See the Chymyst project for a new implementation of join calculus in Scala.

An experimental implementation of join calculus in Objective-C.

Join calculus is a formal model for (mostly) purely functional, concurrent computations. Join calculus is somewhat similar to the actor model but, in some sense, is "more declarative".

There are a few implementations of join calculus in functional programming languages such as OCaml ("JoCaml"), F# ("joinads"), and Scala ("scalajoins").

For a tutorial introduction to join calculus and several examples using JoCaml, see https://sites.google.com/site/winitzki/tutorial-on-join-calculus-and-its-implementation-in-ocaml-jocaml

This project contains the join calculus library and an example iOS application, DinPhil5, that simulates five "dining philosophers" taking turns thinking and eating. The asynchronous logic of this iOS application is implemented as a declarative, purely functional program in join calculus.

• Version 0.3.0

The stop/resume functionality is now implemented.

The "Dining Philosophers" simulation can be restarted; this takes time since we have to wait until all reactions finish and the soup becomes quiet.

• Version 0.2.1

CocoaJoin is now compiled as a static library.

Four tests pass, including a test that does not use macros.

The library will throw a fatal exception at runtime if the user injects a molecule that has not been defined as input molecule for a reaction.

Added a test to make sure that background reactions have been scheduled even if the thread is blocked immediately after injecting some slow molecules.

• Version 0.2

The implementation of macros was changed to allow less verbose syntax: molecules are now injected by a(3).

Avoid unnecessary copying of molecule objects on injection.

Produce compile errors if molecules are declared but not used as input in any reactions.

Define more molecule types (empty, id, int, float).

More documentation about join calculus and more examples.

• Version 0.1

The operational semantics of join calculus is fully implemented.

Molecules are injected with Objective-C syntax, [a put:3] instead of a(3).

"Dining philosophers" is implemented as an iPhone app.

Join calculus realizes asynchronous computations through an "abstract chemical machine". The chemical machine performs computations by simulating "chemical reactions" between "molecules". Molecules are objects labeled by a name (a, b, c, incr, counter and so on). Each molecule carries a value on itself (an integer, a string, an object, etc.). In join calculus, this is denoted by a(123), b("yes"), etc. Molecules can, of course, carry a tuple of several values, which will be denoted by a(1,2) etc.

The programmer defines the names of allowed molecules and the type of value carried by each molecule (say, a carries integer, b carries string, etc.). The programmer also defines all the reactions that can happen to these molecules. Each reaction consumes one or more input molecules, then performs some computation using the values carried by the input molecules, and finally can produce some output molecules with some new values. The input molecules disappear from the "chemical soup" while the reaction is running, and at the end the new output molecules are injected into the "soup".

Reactions start asynchronously and concurrently, whenever the required input molecules become available. One can imagine that the "soup" is constantly being "stirred", so that molecules move around randomly and eventually meet other molecules to start reactions with them.

For now, we will write reactions using an easy-to-understand pseudocode with keywords such as consume, inject, etc. These keywords and the syntax of the pseudocode were chosen only for clarity; they do not exactly correspond to the syntax of any existing implementation of join calculus.

For example, suppose we define a single reaction like this,

consume a(x) & b(y) => print x; print y; inject b(x y)

and let's suppose that no other reactions can consume a or b. After defining this reaction, let us inject 5 copies of the molecule a and 3 copies of the molecule b, each with some random values, for example:

inject a(10), a(2), a(4), a(21), a(156);
inject b(1), b(1), b(1);

Now the "chemical soup" contains the following molecules:

a(10), a(2), a(4), a(21), a(156), b(1), b(1), b(1)

At this point, the chemical machine could start up to three concurrent reactions between some (randomly chosen) pairs of a and b. Reactions are scheduled to run in random order, using some (unspecified number of) concurrent threads. The order in which molecules were injected does not directly affect the order in which reactions are started, and does not determine which of the molecules will be consumed first. All these choices are up to the implementation of the "chemical machine" runtime. So, for example, a possible behavior of the "chemical machine" is that two concurrent reactions are started, the machine prints

21 1
2 1

and the soup then contains the molecules

a(10), b(3), a(4), b(22), a(156), b(1)

The chemical machine will not stop here, because some more reactions between a and b are possible. Reactions will continue to run concurrently in random order, and their input molecules are also chosen randomly. In the present example, three more reactions are possible between some a and b molecules. A possible further behavior is that the machine prints

156 3
10 22
4 1

and the soup now contains

b(32), b(5), b(159)

At this point, no more reactions are possible, because there are only b molecules, but a reaction requires a pair of a and b. So the "chemical machine" will wait. If more a molecules are injected, further reactions will be scheduled to run.

By default all reactions start asynchronously (they are scheduled on a background thread). For this reason, injecting a molecule a does not immediately start a reaction even if some molecules b are already present in the soup.

Once the required input molecules are available, a reaction will be scheduled to start. If several different reactions become possible with the present molecules, one of the reactions will be chosen at random and scheduled. Since the input molecules for that reaction are consumed, other competing reactions may get delayed by this.

If several copies of the input molecules are available, the reaction will consume randomly chosen copies of the input molecules. Injecting molecules is not similar to sending messages to a "mailbox" or a "channel" because reactions do not inspect the molecules in the order they were injected. The chemical machine will always see all molecules present at a given time, regardless of the order of injection. A reaction expecting input molecules a and b can start whenever some copies of a and b are present. Reactions will not preferentially consume the molecules that were injected earliest. Also, several reactions may start simultaneously on different copies of the molecules.

So it is the responsibility of the programmer to design the "chemistry" such that the desired values are computed in the right order, and to organize certain computations sequentially or concurrently as required. The programmer is free to define any number of molecules and reactions.

When a reaction is finished, it may or may not inject any output molecules into the soup. If a reaction does not inject any output molecules, the input molecules will be consumed and will disappear from the soup. However, a reaction must consume at least one input molecule.

