## 3  Pattern matching in join definitions

### 3.1  Principle

Consider the following join definition:
```let A(n) | B() = P(n)
and A(n) | C() = Q(n)
;;
```
This defines three names A, B and C. Name A has arity one, whereas names B and C have arity zero. Names may be synchronous or asynchronous, depending on whether there are replyto… constructs applying to them inside the guarded processes P(n) and Q(n) or not.

According to the general join-calculus semantics, the guarded process P(n) may be fired whenever there are some messages pending on A and B. Similarly, Q(n) may be fired whenever there are some messages pending on A and C. In both cases, the formal parameter n is replaced by (or bound to in the implementation) one of the messages pending on A.

Reactivity information is to be considered at the definition level, since matching is indeed performed at this level. Moreover, in order to use finite-state automata, we want this information to range on a finite set of possible values. As far as matching is concerned and by the linearity of patterns, only the presence or absence of messages matters. Thus, let us call 0 the status of a name without any message pending, and N the status of a name with at least one message pending. Then, the status of a definition is a tuple consisting of the statuses of the names it defines, once some arbitrary order of these names has been adopted.

For instance, if some messages are pending on B and C, whereas none is pending on A, then the status of the A, B, C definition is a three-tuple written 0NN.

A matching status is defined as a status that holds enough N, so that at least one guarded process can be fired.

Definition status evolves towards matching status as messages arrive. This yields a first kind of “increasing” transitions. More specifically, when a message arrives on some name, then this name status either evolves from 0 to N or remains N. Definition status evolves accordingly. In the A, B, C case we get the following transitions. (In this diagram, transitions are labeled by the name that gets a new message and matching statuses are filled in gray.)

Definition status also evolves when matching occurs. This yields new, “decreasing”, transitions that we call matching transitions. According to the join-calculus semantics, matching may occur at any moment (provided of course that matching is possible). As a consequence, matching transitions start from matching states and they are unlabeled. In the A, B, C case, they are as follows:

Observe that there may be several matching transitions starting from a given status. This is not always a consequence of the non-deterministic semantics of the join-calculus.

Often, ambiguity is only apparent. For instance, matching transitions starting from NN0 lead to NN0, N00, 0N0 and 000. When such a matching occurs, two messages are consumed (one pending on A and one pending on B) then, depending on whether there are some messages left pending on A and B or not, status evolves to NN0, N00, 0N0 or 000. From the implementation point of view, this means that a little runtime testing is required once matching has been performed. Here, we pay a price for using finite-state automata.

However, some true non-determinism is still present. Consider status NNN for instance. Then, both guarded processes of the A, B, C definition can now be fired. The choice of firing either P(n) or Q(n) will result in either consuming one message pending on A and one on B, or consuming one message pending on A and one on C.

Finally, a view of join-matching compilation can be given by taking together both kinds of transitions. This yields a non-deterministic automaton.

Note that matching of non-linear patterns can also be compiled using automata. For instance, if a name appears at most twice in one or more pattern, then it status will range on 0, 1 and N. We do not present this extension in greater detail, for simplicity, and because we do not implement non-linear patterns.

### 3.2  Towards deterministic automata

For efficiency and simplicity reasons we choose to implement matching using deterministic automata that react to message reception.

Fortunately, it is quite possible to do so. It suffices to perform matching as soon as possible. More precisely, when a message arrives and carries definition status into matching status, matching is performed immediately, while definition status is adjusted to reflect message consumption. This results in pruning the status space just below matching statuses.

In practise, in the A, B, C case, we get the automaton of figure 1. Observe that all transitions are now labeled and that a name labels a transition when message reception on this name triggers that transition. Furthermore, matching transitions that correspond to firing P(n) or firing Q(n) are now represented differently (the former by a dotted arrow, the latter by a dashed arrow). This highlights the difference between false and true non-deterministic transitions: real non-determinism is present when there are both dotted and dashed edges with the same label starting from the same node.

For instance, there are two B-labeled dotted transitions starting from N00. Non-determinism is only apparent here, since P(n) is fired in both cases and that the selected transition depends only on whether there is at least one message left pending on A or not after firing P(n).

By contrast, from status 0NN, the automaton may react to the arrival of a message on A in two truly different manners, by firing either P(n) or Q(n). This is clearly shown in figure 1 by the A-labeled edges that start from status 0NN, some of them being dashed and the others being dotted. A simple way to avoid such a non-deterministic choice at run-time is to perform it at compile-time. That is, here, we suppress either dotted or dashed A-labeled transitions that start from 0NN.

In the rest of the paper, we take automata such as the one of figure 1 as suitable abstractions of join-pattern compilation output.

### 3.3  Automata and semantics

Both the “match as soon as possible” behavior and the deletion of some matching transitions have a price in terms of semantics. More precisely, some CHAM behaviors now just cannot be observed anymore. However, the CHAM semantics is a non-deterministic one: an initial configuration of the CHAM may evolve into a variety of configurations. Furthermore, there is no fairness constraint of any kind and no particular event is required to occur.

As an example of the consequence of the “match as soon as possible” behavior, consider this definition:
```let A() = print(1);
and B() = print(2);
and A() | B() = print(3);
;;
```
Then, we get the following automaton:
Status NN that is preceded by the two matching statuses 0N and N0 cannot be reached. As a consequence, the above program will never print a 3, no matter how many messages are sent on A and B.

Next, to illustrate the effect of deleting ambiguous matching transitions, consider the following definition:
```let A() = print(1);
and A() | B() = print(2);
```
Such a definition will get compiled into one of the following deterministic automata:
In the case of the left automaton, only 1 will ever get printed. In the case of the right automaton, 2 will be printed when some messages arrives on B and then on A. Both automata lead to correct implementations of the semantics. However the second automata looks a better choice than the first one, since it yields more program behaviors.