\[\newcommand{\seq}{\cdot} \newcommand{\alt}{+} \newcommand{\nat}{\mathbb{N}} \newcommand{\bool}{\mathbb{B}} \newcommand{\true}{\mathbf{true}} \newcommand{\false}{\mathbf{false}} \renewcommand{\a}[1]{\textit{#1}} \renewcommand{\implies}{\mathop{\Rightarrow}} \newcommand{\act}{\mathit{Act}} \newcommand{\mccan}[1]{\langle #1 \rangle} \newcommand{\mcall}[1]{[ #1 ]} \newcommand{\sem}[1]{[\![ #1 ]\!]} \newcommand{\R}{\mathrel{R}} \newcommand{\bisim}{\stackrel{\leftrightarrow}{-}} \newcommand{\oftype}{\smash{\,:\,}} %\]

Basic modelling with mCRL2

Behaviour and transition systems

In process algebras such as mCRL2, the basic units of computation are called actions. The philosophy is that any system can be described in terms of the observations that you can do about that system. The observations one can make with respect to a system are dictated only by its behaviour. By extension, every two systems that one might wish to distinguish must behave differently, or one would not be able to distinguish them by observing them.

Actions correspond to observable events in the behaviour of a system. As a running example, we will be introducing various coffee machines, the behaviour of which will differ, but of which the functionality will usually be expressed in terms of inserting coins into the machines, and retrieving coffee from it. Obvious choices for actions that a coffee machine might perform are ‘accepting a coin’, and ‘dispensing coffee’.

As we already mentioned, coffee machines will usually both accept coins and provide coffee. In fact, most coffee machines will only provide coffee after accepting coins. Before we continue, we formalise this notion of behaviour by saying that the behaviour of a system can be described by a labelled transition system, or LTS for short. These are relational structures that have a convenient graphical representation, and we will use them to give a semantics to the mCRL2 processes we present.

Definition (LTS)

A labelled transition system (LTS) is a tuple \(\langle S, \act, \rightarrow, i \rangle\), where

• \(S\) is a set of states,
• \(\act\) is a set of action labels,
• \({\rightarrow} \subseteq S\times \act \times S\) is a transition relation and
• \(i\in S\) is the initial state.

In graphical depictions of LTSs, states are shown as circles. The initial state is marked by an incoming arrow that has no source state. For \(s,s'\in S\) and \(a\in\act\), we will write \(s \stackrel{a}{\longrightarrow} s'\) instead of \((s,a,s') \in \rightarrow\).

Example

Consider a simple coffee machine that accepts a single coin and then dispenses coffee. This system could be modelled by the LTS with \(S=\{s_0, s_1, s_2\}\), \(\act=\{\a{coin}, \a{coffee}\}\), \(\rightarrow = \{(s_0, \a{coin}, s_1), (s_1, \a{coffee}, s_2)\}\), \(i=s_0\) and \(f=s_2\). Its graphical representation is as follows:

Sequences and choices

The coffee machine in the example above is not very exciting. It performs only two actions, and it performs them sequentially. More interesting behaviour usually involves some form of choice. Choices in mCRL2 are nondeterministic, that is to say, if a system can choose between two actions to perform, then we don’t know anything about the probability with which it will choose either.

Let us consider a slightly more advanced coffee machine. It provides two kinds of coffee. The first type costs only one coin, and is of the undrinkable, asphalty kind. The event that the coffee machine dispenses this bad coffee is modelled by the action \(\a{bad}\). The machine may also dispense nice coffee, modelled by the action \(\a{good}\), but this will cost you an extra coin. We will create a model of this machine in the mCRL2 language.

We start by giving an mCRL2 specification of our simple coffee machine, with on the left the semantics of the specification.

A simple coffee machine.

act coin, coffee; init coin . coffee;

Note that the act statement explicitly defines the set \(\act\) of the LTS on the left. The init statement says that the initial state \(i\) is a state that can first perform a \(\a{coin}\) action, followed by a \(\a{coffee}\) action. This sequential behaviour is expressed by the . operator.

We now wish to express that after inserting a single coin, we can either retrieve bad coffee, or we can insert another coin and get good coffee. This can be expressed as follows:

Another coffee machine.

act coin, good, bad; init coin . (bad + coin . good);

Again the . operator is used to indicate sequential execution (after inserting a coin, the machine can perform bad + coin . good). Now is a good time to note that . binds stronger than +, so bad + coin . good is equal to bad + (coin . good). The + operator expresses the choice between either dispensing bad coffee (bad), or accepting another coin and then dispensing good coffee (coin . good). In the corresponding LTS, this choice is visualised as a state that has two outgoing arrows.

Exercise

Give a specification for a machine that sells tea for 10¢ and coffee for 20¢.