The reaction's body is written as a function that takes each input molecule's value as an argument. The reaction body can then compute some values and inject new molecules carrying these values.

For instance, consider the reaction

consume a(x) & b(y) => 
	int r = compute_whatever(x,y);
	inject c(r), a(x), a(y), a(22); // whatever

This reaction consumes two input molecules a and b. These molecules carry values that are denoted by the pattern variables x and y in the consume pattern. When the reaction starts, its body takes x and y as arguments and computes something, then injects some new molecules back into the soup. The values carried by the output molecules are functions of the values carried by the input molecules.

Also note that, in this example, the reaction injects a molecule c, - a molecule that was not consumed by this reaction. This is freely permitted, as long as the molecule name c is defined within the lexical scope of this code. For instance, c could be an input molecule defined by another reaction jointly defined with this one; also, c could be a function parameter passed to the local scope. Molecule names such as c are ordinary (and immutable) values in the program.

The names of the input molecules of a reaction must be all different, and the argument names also must be all different. Thus it is not allowed to define reactions such as

consume a(x) & a(y) & a(z) => ... (wrong)

or

consume a(x) & b(x) & c(x) => ... (wrong)

This limitation ("reactions must be linear in the input") is not really restricting the computational power of join calculus.

When defining a reaction, we at the same time define the input molecules. So if one reaction uses the new input molecules defined by another reaction, these reactions need to be defined together. A set of reactions defined together is called a join definition. More precisely, a join definition defines at the same time the names of new input molecules and all the reactions involving these input molecules. The names of the input molecules are treated as new local values, shadowing any previously visible definition of these names.

In join calculus, reactions involving (some combinations of) the same input molecules always have to be defined together in a single join definition. For example, consider the following two reactions,

consume a(x) & b(y) => ...
consume b(y) & c(z) => ...

Both reactions have the molecule b as an input molecule, so we need to define these reactions together in a single join definition. This join definition will also declare the new molecule names a, b, c.

consume a(x) & b(y) => inject b(x y)
consume c(x) & d(y) => inject a(x-y)

These two reactions do not have a common input molecule. For this reason, we do not necessarily have to define them together in a single join definition. However, note that the first reaction defines the name a while the second reaction uses this name to inject a molecule a(x-y). It is important to realize that a molecule can be injected only if its name is visible in the current lexical scope. For this reason, the second reaction needs to be defined after the first one, and the definition of the second reaction must be made within a scope where the name a is still visible.

Defining a single reaction like this, which defines a and b but uses the value c,

consume a(x) & b(y) => inject c(x y 1)

is valid only in a scope where c is a visible local value of the correct type (the type of molecule name with value of integer type). This is quite similar to the fact that an ordinary function

function (x,y) { return c(x y); }

is valid only in a scope where c is already defined.

Once a join definition is made and some new molecules have been defined, the "chemistry" of the new molecules is set in stone - no further reactions can be added to the same input molecules. In other words, reactions and molecule names are defined statically.

In join calculus, the molecule names (a, b, etc.) are syntactically functions and play the role of "injectors" for molecules. By writing inject a(2), the programmer performs an injection of the molecule a with value 2 into the soup. The operator inject a(2) does not return any value (or, formally, you can say it returns an empty value). So, by itself, the molecule a(2) is not a value accessible to the programmer. Rather, the syntax a(2) represents injecting a "fully constructed" molecule into the chemical soup.

On the other hand, the molecule name a is a local (and immutable) value in the program. A molecule name can be used as an argument to a function, stored in an array, and so on.

Footnote: In the Objective-C implementation, the keyword inject is not used, and the syntax a(2) directly injects the molecule a with value 2. An object representing a "fully constructed molecule" is not directly available to the programmer, which is as it should be in join calculus.

The operation of injecting a "slow" molecule looks like a function call that returns no value. Injection is performed right away and does not block the execution thread. However, reactions do not necessarily start at the same time since this is a "slow" (i.e. asynchronous) molecule.

There is a second type of molecules that are "synchronous" or "fast". A fast molecule has some special features compared with "slow" molecules:

• when injected into the soup, a fast molecule will force some reaction to start right away (or as soon as possible)
• injecting a fast molecule is a blocking operation that remains blocked until some reaction can start involving the newly injected fast molecule
• a fast molecule can return a value to the injecting thread

So, the operation of injecting a fast molecule into the soup looks like an ordinary function call. It blocks the execution thread until some computation is finished and a value is returned to the caller. If no reactions are ever available for the fast molecule molecule, the injecting thread will be blocked indefinitely.

From the point of view of a reaction, a fast molecule is consumed as an input molecule, just like any other input molecule. A special operator reply is used on the "fast" molecule, in order to deliver a return value to the injecting call. The operator reply looks like this,

reply x to m

This assigns the return value x to the "fast" molecule m. The reply operator cannot be used on a "slow" molecule. (So, if we see a reply operator used with a molecule name, we know this is a fast molecule.)

Once the reply operator is called with a value, the injecting call unblocks and the value is returned. Thus, the call m() will return the value x, like an ordinary, blocking function call. The reaction, meanwhile, continues (perhaps on a different thread) and may inject other molecules into the soup or "reply" to other fast molecules - or do whatever else. The fast molecule m does not remain in the soup, as if the caller has pulled the fast molecule out when the reply operation returned a value to the caller.

Here is an example. Suppose we have some slow molecule enabled that carries a boolean value, so that we can have either enabled(true) or enabled(false) in the soup. We would like to find out what is the current value on the enabled molecule, and we would like to have this information synchronously (to know what is the status right now). For this, we define a reaction,

consume enabled(s) & getStatus() => reply s to getStatus; inject enabled(s)

After this, we can write code like

if (getStatus()) then ....

