = \frac{1}{3}\).

The multinomial opinion \( \omega_{Y}^{S} \) is developed through evidence observation within a particular interval window \( \mathcal{I} \) and evaluates of the system's trustworthiness at that point in time. It does not consider the past or future conduct of the system.

The trust snapshot (\( T_{Y}^{S} \)) value represents the expected likelihood of (\( Y = +1 \)) for the opinion \( \omega_{Y}^{S} \), and is determined using  Eq.~\eqref{eq:mn_prob}.
 

\subsection{Trust Index Opinion (\( \omega_{Y}^{I} \))}
\label{sec:trust_index}

The \textit{trust index opinion} (\( \omega_{Y}^{I} \)) is a pivotal metric within the trust management framework, offering a comprehensive assessment of the system's trust. The multinomial opinion \( \omega_{Y}^{I} \) encapsulates the aggregated opinions derived from the trust snapshot opinion and undergoes continuous updating to reflect the evolving trust value. The cumulative fusion operator within \ac{sl} described in Section~\ref{def:cum_fus} is employed to integrate the two multinomial opinions into an overarching trust index opinion:

\begin{equation}
\omega_{Y}^{I} = \omega_{Y}^{I} \oplus \omega_{Y}^{S}
\label{eq:trust_idx}
\end{equation}

The trust index opinion is subject to continuous updating to ensure its alignment with the evolving state of the system. This updating mechanism integrates new evidence from subsequent trust snapshot opinions, enabling the index to adapt and reflect the most current system behaviour. 

The ``trust index (\( T_{Y}^{I} \))'' value represents the expected likelihood of (\( Y = +1 \)) for the opinion \( \omega_{Y}^{I} \), and is determined using  Eq.~\eqref{eq:mn_prob}.

\subsection{\acexp{tca}}
\label{sec:trust_calib_action}

\textit{\ac{tca}} refer to measures that users can undertake to boost trust in the system if trust drops below a set threshold. Domain experts familiar with the system provide these actions to support troubleshooting and enhance system performance. \ac{tca} are not included in the trust management framework but rather complement its use case by tackling the issue of undertrust. \ac{tca} are recommendations offered to the user, which can be disregarded at their discretion.

For example, if the framework outputs a trust snapshot opinion with a very low trust value for a particular interval window \( \mathcal{I} \), possible \ac{tca} could be to run the anti-virus scan on the system. Alternatively, if the trust index opinion shows a decreasing trend over time, \ac{tca} action could include rebooting the system. Detection of \ac{tca} in the power trace leads to the satisfaction of an \ac{stl} property, increasing trust.

The framework generates a trust snapshot opinion (\( \omega_{Y}^{S} \)) and trust index opinion (\( \omega_{Y}^{I} \)) to assess system trustworthiness. The trust snapshot opinion offers a concise assessment for a particular interval window \( \mathcal{I} \), while the trust index opinion provides an overall evaluation considering past and present evidence. This allows users to evaluate trustworthiness at a specific point in time and track changes over time. 

Both values are provided because, following extensive evidence collection, a trust snapshot opinion (\( \omega_{Y}^{S} \)) with a low trust value may not exert considerable influence on the trust index opinion (\( \omega_{Y}^{I} \)) due to \ac{sl} formalism. However, it remains crucial for the user to be aware that a particular trust snapshot opinion exhibits a low trust value as this signifies potential unreliability in the system at that specific moment. Thus, both values are presented to give users a comprehensive understanding of the system's trustworthiness.