Specifying systems

We mentioned in the introduction that we are interested in that part of the behaviour of systems which we can observe. In the realm of modelling, we therefore want to have a means of describing properties of systems such as ‘this coffee machine will always dispense good coffee after inserting two coins’. To this end, we introduce Hennessy-Milner logic, an extension of Boolean logic that introduces a modality \(\mcall{a} \varphi\) that expresses that if a system performs an \(\a{a}\) action, then directly afterwards, the property \(\varphi\) always holds. Note that in particular this is true if the system cannot do an \(\a{a}\) action. Its dual is the modality \(\mccan{a} \varphi\), which says that a system is able to (as opposed to must) do an \(\a{a}\) action, after which \(\varphi\) holds.

Definition (Action formula)

An action formula over a set of actions \(\act\) is an expression that adheres to the following syntax in pseudo-BNF, where \(a\in \act\).

\(A,B ::= \false ~|~ a ~|~ \overline{A} ~|~ A \cup B\)

The following abbreviations may also be used.

\[\begin{split}\true &= \overline{\false} \\ A \cap B &= \overline{\overline{A} \cup \overline{B}}\end{split}\]

An action formula \(A\) over \(\act\) is associated with a set \(\sem{A} \subseteq \act\) in the following manner.

\[\begin{split}\sem{\false} &= \emptyset \\ \sem{a} &= \{ a \} \\ \sem{\overline{A}} &= \act \setminus \sem{A} \\ \sem{A \cup B} &= \sem{A} \cup \sem{B}\end{split}\]

Example

Let \(\act=\{a, b, c\}\). Then the formula \(\overline{a}\cap\overline{b}\) corresponds to \(\{ c \}\).

Definition (HML)

A Hennessy-Milner logic formula interpreted over an LTS with action labels \(\act\) adheres to the following syntax in pseudo-BNF, where \(A\) is an action formula over \(\act\).

\[\begin{split}\varphi,\chi ::= \mccan{A}\varphi ~|~ \varphi \land \chi ~|~ \neg \varphi ~|~ \true \\\end{split}\]

The following common abbreviations are allowed:

\[\begin{split}\begin{align*} \false &= \neg \true & \varphi \lor \chi &= \neg(\neg \varphi \land \neg \chi) \\ \mcall{A}\varphi &= \neg \mccan{A} \neg \varphi & \varphi \implies \chi &= \neg \varphi \lor \chi \end{align*}\end{split}\]

An HML formula \(\varphi\) is interpreted over an LTS \(T = \langle S, \act, \rightarrow, i \rangle\). Its semantics is given as the set of states \(\sem{\varphi}_T \subseteq S\) of the LTS in which the formula holds. It is defined as follows.

\[\begin{split}\begin{align*} \sem{\true}_T &= S \\ \sem{\neg\varphi}_T &= S \setminus \sem{\varphi}_T \\ \sem{\varphi \land \chi}_T &= \sem{\varphi}_T \cap \sem{\chi}_T \\ \sem{\mccan{A}\varphi}_T &= \{ s \in S ~|~ \exists_{s'\in S, a \in \sem{A}}~ s \stackrel{a}{\longrightarrow} s' \land s' \in \sem{\varphi}_T \} \end{align*}\end{split}\]

We say that \(T\) satisfies \(\varphi\), denoted \(T \models \varphi\), if and only if \(i \in \sem{\varphi}_T\).

Example

The formula \(\mccan{\a{coin}}\mccan{\a{good}}\true\) can be used to express that there is a possibility that a system dispenses good coffee after accepting one coin. This formula does not hold for the machine in figure with another coffee machine, because when you insert one coin, the machine will only provide you with bad coffee.

Example

The formula \(\mcall{\a{coin}}\mccan{\a{bad} \cup \a{coin}}\true\) does hold for the machine in another coffee machine; it says that always after accepting a coin, the machine might (is able to) dispense bad coffee or accept another coin.

Exercises

1. Show that an arbitrary LTS can never satisfy \(\mccan{a}\false\), and that it will always satisfy \(\mcall{a}\true\).
2. Describe in English what the formula \(\mcall{a}\false\) means.
3. Give HML formulae expressing the following properties:
   1. The coffee machine can dispense good coffee after two coins have been inserted.
   2. The coffee machine will not dispense bad coffee after two coins have been inserted.

Comparing systems

Given two system models, one might wonder whether they are in some sense interchangeable. This calls for a natural notion of behavioral equivalence that relates systems that cannot be distinguished by observing their behaviour. As we shall see, such an equivalence can be quite straightforward. Not quite coincidentally, HML was originally designed as an alternative way to distinguish systems. In fact, it was shown that two systems are related by the equivalence described below if and only if there is no HML formula that is true for one and false for the other.

Definition (Strong bisimulation)

Let \(\langle S_1, \act, \rightarrow_1, i_1 \rangle\) and \(\langle S_2, \act, \rightarrow_2, i_2 \rangle\) be labelled transition systems. A relation \({\R} \subseteq S_1\times S_2\) is a strong bisimulation relation if and only if for \((s, s') \in {\R}\) (also written \(s \R s'\)) we have for all \(a \in \act\):

• if \(s \stackrel{a}{\longrightarrow}_1 t\), then there is a \(t'\in S_2\) such that \(s' \stackrel{a}{\longrightarrow}_2 t'\) and \(t \R t'\).
• if \(s' \stackrel{a}{\longrightarrow}_2 t'\), then there is a \(t\in S_1\) such that \(s \stackrel{a}{\longrightarrow}_1 t\) and \(t \R t'\).