In other words, injecting getStatus() looks like an ordinary function call. By design, there is always at most one instance of enabled in the soup, so the chemical machine will run at most one instance of this reaction. Each time we call getStatus(), the reaction will run; the calling thread will wait for completion of that reaction, and the current value of s will be returned to the caller.

Within the scope of the reaction, however, the reply operation is a side effect that does not produce any value. Except for this side effect, the body of the reaction is a pure function, and values on the molecules are immutable. To change the "enabled status" in this example, the programmer can write a reaction that consumes the enabled molecule and then injects another instance of the enabled molecule with a different boolean value.

In this way, the programmer is automatically protected from race conditions and deadlocks, while being able freely to call getStatus() from several different threads (i.e. from different reactions) concurrently. The molecule enabled(...) is consumed by the reaction and disappears from the soup until the reaction is finished, because this is the operational semantics of join calculus: All input molecules disappear when a reaction starts running. For this reason, it is not possible that some other reaction consumes enabled and injects a different enabled while a reply to getStatus is still being sent.

It is important to keep in mind that molecule names are local values in the program. These values are declared whenever a new reaction is defined that uses these names as input molecules. The definition of the new reaction is local in the current scope.

For example, we may write (pseudocode)

// define reaction
consume a(x) & b(y) => ...
// now "a" is defined as a local variable
inject a(2);
let f = a; // now "f" is the same as "a"
call_some_function(f, b); // same as call_some_function(a,b)
inject f(3); // same as "inject a(3)"

It may become confusing if we write code like this, because it is easy to forget which reaction has been defined with the molecule name a if we alias it to another local variable, f. Nevertheless, technically this is valid code.

The local scoping property has important uses for programming in join calculus.

The first major use case is encapsulation. Suppose we defined certain molecules and useful reactions within a function (which has its local scope). The correct functionality of this "fragment of chemistry" will often depend on having a certain number of molecules injected but no other copies of these molecules, and a certain number of reactions defined but no other reactions. The functionality can be broken if just one more molecule were erroneously injected.

Now, the new molecule names and the reactions defined for these molecules are visible within the local scope but not visible outside that scope. The outside scope cannot break the functionality by modifying these reactions or directly injecting the molecules defined inside.

If some of these new molecules are needed outside the local scope, their names must be returned to the outer scope, say as return values of the function that defines the reactions. The outer scope will then be able only to inject these new molecules into the soup. It will remain impossible to define any new reactions for these molecules, or to inject molecules whose names were not exposed. In this way, we guarantee that the functionality can be used safely within any outer scope. There will be examples of this encapsulation below.

The second major use case of locally scoped molecules is creating a dynamic structure of reactions. This is a more advanced technique where a recursive function defines new reactions using some new and some old molecules. The new molecules are passed again to a recursive call of the same function, defining yet other new reactions, and so on. In this way, one can create a recursive structure of reactions, such as a binary tree or a linked list. This kind of structure may be used for computations where the number of concurrent threads is not known in advance; this can be useful for, say, concurrent binary search or concurrent sorting. Join calculus is able to create at run time a dynamically computed number of interlinked reactions, while keeping the requirement that all reactions and molecule names must be defined statically.

Here is how one can implement an "asynchronous counter".

Define molecule inc with empty value and counter with integer value. Define a fast molecule get with empty value, returning int. Define two reactions:

consume inc() & counter(n) => inject counter(n 1)
consume get() & counter(n) => inject counter(n), reply n to get

Initially, we inject counter(0). Then, at any time inject inc() to increment the counter and get() to obtain the current value.

This pair of reactions works as follows. Whenever a molecule inc() is injected, the counter molecule is consumed and then injected into the soup with a new value. Whenever the get molecule is injected, the current value of n is returned.

For example,

inc(); inc(); 
usleep(200000); // wait until counter is asynchronously incremented 
int x = get();

will assign 2 to x, as long as we wait long enough for the reactions to start.

The operational semantics of join calculus guarantees that the molecule counter disappears from the soup whenever each reaction starts, and appears only after incrementing the value. For this reason, it is possible to inject many copies of inc() simultaneously, and there is no problem with concurrent updates of the counter value. (Of course, this depends on the commutativity of addition: it does not matter in which order the reactions are started.) Each reaction consumes counter and injects it back, ready to be consumed by another reaction. Eventually, all inc() molecules will be consumed, one by one.

Nevertheless, this implementation is "brittle" because the user could forget to inject the counter(0) molecule at the beginning of the program, and then no reactions will ever run, and the call to get will block forever. The user could also, by mistake, inject several copies of counter into the soup, and then the results of get will be unpredictable.

In order to fix this problem, we can define the reactions within a local scope. Pseudocode:

define_counter = function(initial_value) {
	// define reactions
		consume inc() & counter(n) => inject counter(n 1)
		consume get() & counter(n) => inject counter(n), reply n to get
		
	inject counter(initial_value)
	return (inc, get)
}
// outer scope

(inc, get) = define_counter(0)
// counter(0) has been already injected, can use it now
inject inc()
inject inc()
usleep(200000) // let the threads churn for 0.2 seconds, should be enough.
int x = get(); // most probably, this returns 2

The function define_counter returns a pair of two molecule names, inc and get, defined within the local scope but now usable outside. The outer scope can then inject inc() to increment the counter asynchronously, or call get() to obtain the current value synchronously.

The molecule counter is also defined within the local scope of define_counter, but the name counter is not returned to the outer scope. So the outer scope cannot inject counter. This prevents the user from injecting counter incorrectly or making any other mistakes using this functionality.

The function define_counter defines two reactions, which constitute the "join definition" that determines the "chemistry" of the input molecules inc, get, and counter. The "chemistry" is defined statically, which means that:

• after this definition, the reactions cannot be modified

• the user cannot define any new reactions that consume inc, get, or counter as input molecules

The user can certainly define new reactions that consume other molecules and inject inc or get. However, the user cannot define a new reaction that consumes inc or get.

