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b/semester4/fmfp/formal-methods-functional-programming-summary.tex @@ -138,7 +138,9 @@ \input{parts/04_modelling/01_promela/01_expressions.tex} \input{parts/04_modelling/01_promela/02_statements.tex} \input{parts/04_modelling/01_promela/03_macros.tex} -% \input{parts/04_modelling/01_promela/} +\input{parts/04_modelling/02_linear-temporal-logic/00_linear-time-properties.tex} +\input{parts/04_modelling/02_linear-temporal-logic/01_linear-temporal-logic.tex} +% \input{parts/04_modelling/02_linear-temporal-logic/} % \input{parts/04_modelling/} diff --git a/semester4/fmfp/parts/04_modelling/02_linear-temporal-logic/00_linear-time-properties.tex b/semester4/fmfp/parts/04_modelling/02_linear-temporal-logic/00_linear-time-properties.tex new file mode 100644 index 0000000..d3ff7fa --- /dev/null +++ b/semester4/fmfp/parts/04_modelling/02_linear-temporal-logic/00_linear-time-properties.tex @@ -0,0 +1,66 @@ +% TOOD: Some more intuition +\subsection{Linear Temporal Logic} +\subsubsection{Transition System of a Promela Model} +For the transition systems, compared to earlier, we use a fixed initial configuration because we only model one program or system, +as opposed to all programs of a programming language, as previously. +In addition, we omit terminal configurations from the definition, which simplifies the theory, and termination can be modelled by a transition into a special state, +the \bi{sink state}, which only allows further transitions back to itself. + + +\paragraph{Converting a Promela model into a transition system} + +The configurations are the states of the model, such as global variables and channels, as well as per active process the local variables, channels, and location counters. + +The initial configuration is the initial state of the model. + +The transition relation is defined by the operational semantics of the statements. They are kept informal in this course. + +A Promela model has a finite number of states, since there are a finite number of allowed active processes (255), as well as finite numbers of variables and channels, +finite ranges of variables and finite buffers of channels. + + +\shade{gray}{Computations} + +$S^\omega$ is the set of infinite sequences of elements of set $S$ and $s_{[i]}$ denotes the $i$-th element of the sequence $s \in S^\omega$ + +$\gamma \in \Gamma^\omega$ is a \bi{computation} of a transition system if: +\begin{itemize} + \item $\gamma_{[0]} = \sigma_I$ + \item $\gamma_{[i]} \rightarrow \gamma_{[i + 1]}$ (for all $i \geq 0$) +\end{itemize} +Note that we use $\sigma$ to range over the states $\Gamma$ of a transition system. Here $\gamma_{[i]}$ denotes the $i$-th element in $\gamma = \sigma_0 \sigma_1 \ldots$ + +We use $\cC(TS)$ to denote the set of all computations of a transition system $TS$ + + + +\subsubsection{Linear Time Properties} +LT-Properties can be used to specify the permitted computations of a transition system. +A linear time property $P$ over $\Gamma$ is a subset of $\Gamma^\omega$, where $P$ specifies a particular set of infinite sequences of configurations. + +A transition system $TS$ \bi{satisfies} the LT-Property $P$ (over $\Gamma$), denoted $TS \models P$, if and only if $\cC(TS) \subseteq P$. +Intuitively, this means that all computations of $TS$ belong to the set $P$. + +LT-Properties \bi{precisely} express properties of computations. +Non-termination is handled using infinite sequences and non-determinism is handled by considering each computation separately. + +To make notation cleaner and quicker, we define \bi{atomic propositions} (AP) for a transition system $TS$, which is a proposition that does not contain any logical connectives. +For example, $AP = \{ open, closed \}$ (for files). + +We then provide a \bi{labeling function} $L : \Gamma \rightarrow \cP(AP)$ that maps configurations to sets of atomic propositions from AP. +$L(\sigma)$ is called an \bi{abstract state}. AP and $L$ are both considered to be \hl{part of the transition system}. + + +\paragraph{Traces} +A trace is an abstraction of a computation, which only observes the propositions of each state, not the concrete state itself. +It is an infinite sequence of abstract states $\cP(AP)^\omega$. + +$t \in \cP(AP)^\omega$ is a trace of a transition system $TS$ if $t = L(\gamma_{[0]} L(\gamma_{[1]}) \ldots$ and $\gamma$ is a computation of $TS$. +The set of all traces of $TS$ is $\cT(TS)$. +The LT-Properties are typically specified over infinite sequences of abstract states, rather than over sequences of configurations. + + +\subparagraph{Safety Properties} + + +\subparagraph{Liveness Properties} diff --git a/semester4/fmfp/parts/04_modelling/02_linear-temporal-logic/01_linear-temporal-logic.tex b/semester4/fmfp/parts/04_modelling/02_linear-temporal-logic/01_linear-temporal-logic.tex new file mode 100644 index 0000000..e69de29