Two states \(s, s'\) are said to be strongly bisimilar, denoted \(s \bisim s'\), if there is a strong bisimulation relation \(\R\) such that \(s \R s'\). Two LTSs are strongly bisimilar iff their initial states are bisimilar.

Example

In the following diagram, the dotted lines indicate the pairs of nodes that are related by a relation \(R\).

\(R\) is a bisimulation relation that relates the initial states of the two transition systems, hence they are bisimilar.

Note that the definition also allows you to compare states within a single transition system (i.e., \(\langle S_1, \act, \rightarrow_1, i_1 \rangle = \langle S_2, \act, \rightarrow_2, i_2 \rangle\)). If two states are found to be bisimilar, then for all intents and purposes it is reasonable to see them as only one state, thus giving rise to a natural statespace reduction: if in an LTS \(T\) we merge all states that are bisimilar, the resulting LTS \(T'\) is bisimilar to \(T\).

Example

In the following diagram, the dotted lines indicate the pairs of states that are related by a relation \(R\).

\(R\) is a bisimulation relation, so merging all related states will yield a smaller, bisimilar transition system (namely the left transition system of the previous bisimulation example).

Exercise

Are the following two process definitions bisimilar?

act coin, good, bad;
init coin . (bad + coin . good);

act coin, good, bad;
init coin . bad + coin . coin . good;

A dash of infinity

Recursion

So far, our models of coffee machines only modelled a single transaction–after dispensing a single cup of coffee, the machine terminated. In many situations in real life, however, we wish to model systems that repeatedly perform the same procedures.

The recipe for this is simple. We give a process a name, say \(P\), and then say that \(P\) may exhibit some finite behaviour, after which it will once more behave like \(P\). Let us create an optimistic model of a coffee machine, that lets us operate the machine forever.

An ever-lasting coffee machine.

act coin, good, bad; proc P = coin . (bad . P + coin . good . P); init P;

In the specification, we see that the proc operator accepts equations that associate processes with process names. In this case, the process P is defined. By using it in the right-hand side of its own definition, we introduced infinitely repeating behaviour. To illustrate this, we could unfold this repetition once and obtain a bisimilar system, as shown in the figure of the unfolded ever-lasting coffee machine.

The ever-lasting coffee machine, unfolded once.

act coin, good, bad; proc P = coin . (bad . P + coin . good . P); init coin . (bad . P + coin . good . P);

Note that the definition of bisimilarity does not have to be changed to deal with recursive systems; the co-inductive definition guarantees that the future behaviour stays the same.

Exercise

Show that the statespaces from figures the ever-lasting coffee machine and the unfolded ever-lasting coffee machine are bisimilar.

Regular HML

We saw that recursion does not require the definition of bisimilarity to be changed. Similarly, HML is still adequate to distinguish recursive systems: if two finite state systems are not bisimilar, then there is a finite HML expression that distinguishes the two. However, when we are dealing with infinitary systems, we often want to express properties of a system that say that it will always keep doing something, or that it will eventually do something. Such properties cannot be expressed by HML expressions of finite length.

To remedy this shortcoming, HML can be extended to allow regular expressions over action formulas inside the \(\mccan{\cdot}\) and \(\mcall{\cdot}\) modalities. In particular, the Kleene star is a powerful operator that effectively abbreviates certain HML formulas of infinite size.

Definition (Regular HML)

A regular HML formula interpreted over an LTS with action labels \(\act\) adheres to the following syntax in pseudo-BNF, where \(A\) is an action formula over \(\act\).

\[\begin{split}\varphi,\chi &::= \mccan{\alpha}\varphi ~|~ \varphi \land \chi ~|~ \neg \varphi ~|~ \true \\ \alpha,\beta &::= A ~|~ \alpha ^* ~|~ \alpha \cdot \beta ~|~ \alpha + \beta\end{split}\]

The usual abbreviations are allowed, as well as writing \(\alpha^+\) for \(\alpha\cdot\alpha^*\). We will informally give the semantics by rewriting regular HML formulas to infinitary HML formulas:

\[\begin{split}\mccan{\alpha \cdot \beta}\varphi &= \mccan{\alpha}\mccan{\beta}\varphi \\ \mccan{\alpha + \beta}\varphi &= (\mccan{\alpha}\varphi) \lor (\mccan{\beta}\varphi) \\ \mccan{\alpha^*}\varphi &= \varphi \lor (\mccan{\alpha}\varphi) \lor (\mccan{\alpha\cdot\alpha}\varphi) \lor \ldots\end{split}\]

Exercise

Rewrite the regular HML formula \([a+b]\false\) to a pure HML formula.

The modal µ-calculus

Although regular HML is a powerful tool to specify properties over infinite systems, it is still not expressive enough to formulate so-called fairness properties*. These are properties that say things like if the system is offered the possibility to perform an action infinitely often, then it will eventually perform this action.