What happens if the user tries to define a new reaction that consumes inc, say

consume inc() => print "gotcha"

is that a new local variable with name inc is defined, representing a new molecule. This new molecule inc belongs to a new join definition and cannot react with the old counter molecule. This is so because a new join definition always defines not only the reactions, but also the input molecule names as new values in the local scope.

Trying to define a new reaction that consumes a previously defined molecule is similar to writing this code:

int x = 2;

{ 
	int x; // x is a new variable now, not equal to 2
		// the old x=2 is shadowed here.
}

Due to this feature, local reactions are encapsulated and can be safely used from an outer scope.

Suppose we need to run, say, 100 concurrent computations and wait until they are all done, then call a certain function, all_done().

For each of the computations, we define a reaction with input molecule begin(f) where f is a closure that needs to be evaluated. The reaction will compute f() and produce a molecule done(). In order to initiate the computation, we will just have to inject 100 begin molecules, specifying the required computations as values carried by these molecules.

To simplify our example, we assume that f does not take an argument and that it is not necessary to collect any results of the computations. (If this is not the case, the results will have to be put onto the done molecule; this modification is straightforward.) The reaction for begin looks like this:

	consume begin(f) => f(); inject done();

Now, it remains to wait until all reactions are finished. How can we do this? We know how many begin molecules we injected, and we need to make sure that exactly as many done() molecules have been produced.

Join calculus does not allow us to query the global state of the chemical soup, to ask how many molecules of type done are available, or to start a reaction when exactly 100 are available. There is always only one way to do any such bookkeeping: through new reactions.

So we need to introduce a new molecule, say remain(x), that counts how many computations remain. Here is a simple way of doing this:

	consume remain(x) & done() => inject remain(x-1)

Now, if we inject a single instance of remain(100) at the beginning, we can be sure that at most one copy of remain is available in the soup at any time. The reaction will consume the done() molecules one by one, i.e. sequentially, without any possibility for a race condition or deadlock. (We get this functionality from join calculus "for free".)

The entire code now becomes two coupled reactions:

	consume begin(f) => f(); inject done()
	consume remain(x) & done() => if x==1 then all_done() else inject remain(x-1)
	inject remain(100)
	inject begin(...), begin(...), ...

This code will work, but there are some minor problems with it:

• we would like to avoid injecting the begin molecules by hand
• the code is "brittle": if the programmer mistakenly forgets to inject remain(100), or injects remain(x) with another value of x, or later injects several more copies of remain or done(), the reactions will not work as desired! The closure all_done could be called too early, or called several times, or not called at all.

This useful functionality can be protected from change and at the same time encapsulated by a function that receives as arguments, say, a collection of closures to be evaluated and an all_done closure. The function will hide the molecule and reaction definitions within its scope.

run_all_and_report(array, all_done) =
	{
		consume begin(f) => f(); inject done()
		consume remain(x) & done() => if x==1 then all_done() else inject remain(x-1)
		inject remain(array.length)
		inject begin(f) for each f in array
	}

It is important to note that the molecule remain is invisible outside this function. So the user will not be able to inject remain with an incorrect value or too many times.

In fact, the user of this function will not be able to inject remain, or begin, or done at all. The only scope where these molecule names are visible is the local scope of the function run_all_and_report.

"Map": We need to schedule n computational tasks compute_something(x) concurrently on each element x of a collection C. "Reduce": as soon as each task is finished, we need to collect the intermediate results and merge them repeatedly together with the function reduce(a,b) in order to compute the final value.

We assume that the reducer is associative:

reduce(a,reduce(b,c)) = reduce(reduce(a,b),c)

Thus, we are allowed to reduce the intermediate results in any order and even concurrently, as long as no intermediate values are lost.

We design the "chemistry" as follows:

• each task is initiated by a molecule begin(x) by itself

• when the computation is finished, a molecule done(result) is injected, carrying the result value of the computation

    consume begin(x) =>  inject done(result), where result = compute_something(x) 
  

• ideally we would like to define the "reduce" reaction like this:

    consume done(x) & done(y) =>  (wrong!)
    	inject done(z) where z = reduce(x,y)
  

If this were possible, we would achieve the result that all reducing operations start concurrently. However, we are not allowed to define reactions that consume two copies of the same molecule. We need to use a different molecule instead of done(y), so we change this reaction to

	consume done(x) & done'(y) => inject done(z) where z = reduce(x,y)

To convert done into done', we use another reaction with a primer molecule:

	consume done(x) & primer() => inject done'(x)

Now we just need to make sure that there are enough primer molecules in the soup, so that all intermediate results get reduced. Here is how we can reason about this situation. If we have n tasks, we need to call the reducer n-1 times in total. The reducer is called once per a "primed" molecule done'. Therefore, we need to create n-1 primed molecules, which is possible only if we have n-1 copies of primer() in the soup. If we inject n-1 copies of primer() into the soup at the beginning, the result at the end will be a single done(z) molecule, regardless of the order of intermediate reactions.

• finally, we need to signal that all jobs are finished. A single done(z) molecule will carry our result z, but it will stay in the soup indefinitely and will not start any reactions by itself. In join calculus, we cannot define a reaction with a "guard condition", such as

    consume done(x) if (x > 100) => ... (wrong)
  

Reactions start whenever input molecules are present, regardless of the values on the molecules. Guard conditions are not part of the join calculus.

Therefore, we need to be able to detect, before injecting the last done molecule, that this molecule is going to be the last one. The only way to know that is if the done molecule carries on itself the number of already performed reductions. Thus, we let the done molecule carry a pair: done(x,k) where k shows how many reductions were already performed. When a single task is done, we inject done(x,1) into the soup:

	consume begin(x) => inject done (z, 1) where z = compute_something(x)

When we reduce two values to one, we add their k values:

	consume done(x,k) & done'(y,l) => inject done( reduce(x,y), k l )

Just one more refinement of this: when k l becomes equal to n, we need to stop and signal completion. For instance, like this:

	consume done(x,k) & done'(y,l) => 
		if m==n then inject all_done(z) 
		else inject done(z,m) 
		where
			z=reduce(x,y)
			m=k l

This completes the implementation of map/reduce with fully concurrent computations. The full pseudocode looks like this (assuming integer values):

function map_reduce(initial_array, compute_something, reduce, all_done) {
	let n = length of initial_array
	
	define molecules begin(integer), done(integer, integer), done'(integer),
		primer(void);
	consume begin(x) => inject done(z, 1) where z = compute_something(x);
	consume done(x,k) & primer() => inject done'(x,k);
	consume done(x,k) & done'(y,l) => 
		if k l==n then inject all_done(z) 
		else inject done(z,k l) 
		where z=reduce(x,y);
	inject begin(x) for x in initial_array;
	inject (n-1) copies of primer();
}

This function receives a previously defined molecule name, all_done, to signal asynchronously that the job is complete and to deliver the final result value. All reactions and newly defined molecules remain hidden in the local scope of the function.

The function map_reduce can be seen as part of a "standard chemical library" of predefined molecules and reactions that can be reused by programmers.

With a slightly different set of "chemical laws", it is possible to signal completion synchronously, or to limit the number of concurrently running tasks, or to provide only a fixed number of concurrent reducers.

In this way, the programmer can organize the concurrent computations in any desired manner.

In an interactive application, we might have a button that starts an asynchronous computation. This can be implemented in join calculus by making the button inject a slow molecule that starts an asynchronous reaction. Now, suppose we would like to "enable" or "disable" this response: when "disabled", the molecule should not start the computation.

Here is how this functionality can be implemented in a "chemical library".

Reactions are defined statically, so there is no way to modify a reaction so that, say, a new input molecule is required. All input molecules for the reaction have been already defined; now we would like to control whether this reaction starts. The only way of doing this is to control whether the required input molecule has been injected. Instead of enabling/disabling a reaction, we will enable/disable the injection of a molecule.

Given a molecule name m, we define new molecule names m_on, m_off, m_state, request_m and the reactions,

	consume m_on() & m_state(_) => inject m_state(true)
	consume m_off() & m_state(_) => inject m_state(false)
	consume request_m(x) & m_state(is_on) =>
		inject ( if is_on then m(x) else () );
		inject m_state(is_on)

Injecting m_on or m_off switches the molecule state. Te user is supposed to inject request_m(x) instead of directly injecting m(x). This will result in injecting m(x) only if the molecule m is in the state "on". Otherwise the request to inject m is ignored.

Other variations of this technique could put more values on the molecule request_m. For example, this molecule could carry not just the value x for the molecule m, but also some closures to be evaluated when the request was granted or refused. The molecule request_m could even carry the molecule name m as value and inject that molecule:

	consume request(m,x) & m_state(is_on) =>
		inject ( if is_on then m(x) else () );
		inject m_state(is_on)

Now, we can create these reactions in a local scope and return just the molecule names m_on, m_off, and request.

make_switch(initial) = 
	consume m_on() & m_state(_) => inject m_state(true); reply () to m_on
	consume m_off() & m_state(_) => inject m_state(false); reply () to m_on
	consume request(m,x) & m_state(is_on) =>
		inject ( if is_on then m(x) else () );
		inject m_state(is_on)
	inject m_state(initial) // initial value of the switch
	(m_on, m_off, request) // return these names, but do not return m_state

The user of this function is then free to pass any molecule name as the value on request. This molecule will be either injected or not, according to the current state of the switch.

To prevent undetermined behavior in case several m_on and m_off molecules are injected at once, e.g.

m_on(); m_off(); request(...)

we have designated m_on and m_off as fast molecules.

Note that the current implementation allows us to use several instances of request at the same time. The checking of the switch state is automatically non-concurrent because there is only one copy of m_state, so only one reaction request & m_state => ... can run at any one time.

When a computation takes a long time, we may need to cancel it in the middle. Since it is impossible to stop a running thread from outside, what we need to do is to split the computation into several steps and check, after every step, whether we need to proceed to the next step or the computation has been cancelled. We may also need to notify an outside scope that the computation has been aborted after a certain step, and send the partial results to the outside scope.

Here is how we can implement this functionality in a "chemical library".

Splitting a long computation into steps needs to be performed by the programmer. Once this is done, we can imagine having the pseudocode

y = step_1(x)
z = step_2(y)
t = step_3(z)
...

The desired functionality should now allow programmer to rewrite this into

y = step_1(x);
continue_next( () => 
	z = step_2(y);
	continue_next( () =>
		t = step_3(z);
		...
	)		
)

and achieve a computation that will not proceed to the next step once a cancellation molecule has been injected.

It seems that continue_next should not be a slow molecule; otherwise we will be inserting unnecessary thread switching into a computation. Let us guard the call to continue_next by using the make_switch function defined in the previous example:

(m_on, m_off, request) = make_switch(true);
continue_next f = inject request(f, ())

Any calls to continue_next will result in first injecting request(f, ()). This molecule will do nothing if the state of the switch is "off". Otherwise the closure f will be evaluated on its argument (). To cancel the computation at any time, it is sufficient to inject the (fast) molecule m_off.

Join calculus is a (mostly) purely functional, declarative model of concurrent computation. This model does not depend on any particular programming language and can be embedded as a library in almost any host language.