Another way of extending HML to deal with infinite behaviour is to add a least fixpoint operator. This extension is called the modal µ-calculus, named after the least fixpoint operator \(\mu\). The µ-calculus (we often leave out the modal, as no confusion can arise) is famous for its expressivity, and infamous for its unintelligability. We will therefore first give the definition and the formal semantics, and then elaborate more informally on its use.

Definition (µ-calculus)

A µ-calculus formula interpreted over an LTS with action labels \(\act\) adheres to the following syntax in pseudo-BNF, where \(A\) is an action formula over \(\act\), and \(X\) is chosen from some set of variable names \(\mathcal{X}\).

\[\varphi,\chi ::= \true ~|~ X ~|~ \mu X\,.\, \varphi ~|~ \mccan{A}\varphi ~|~ \varphi \land \chi ~|~ \neg \varphi\]

We allow the same abbreviations as for HML, and we add the greatest fixpoint operator, which is the dual of the least fixpoint operator:

\[\begin{align*} \nu X \,.\, \varphi &= \neg \mu X\,.\, \neg\varphi[\neg X / X] \end{align*}\]

In the above, we use \(\varphi[\neg X/X]\) to denote \(\varphi\) with all occurrences of \(X\) replaced by \(\neg X\).

For technical reasons, we impose an important restriction on the syntax of µ-calculus formulas: only formulas in which every use of a fixpoint variable from \(\mathcal{X}\) is preceded by an even number of negations are allowed. The formula is then in positive normal form, allowing us to give it a proper semantics [1].

A µ-calculus formula \(\varphi\) is interpreted over an LTS \(T = \langle S, \act, \rightarrow, i, f \rangle\). To accomodate the fixpoint variables, we also need a predicate environment} `rho: mathcal{X} to 2^S`, which maps variable names to their semantics (*i.e., sets of states from \(T\)). We use \(\rho[X\mapsto V]\) to denote the environment that maps \(X\) to the set \(V\), and that maps all other variable names in the same way \(\rho\) does.

The semantics of a formula is now given as the set of states \(\sem{\varphi}_T^\rho \subseteq S\), defined as follows.

\[\begin{split}\begin{align*} \sem{\true}_T^\rho &= S \\ \sem{X}_T^\rho &= \rho(X) \\ \sem{\neg\varphi}_T^\rho &= S \setminus \sem{\varphi}_T^\rho \\ \sem{\varphi \land \chi}_T^\rho &= \sem{\varphi}_T^\rho \cap \sem{\chi}_T^\rho \\ \sem{\mccan{a}\varphi}_T^\rho &= \{ s \in S ~|~ \exists_{s'\in S}~ s \stackrel{a}{\longrightarrow} s' \land s' \in \sem{\varphi}_T^\rho \} \\ \sem{\mu X\,.\,\varphi}_T^\rho &= \bigcap \{V \subseteq S ~|~ \sem{\varphi}_T^{\rho[X \mapsto V]} \subseteq V \} \end{align*}\end{split}\]

We say that \(T\) satisfies \(\varphi\), denoted \(T \models \varphi\), if and only if \(i \in \sem{\varphi}_T^\rho\) for any \(\rho\).

Using the µ-calculus

To understand how the µ-calculus can be used to express properties of systems, it is instructive to see that regular HML can be encoded into the modal µ-calculus by using the following equalities.

\[\begin{align*} \mccan{\alpha^*}\varphi &= \mu X\,.\, \varphi \lor \mccan{\alpha}X & \mcall{\alpha^*}\varphi &= \nu X\,.\, \varphi \land \mcall{\alpha}X \end{align*}\]

Intuitively, the least fixpoint operator \(\mu\) corresponds to an eventuality, where the greatest fixpoint operator says something about properties that continue to hold forever.

We can read \(\mu X\,.\, \varphi \lor \mccan{\alpha}X\) as \(X\) is the smallest set of states such that a state is in \(X\) if and only if \(\varphi\) holds in that state, or there is an \(\alpha\)-successor that is in \(X\)’. Conversely, \(\nu X\,.\, \varphi \land \mcall{\alpha}X\) is the largest set of states such that a state is in \(X\) if and only if \(\varphi\) holds in that state and all of its \(\alpha\)-successors are in \(X\).

A good way to learn how the µ-calculus works is by understanding how the semantics of a formula can be computed. To do so, we use approximations. For each fixpoint we encounter, we start with an initial approximation, and then keep refining the approximation until the last two refinements are the same. The current approximation is then a fixpoint of the formula, which is what we were after. The first approximation \(\hat{X}^0\) for a fixpoint \(\mu X \,.\, \varphi\) is given by \(\varphi[\false / X]\). For a greatest fixpoint \(\nu X \,.\, \varphi\), it is given by \(\varphi[\true / X]\). In other words, for a least fixpoint operator the initial approximation represents the empty set of states, and for a greatest fixpoint operator we initially assume the formula holds for all states. Each next approximation \(\hat{X}^{i+1}\) is given by \(\varphi[\hat{X}^i / X]\). If \(\hat{X}^{i+1} = \hat{X}^i\), then we have reached our fixpoint.