Join calculus only needs a few features of the host language:

• functions returning locally defined functions as values (molecule names are syntactically functions)
• locally scoped values (no access to values defined in another scope)
• concurrent threads (for scheduling reactions)
• sending values synchronously to a blocking call in another thread (for implementing fast molecules)
• functions with side effects (for implementing inject, reply, and fast molecules)

For this reason, join calculus can be implemented as a library in most programming languages. Any special features of the programming language (algebraic types, polymorphism, OOP, etc.) can be immediately used by the embedded join calculus library.

Join calculus gives the programmer the following basic functionality:

• define arbitrary names for molecules, with arbitrary types of values
• jointly define several reactions with one or more input molecules
• inject molecules with values into the soup at any time (also within reactions)
• use molecule names as locally defined values, pass them to functions, store them in local data structures

The programmer can use any number of molecules and reactions. By defining the "chemistry" in a suitable way, the programmer can organize concurrent computations in any desired fashion while remaining within the declarative and purely functional paradigm. For instance, the programmer can:

• use "fast" molecules in order to wait synchronously until certain reaction start or end
• use "slow" molecules to receive response asynchronously from other reactions
• use "fast" molecules to receive values synchronously from other reactions
• use locally defined reactions to encapsulate and reuse concurrent functionality
• create new reactions and molecules inside recursive functions, thus creating a dynamic, recursive graph of reactions at run time
• use "higher-order" chemistry: molecules can carry values that contain other molecule names or functions of molecule names, which then become available within a reaction and may be used to inject arbitrary molecules or to perform arbitrary computations with molecule names obtained at run time
• create an abstract library of useful "chemical reactions" with a purely functional API

Join calculus has advantages over other models of concurrent computation:

• concurrency is simple to reason about because the operational semantics is based on easily visualized principles:

1. A reaction can start only if all of its input molecules are present in the soup.
2. A reaction first consumes the input molecules from the soup, then performs some computation and injects the output molecules into the soup.
3. All reactions can start concurrently and in random order.

• reaction and molecule name definitions are locally scoped, immutable, type-checked, and static (fixed at compile time)
• the local scoping enables the safe reuse of reactions: the programmer cannot, by mistake, destroy the functionality of any previously defined reactions (either by modifying the reactions or by injecting molecules at wrong times or in wrong numbers)
• the programmer does not manipulate shared mutable state because each computation is a pure function in its local scope (side effects are limited to molecule injection and the reply operator, i.e. to operations with injection and ejection of molecules)
• state is represented by values carried by the molecules, passed automatically and implicitly from one reaction to another
• all concurrent computations are scheduled implicitly: there is no hand-written code for creating or stopping new threads, scheduling new jobs, or waiting for completion; in other words, all concurrency is declarative
• one core or multicore, one threads or many threads - these low-level details are hidden from the programmer, who merely needs to inject some molecules to initiate concurrent computations
• the programmer does not use error-prone low-level synchronization primitives, such as locks and semaphores; instead the programmer operates with the visually clear "laws of chemistry"

If we imagine a program written entirely in join calculus, the program will be declarative: it is a collection of reaction definitions. Each reaction consists of a limited side effect (consuming the input molecules and injects output molecules. Additionally, a reaction may perform some calculations, which will compute the values carried by the output molecules. These values are functions of the values carried by the input molecules, and the required computations can be pure functions.

Of course, at least one molecule must be injected at some time, so that some reactions can start. This injection can be made implicit if we introduce, say, a main(argv,argc) molecule automatically injected by the runtime system when starting the program. In this way, we can indeed write an entire program in join calculus, and the program will look like a declarative collection of pure functions and reactions (join definitions).

Strictly speaking, join calculus is not fully referentially transparent for several reasons:

• The operators inject and reply, as well as "fast" molecule calls, have implicit side effects: they may start reactions and change the state of the chemical soup.

The operators inject and reply are syntactically functions that return an empty value. However, these operators perform side effects and thus cannot be replaced by their return values, which is a violation of referential transparency. Despite this, the order of inject and reply operations is not significant because these operations do not modify any values in the local scope, and because operational semantics of join calculus says that molecule injections and reactions will be scheduled by the system in random or unspecified order. Thus, if we need to perform several inject and reply operation, together with other calculations, we may put these operations in any order, e.g.

inject a(), inject b(), let x=y z, reply x to c(), inject d()

and so on.

The order of inject operations is significant when injecting fast molecules. For example, consider the following reaction, where f() is a fast molecule and a() is a slow molecule:

consume a() & f() => reply 123 to f()

In this case, the expression inject a(); f() will return 123. However, the expression f(); inject a(); may block forever if no a() molecules are present in the beginning. This is so because f() blocks until some reaction involving f() can start. However, there is only one such reaction, and it can start only when a() is present.

• Injections of fast molecules syntactically look like function calls (m()) but are not pure function calls since different subsequent injections of the same fast molecule may return different values.

• The reply operation (reply x to m) implicitly depends on the particular instance of the fast molecule being injected, not only on the molecule name m and value x.

Several different threads can be calling m() concurrently, and each thread may receive a different resulting value. In this case, the soup could contain several injected copies of the same fast molecule (and/or several copies pending injection from a different thread). When a reaction starts and consumes a fast molecule, it must consume a particular instance of the fast molecule, chosen by the chemical machine among all the existing instances. When this reaction replies to m, it must reply to that particular instance of m. For this reason, the reactions that contain a reply operation must be able to distinguish between the different instances of injected m() molecules. In other words, the reply operation (reply x to m) is not a pure function of the values x and m, but has an additional dependence on the fast molecule instance.

This dependence is implicit in join calculus because the molecule instances are not values that the programmer can manipulate. Nevertheless, this is an important detail of the operational semantics of join calculus.

This implementation of join calculus in Objective-C is called CocoaJoin and consists of a small library and a set of CPP convenience macros. The implementation uses Apple's multithreading library GCD ("grand central dispatch") to schedule reactions. Each "join definition" allocates a new asynchronous queue for reactions (however, GCD will make sure not to overload the available CPU cores). A semaphore and shared state is used to implement "fast" molecules. Molecule names are anonymous closures ("Objective-C blocks" as Apple calls them).