Example

Consider the following formula, which states that a coffee machine will always give coffee after a finite number of steps.

\[\mu X\,.\, \mccan{\true}\true \land \mcall{\overline{`\a{coffee}`}} X\]

Note that this formula cannot be expressed using regular expressions. To see how the formula works, consider \(\hat{X}^0 =\mccan{\true}\true \land \mcall{\overline{\a{coffee}}}\false\). The first conjunct of this first approximation says that an action can be performed, and the second conjunct says that any action that can be performed must be a \(\a{coffee}\) action. The first approximation hence represents the set of states that can–and can only–do \(\a{coffee}\) actions.

The next approximation is \(\hat{X}^1 = \mccan{\true}\true \land \mcall{\overline{\a{coffee}}} \hat{X}^0\). The first conjunct again selects all states that may perform an action, and the second conjunct now selects those that can additionally do only \(\a{coffee}\) actions, or that can do another action and then always end up in the set of states where \(\hat{X}^0\) holds. Continuing this reasoning, it is easy to see that \(\hat{X}^i\) represents the set of states that must reach a state that must do a \(\a{coffee}\) action in \(i\) or less steps. Hence, when we find a fixpoint, this fixpoint represents those states that must eventually reach a state from which a \(\a{coffee}\) action must be performed.

More complicated properties can be expressed by nesting fixpoint operators.

Exercise

What does the formula \(\nu X \,.\, \varphi \land \mccan{a}X\) express? Can it be expressed in regular HML?

Data

Recursion is one way to introduce infinity in system models. It neatly enables us to model systems that continuously interact with their environment. The infinity obtained by recursion is an infinity in the depth of the system. There is another form of infinity that we have not yet explored: infinity in the width of the system. This type of infinity can be obtained by combining processes and data.

We first illustrate the idea of combining processes and data with a simple example. Let us reconsider the coin action of the coffee machine. Rather than assuming that there is only one flavour of coins, there are in fact various types of coins: 2, 5 and 10 cents; these values can be thought of as elements of the structured sort Val, defined as:

sort Val = struct c2 | c5 | c10;

The action coin can be thought of as inserting a particular type of coin, the value of which is dictated by a parameter of the action. Thus, coin(c2) represents the insertion of a 2 cent coin, whereas coin(c10) represents the insertion of a 10 cents coin. Below, we have a state that accepts all possible coins, with on the right the required mCRL2 notation.

A state accepting coins of all flavour.

sort Val = struct c2 | c5 | c10; act coin: Val; init sum v: Val . coin(v);

The statement sum v: Val . coin(v) actually binds a local variable v of sort Val, and, for every of its possible values, specifies a coin action with that value as a parameter. An alternative description of the same process is

init coin(c2) + coin(c5) + coin(c10);

This suggests that the summation is like the plus.

As soon as the sort that is used in combination with the sum operator has infinitely many basic elements, the branching degree of a state may become infinite, as illustrated by figure Transition system with an infinite number of transitions.. Since each mCRL2 expression is finite, we can no longer give an equivalent expression using only the plus operator.

Transition system with an infinite number of transitions.

act num: Nat; init sum v: Nat . num(2 * v);

The sum operator is quite powerful, especially when combined with the if-then construct b -> p and the if-then-else construct b -> p <> q, which behaves as process p if b evaluates to true, and, in case of the if-then-else construct, as process q otherwise. Using such constructs, and a Boolean function even, we can give an alternative description of the infinite transition system above:

map even: Nat -> Bool;
var n: Nat;
eqn even(n) = n mod 2 == 0;

act num: Nat;
init sum v: Nat . even(v) -> num(v);

The Boolean condition even(v) evaluates to true or false, dependent on the value of v. If, the expression even(v) evaluates to true, action num(v) is possible.

Exercise

Give a µ-calculus expression that states that this process cannot execute actions num with an odd natural number as its parameter.

Data variables that are bound by the sum operator can affect the entire process that is within the scope of such operators. This way, we are able to make the system behaviour data-dependent. Suppose, for instance, that our coffee machine only accepts coins of 10 cents, and rejects the 2 and 5 cent coins. The significant states modelling this behaviour, including parts of the mCRL2 description, are as follows:

A picky coffee machine.

sort Val = struct c2 | c5 | c10; act coffee; coin, rej: Val; proc P = sum v: Val . coin(v) . ( (v != c10) -> rej(v) . P + (v == c10) -> coffee . P ); init P;

Data may also be used to parameterise recursion. A typical example of a process employing such mechanisms is an incrementer:

act num:Nat;
proc P(n:Nat) = num(n).P(n+1);
init P(0);

Or we could have written the picky coffee machine as follows:

proc P(v: Val) =
  coin(v) . (
    (v != c10) -> rej(v) . P
  + (v == c10) -> coffee . P
  );

init sum v: Val . P(v);

It may be clear that most data-dependent processes describe transition systems that can no longer be visualised on a sheet of paper. However, the interaction between the data and process language is quite powerful.

Exercises

1. Is there a labelled transition system with a finite number of states that is bisimilar to the incrementer? If so, give this transition system and the witnessing bisimulation relation. If not, explain why such a transition system does not exist.
2. Consider the mCRL2 specification depicted below, defining a rather quirky coffee machine. List some odd things about the behaviour of this coffee machine and give an alternative specification that fixes these.

An mCRL2 specification of a quirky coffee machine.

sort Val = struct c2 | c5 | c10; map w: Val -> Nat; eqn w(c2) = 2; w(c5) = 5; w(c10) = 10; act insert_coin, return_coin: Val; cancel, bad, good; proc Loading(t: Int) = sum v: Val . insert_coin(v) . Loading(t + w(v)) + (exists v: Val. t >= w(v)) -> cancel . Flushing(t) + (t >= 10) -> bad . Loading(t - 10) + (t >= 20) -> good . Loading(t - 20); Flushing(t: Int) = sum v: Val . sum t': Nat . (t == t' + w(v)) -> return_coin(v) . Flushing(t') + (forall v: Val . w(v) > t) -> Loading(t); init Loading(0);

The first-order µ-calculus

With the introduction of data-dependent behaviour and, in particular, with the sum operator, we have moved beyond labelled transition systems that are finitely branching. As you may have found out in this exercise, the logics defined in the previous sections are no longer adequate to reason about the systems we can now describe. This is due to the fact that our grammar does not permit us to construct infinite sized formulae. We mend this by introducing data in the µ-calculus. This is done gently: first, we extend Hennessy-Milner logic to deal with the infinite branching.

Consider the action formulae of Hennessy-Milner logic. It allows one to describe a set of actions. The actions in our LTSs are of a particular shape: they start with an action name a, taken from a finite domain of action names, and they carry parameters of a particular sort, which can possibly be an infinite sized sort. What we shall do is extend the Hennessy-Milner action formulae with the facilities to reason about the possible values these expressions can have. This is most naturally done using quantifiers.

Definition (Action formulae)

An action formula over a set of action names \(\act\) is an expression that adheres to the following syntax in pseudo-BNF, where \(a \in \act\), \(d\) is a data variable, \(b\) is a Boolean expression, \(e\) is a data expression and \(D\) is a data sort.

\[A,B ::= b ~|~ a(e) ~|~ \overline{A} ~|~ A \cup B ~|~ \exists d{:}D. A\]

The following abbreviations may also be used:

\[\begin{align*} A \cap B &= \overline{\overline{A} \cup \overline{B}} & \forall d{:}D. A &= \exists d{:}D. \overline{A} \end{align*}\]

Since our action formulae may now refer to data variables, the meaning of a formula necessarily depends on the value this variable has. The assignment of values to variables is recorded in a mapping \(\varepsilon\). An action formula \(A\) over \(\act\) is associated with a set \(\sem{A}{\varepsilon} \subseteq \{a(v) ~|~ a \in \act \}\) in the following manner.

\[\begin{split}\begin{align*} \sem{b}{\varepsilon} &= \{a(v) ~|~ a \in \act \wedge \varepsilon(b)\} \\ \sem{a(e)}{\varepsilon} &= \{ a(v) ~|~ v = \varepsilon(e) \} \\ \sem{\overline{A}}{\varepsilon} &= \{ a(v) ~|~ a \in \act \} \setminus \sem{A}{\varepsilon} \\ \sem{A \cup B}{\varepsilon} &= \sem{A}{\varepsilon} \cup \sem{B}{\varepsilon} \\ \sem{\exists d{:}D. A}{\varepsilon} &= \bigcup\limits_{v \in D} \sem{A}{\varepsilon[d := v]} \end{align*}\end{split}\]

Remark

Note that the function \(\varepsilon\) is used to assign concrete values to variables and extends easily to expressions. Consider, for instance, the Boolean expression \(b \wedge c\), where \(b\) and \(c\) are Boolean variables. Suppose that function \(\varepsilon\) states that \(\varepsilon(b) = \varepsilon(c) = \true\). Then \(\sem{b \wedge c}{\varepsilon} = \varepsilon(b \wedge c) = \varepsilon(b) \wedge \varepsilon(c) = \true \wedge \true = \true\).

The extension of our action formulae with data is sufficiently powerful to reason about the infinite branching introduced by the sum operator over infinite data sorts. However, it still does not permit us to reason about data-dependent behaviour. Consider, for instance, the LTS described by the following process:

A transition system with nested trees.

act num: Nat; proc P(n: Nat) = sum m: Nat . (m < n) -> num(m) . P(m); init sum m: Nat . P(m);

Each num(v) action leads to a state with branching degree \(v\), in which the only actions num(w) possible are those with w < v. Using Hennessy-Milner logic combined with our new action formulae fails to allow us to express that from the initial state, no action num(v) can be followed by an action num(v') for which v <= v'. We can mend this by also extending the grammar for Hennessy-Milner logic.