To use the library, import CJoin.h. Compile the .m files in CocoaJoin/. The implementation uses ARC and should work on iOS 6.0 and up.

For convenience, macros are provided. The example of "asynchronous counter" is implemented in the tests and looks like this:

cjDef(
   	// define the new input molecules
      cjAsync(inc, empty) // define slow molecule, inc()
      cjAsync(counter, int) // define slow molecule, counter(int x)
      cjSyncEmpty(int, getValue) // define fast molecule, int getValue()
      
       // define reaction: consume counter(n) & inc(), inject counter(n 1)
      cjReact2(inc, empty, dummy, counter, int, n, // using the name "dummy" for empty value
       counter(n 1); ); // define reaction: consume inc(), counter(n), inject counter(n 1)
       // define reaction: consume counter(n) & getValue(), inject counter(n) and reply n to getValue()
      cjReact2(counter, int, n, getValue, empty_int, dummy,
      { counter(n), cjReply(getValue, n); } );
);
counter(0), inc(), inc(); // inject some molecules
[CJoin cycleMainLoopForSeconds:0.2];	// allow enough time for reactions to run
int v = getValue(); // getValue returns 2

CocoaJoin modifies the model of join calculus in some inessential ways:

• Only a subset of primitive types are supported for molecule values: empty, int, float, id. Similarly, the return values of fast molecules can have only these types. Here empty is the functionally same as NSNull.
• Molecule names are local values of certain predefined types such as CjM_int, CjM_empty, CjM_id_id, etc., depending on the types of values. (Fully constructed molecules are not directly available as objects, as in JoCaml.)

Available types:

CjM_empty -- name of a slow molecule with empty value, such as inc()
CjM_id -- name of a slow molecule with id value, such as s(@"a")
CjM_int -- name of a slow molecule with int value
CjM_float -- name of a slow molecule with float value
CjM_empty_int -- name of a fast molecule with empty value, returning int

Other types of this form: CjM_t_r represents a name of a fast molecule carrying value of type t returning a value of type r. In ordinary join calculus, this type would be a function t -> r.

• The syntax of molecule injection is simply a(x) or b(). (These function calls return void and inject the molecule. A special keyword such as inject is not used. A fully constructed molecule is not available as a separate object.)
• The syntax of reply is cjReply(name, value), where name must be the name of a fast molecule. (Otherwise there will be a compile-time error, since the reply method is only defined for fast molecules.)
• It is not possible (due to limitations of the CPP macro processor) to make two join definitions one after another in the same local scope. Separate them with { ... } or define them within different function/method scopes.
• To make a new join definition, each new molecule name must be defined with its explicit type. If you do not define some of the new input molecules, or if you define a new input molecule but do not use it in the input, there will be a compiler error (undefined variable, or unused variable).

We need to list explicitly all the newly declared input molecule names, because otherwise we cannot generate code for defining them. (The macro processor is unable to process arrays of parameters or keep track of which names were used in a list of parameters.)

• Defining a reaction with an input name that has already been defined in the same local scope is impossible (it may result in a compiler error due to name clash).

This is so because the definition of an input molecule name, e.g. counter with integer value, is equivalent in Objective-C to the declaration of a new local variable,

  CjM_int counter = ...

If the name counter is already locally defined, it is a compile-time error to define the same name again in the same scope. No error will result if the name is redefined within another local scope.

• A join definition is represented by an object of class CJoin (the "join object").

The join object is not visible directly, and should not be manipulated by the programmer. (It is not possible to hide it entirely, without making the Objective-C syntax of join calculus extremely verbose.) After performing a join definition, the join object initializes its job queue and is ready to accept any injected molecules known to it, i.e. any of its input molecules. When the join object is destroyed, it also destroys the queue it has created.

Each molecule name carries a strong reference to the join object to which it belongs, and there are no other references to the join object. Once you destroy the last molecule name that uses the join object, the join is gone.

Together with the join object, the molecule names are created in the local scope, and their reactions have been defined and stored in the join object.

• A reaction may be designated for the "UI thread".

By default, all computations run asynchronously on a background thread, but updating UI on the iOS platform requires to call certain methods on the "UI thread" (or "main thread").

For this reason, a special feature is added to join calculus: the user can designate some reactions to run on the "UI thread". In addition, the user can specify that some join definitions to run on the "UI thread".

• A join definition may be designated for the "UI thread".

Each join definition runs concurrently in order to schedule its reactions independently of other join definitions. The user can specify that the UI thread should be used for the code that decides which reactions can be started, i.e. for the "decision" code of the join definition. By default, this decision code will be executed on a background thread, which may cause additional delay if the molecules are injected from the UI thread and an immediate synchronous reaction is desired (as could be the case for UI-intensive computations). In this case, both the reaction and decision code for the join definition can be designated for the UI thread.

Note that join calculus intentionally restricts the tasks that the decision code for join definitions needs to perform. The decision code only needs to check which molecules are present and which reactions can be scheduled to start. (In join calculus, reactions start whenever the input molecules are present, regardless of the molecule values.) So, the decision code does not check the values of any molecules and does not call any user-defined functions. In the worst case (many molecules are present and all possible reactions can be started), the decision code will run in time linear in the total number of locally defined molecules and reactions. (Each join definition decides only local reactions, not all reactions defined anywhere in the program!) For this reason, it is often acceptable to designate some join definitions for the UI thread, especially if the number of local reactions is small for these join definitions.

• Weak typing

When the user defines a molecule name with some type such as int, the compiler will check that the name is used with values of the correct type. So, after defining cjAsync(counter, int) it will be an error to inject this molecule as counter(@"x"). However, this error becomes merely a warning with the type id since that type is compatible with any other object type.