Definition (First-order HML)

A First-order Hennessy-Milner logic formula interpreted over an LTS with action labels \(\act\) adheres to the following syntax in pseudo-BNF, where \(A\) is an action formula over \(\act\), \(b\) is a Boolean expression, \(d\) is a data variable and \(D\) is a data sort.

\[\varphi,\chi ::= \mccan{A}\varphi ~|~ \exists d{:}D.~\varphi ~|~ \varphi \land \chi ~|~ \neg \varphi ~|~ b ~|~\]

The following common abbreviations are allowed:

\[\begin{split}\begin{align*} \exists d{:}D. \varphi&= \neg \forall d{:}D. \neg \varphi & \varphi \lor \chi &= \neg(\neg \varphi \land \neg \chi) \\ \mcall{`\a{A}`}\varphi &= \neg \mccan{`\a{A}`} \neg \varphi & \varphi \implies \chi &= \neg \varphi \lor \chi \end{align*}\end{split}\]

An HML formula \(\varphi\) is interpreted over an LTS \(T = \langle S, \act, \rightarrow, i, f \rangle\), and in the context of a data variable valuation function \(\varepsilon\). Its semantics is given as the set of states \(\sem{\varphi}_T^\varepsilon \subseteq S\) of the LTS in which the formula holds. It is defined as follows.

\[\begin{split}\begin{align*} \sem{b}_T^\varepsilon &= \{s \in S ~|~ \varepsilon(b) \}\\ \sem{\neg\varphi}_T^\varepsilon &= S \setminus \sem{\varphi}_T^\varepsilon \\ \sem{\varphi \land \chi}_T^\varepsilon &= \sem{\varphi}_T^\varepsilon \cap \sem{\chi}_T^\varepsilon \\ \sem{\exists d{:}D. \varphi}_T^\varepsilon &= \bigcup\limits_{v \in D} \sem{\varphi}_T^{\varepsilon[d := v]} \\ \sem{\mccan{A}\varphi}_T^\varepsilon &= \{ s \in S ~|~ \exists_{s'\in S, a \in \sem{A}{\varepsilon}}~ s \stackrel{a}{\longrightarrow} s' \land s' \in \sem{\varphi}_T^\varepsilon \} \end{align*}\end{split}\]

We say that \(T\) satisfies \(\varphi\), denoted \(T \models \varphi\), if and only if for all \(\varepsilon\), \(i \in \sem{\varphi}_T^\varepsilon\).

Example

The property that from the initial state the \(\a{num}(v)\) action cannot be followed by a \(\a{num}(v')\) action with \(v' \geq v\) can now be written in a number of ways, one of them being \(\forall_{v,v'\oftype\nat} \mcall{\a{num}(v)}\mcall{\a{num}(v')} v' < v\).

The regular first-order Hennessy-Milner logic extends the first-order Hennessy-Milner logic in the same way as regular Hennessy Milner logic extends Hennessy-Milner logic. This allows us, for instance, to express that along all paths of the LTS described by this transition system, the parameters of the action num are decreasing:

\[\mcall{\true^*} \forall_{v,v'\oftype\nat} \mcall{\a{num}(v) . \a{num}(v')} v' < v\]

In a similar vein, the µ-calculus can be extended with first-order constructs, allowing for parameterised recursion. This allows one to pass on data values and use these to record events that have been observed in the past.

Compositionality

We have seen that systems can be described by means of a labelled transition systems. In this section, we will take a closer look at how to describe labelled transition systems using the process algebra mCRL2. To this end, we need to extend our definition of a labelled transition system a bit by adding a final state.

Definition (LTS)

A labelled transition system (LTS) is a tuple \(\langle S, \act, \rightarrow, i, f \rangle\), where

• \(S\) is a set of states,
• \(\act\) is a set of action labels,
• \({\rightarrow} \subseteq S\times \act \times S\) is a transition relation,
• \(i\in S\) is the initial state and
• \(f\in S\) is the final state.

In graphical depictions of transition systems, final states will be marked by a double circle. In mCRL2, final states are marked by a \(\a{Terminate}\) action.

We will now discuss how transition systems can be built up from basic building blocks: the deadlock process, actions and operators on processes. Let us start with the two building blocks that themselves represent processes.

The deadlock process (delta)

is the process that cannot do anything. In particular, it cannot terminate.

The deadlock process.

init delta;

Actions

A single action is a process. More precisely, it represents the transition system that can perform that action and then terminate.

Actions are processes.

act coffee; init coffee;

All other processes in mCRL2 are created using these two basic building blocks. To arrive at more complicated processes, they are combined using operators that create new processes by applying a transformation to one or more given processes.

Operators

We continue by giving a short description of each of the most important operators on processes in mCRL2.

Sequential composition (.)

identifies the final state of its first argument with the initial state of its second argument. If the first argument does not have a final state (e.g., because the first argument is the deadlock process), then the sequential composition is equal to its first argument.

Sequential composition.

act coin, coffee; proc P = coin; Q = coffee; R = P . delta; init P . Q; init P . R . Q;

Alternative composition (+)

chooses an initial action from the initial actions of its arguments, and then continues to behave like the argument it chose its first action from. Note that the deadlock process is the neutral element for +; it has no initial actions, so P + delta can only choose to behave like P. Another useful property is that any process P is bisimilar to P + P.

Alternative composition.

act coin, coffee; proc P = coin; Q = coffee . Q; init P + Q;

Conditional choice (C -> P <> Q)}

behaves like P if the boolean expression C evaluates to true, and behaves like Q otherwise. It is allowed to write C -> P for C -> P <> delta.

Example

The process true -> coin <> coffee is bisimilar to coin, and false -> coin <> coffee is bisimilar to coffee.

Summation (sum v: T . P)}

is the (possibly infinite) alternative composition of all those processes P' that can be obtained by replacing v in P by a value of type T. Similar to alternative composition, if P is the same, regardless of the value of v, then sum v: T . P is bisimilar to P.

Summation.

proc P = sum b: Bool . coffee; Q = sum b: Bool . b -> good <> bad; R = sum b: Bool . !b -> good;

Rather than writing sum x: T1 . sum y: T1 . sum z: T2 . P, it is also allowed to write the shorter sum x, y: T1, z: T2 . P.

Parallel composition (P || Q)}

is the denotation for the combined state space (transition system) of independently running processes P and Q. It represents the process that can behave like P and Q simultaneously, and therefore we need a device to represent simultaneous execution of actions. This device is the multi-action operator; if a and b are actions, then a|b represents the simultaneous execution of a and b.

Parallel composition.

act a, b; proc M = a || b; init M;

If the state space of P counts \(n\) states, and that of Q counts \(m\) states, then the state space of P || Q will have \(n \cdot m\) states. It is important to realise this, because it implies that the parallel composition of \(N\) processes will yield a statespace of a size that is exponential in \(N\).

Communication (comm(C, P))

is an operator that performs a renaming of multi-actions in which every action has identical parameters. The set C specifies multi-action names that should be renamed using the following syntax for each renaming: a1|...|aN->b, where b and a1 through aN are action names. The operation is best understood by looking at the example about the communication operator.

Communication operator.

act a, b, c: Nat; proc P = a(1) || b(1); init comm({a|b->c}, P); act a, b, c: Nat; proc Q = a(1) || b(2); init comm({a|b->c}, Q);

Rename (rename(R, P))}

works exactly like the communication operator, except that only single action names can be renamed (not multi-actions).

Allow (allow(A, P))

removes all multi-actions from the transition system that do not occur in A. Any states that have become unreachable will also be removed by mCRL2, as the resulting system is smaller and bisimilar.

Allow operator.

act a, b, c; proc P = a || b; init allow({a,c}, comm({a|b->c}, P)); init allow({b|a,a}, P);

Note that the multi-action operator is commutative, so the order in which the actions appear does not matter.

Hide (hide(H, P))

performs a renaming of the actions in the set H to the special action name tau (which takes no parameters). This special action represents an event that is invisible to an outside observer, and therefore has some special properties. For instance, allow(A, tau) is always equal to tau per definition, regardless of the contents of A (so effectively, internal cannot be blocked).

Hiding can be used to abstract away from events and gives rise to coarser notions of behavioural equality, such as branching bisimulation. Such equalities again compare systems based on what can be observed of a system, but this time taking into account that the tau action cannot be observed directly.

Communicating systems

To conclude, we give an example of how the operators from the previous section can be used to model interacting processes.

Communication is modelled by assigning a special meaning to actions that occur simultaneously. To say that an action \(\a{a}\) communicates with an action \(\a{b}\) is to say that \(\a{a}|\a{b}\) may occur, but \(\a{a}\) and \(\a{b}\) cannot occur separately. Usually this scenario will correspond to \(\a{a}\) and \(\a{b}\) being a send/receive action pair. This standard way of communicating is synchronous, i.e. a sender may be prevented from sending because there is no receiver to receive the communication.

Going back to the coffee machine, we now show how we can model a user that is interacting with such a machine. The user is rather stingy, and is not prepared to pay two coins. In mCRL2, it looks like this:

act coin, good, bad,
    pay, yay, boo;
proc M = coin . (coin . good + bad) . M;
proc U = coin . (good + bad) . U;
init allow({pay, yay, boo},
       comm({coin|coin->pay,good|good->yay,bad|bad->boo},
         U || M
     ));

The corresponding statespaces are shown below. On the right hand side, the statespace of the parallel composition is shown. The two black transitions are all that remain when communication and blocking are applied.

It is obvious from this picture that you get what you pay for: good coffee is not achievable for this user. The picture also illustrates that synchronicity of systems helps reduce the state space tremendously. If the actions of the machine and the user had been completely independent, then the resulting state space would have had 3 times more states, and 15 times more transitions.

Note

In our example, we only have two communicating parties, so that no confusion can arise as to who was supposed to be communicating with who. For larger systems, more elaborate naming schemes for actions are often used in order to avoid mistakes, so for instance the sender of a message will perform a s_msg action, and the receiver a r_msg action, rather than both using an action called msg.

It should also be noted that for instance \(n\)-way communication is also possible, which can be useful to model, e.g., barrier synchronisation or clock ticks.

Footnotes

[1]	If a formula is not in positive normal form, then its least and greatest fixpoint are not guaranteed to exist, hence the requirement.