• Thread safety

Each molecule carries a value that may cross thread boundaries. This can happen in several cases:

• the molecule is injected on a background thread but a reaction is designated for the UI thread
• both the injection and the reaction are on background threads, but the GCD system running on a multicore CPU decides to run them on different background threads

Not all Objective-C values are thread-safe in the sense of being able to pass from one thread to another without crashing. In particular, UIKit view objects and Core Data values cannot pass thread boundaries without severe problems (crash, incorrect visual display, or loss of data). Join calculus is designed to operate on immutable values; therefore, Objective-C anonymous closures, immutable objects, and immutable collections should be safe.

If it is required to use a mutable object or collection, join calculus can easily guarantee that only one thread will ever modify the object. To achieve this, define a reaction that consumes a certain input molecule, and then hide this reaction and the molecule in a local scope that injects a single copy of the molecule into the soup. This will make sure that there is ever at most one copy of this molecule in the soup. Thus, there will never be several concurrent reactions of this type. The example of "asynchronous counter" uses this technique.

CocoaJoin uses ARC (automatic reference counting), which means that injecting heap-allocated values is mostly unproblematic. An exception is the case of anonymous closures as molecule values. This is still problematic due to the limitations of Objective-C (no garbage collection and no generics). Because of lack of generics, the closure must be represented by the type id, and then the memory management rests with the programmer. The problem may be that injecting a molecule carrying a closure requires that the closure should be first manually copied to the heap - otherwise the memory management system will lose the closure, causing a crash later (without a stack trace!). A solution can be to cast a closure manually to some fiducial object type (such as NSBlock), so that the memory management system thinks it's an Objective-C object with standard memory management, and then to cast the closure back to its correct type (so that the system allows the programmer to use the closure at all).

In the current implementation, molecule names are anonymous closures, so the users of "higher-order chemistry" must confront this problem!

Finally, Objective-C cannot fully guarantee static typing or fully hide private variables. Nevertheless, Objective-C has local scope and weak typing. It will be certainly possible to break the functionality of CocoaJoin; the compiler cannot prevent using private methods or calling some methods incorrectly. But the library should work correctly as long as the user does not go outside the provided macros.

DinPhil5 is a test project that shows a simple solution of the "dining philosophers" problem in a barebones UI.

The core logic is implemented in "DiningPhilosophicalLogic.m"; consult that file for the actual working code.

There is a single join definition, which is is asynchronous, and all other reactions are also asynchronous, except for a single reaction designated for the UI thread. This reaction updates the visual display of the philosophers.

Make a local join definition:

• a join definition with decision code on a background thread

cjDef(...)

• a join definition with decision code on the UI thread

cjDefUI(...)

What goes inside the cjDef macro:

• definitions of names and types for all new input molecules using cjAsync or cjSync
• definitions of all reactions using cjReact1, cjReact2, etc.

Define a new input molecule name:

• slow molecule with value of type t

cjAsync(name, t)

• fast molecule with value of type t_in, returning value of type t_out

cjSync(t_out, name, t_in)

• for molecules that carry no value, special macros are used:

cjAsyncEmpty(name)

cjSyncEmpty(t_out, name)

For "fast" molecules that return no value, use the empty type, e.g. cjSync(empty, f, int). These molecules will be declared as blocks returning an empty value (in Objective-C, this is NSNull *null).

Define a new reaction:

• reactions taking a single input molecule:

    cjReact1(name1, type1, var1, code...)
    cjReact1UI(name1, type1, var1, code...)
  

name1 must be a newly defined molecule name (value will be created in local scope). type1 is the type of the value of that molecule. For fast molecules, separate type with underscore: for example, empty_int or id_id. var1 is the name of the formal parameter bound to the value of that molecule within the reaction body. code is the body of the reaction; this may use the locally defined names name1 and var1.

Note: for molecules that carry no values, the type empty is used here.

• reactions taking two input molecules:

    cjReact2(name1, type1, var1, name2, type2, var2, code...)
    cjReact2UI(name1, type1, var1, name2, type2, var2, code...)
  

Similar macros cjReact3, cjReact3UI, cjReact4, and cjReact4UI are available. Further such macros are straightforward to implement. (See CJoin.h.)

The "UI" versions of the macros define reactions that are designated for the UI thread.

Here is an example of using the reaction macros. To convert the pseudocode such as

consume a(x) & b(y) & c() => do_computations(x,y); inject a(x y), c()

into a macro call, we need to specify the names of the input molecules (a, b, c), the types of their arguments (int, int, empty), and the names of the formal parameters (x, y, dummy), and finally we need to write the function code for the reaction. Since there are three input molecules, we use the macro cjReact3:

cjReact3(a, int, x, b, int, y, c, empty, dummy,
{   do_computations(x,y); a(x y), c();  }
)

Here we have put the reaction body into its own block for visual clarity, but this is not necessary. Also, it is optional whether to inject the molecules with the comma operator or through separate statements a(x y); c();. The reaction block always returns nothing.

Note also that we have used the name dummy for the argument of the empty type. This is necessary for technical reasons (the CPP macros are insufficiently powerful here). We are required to specify names for all arguments, even if the value is empty. So the reaction body will see a local variable named dummy with value equal to [NSNull null].

The reply operator:

• reply with value val to a fast molecule named name

cjReply(name, val)

The reply operator, as well as injections of known molecules, can be used anywhere in the reaction block.

The CocoaJoin library was tested on a few examples shown above. In addition, the "dining philosophers" problem is implemented with a spartan GUI for 5 philosophers.

Roadmap for the future:

• Urgent: need a mechanism to "deactivate" a local join definition, so that it drops all molecules and pending reactions, stops all running reactions (if possible), and ignores all injected molecules - until the join definition is again "activated".

This is necessary for functionality such as add/remove listener. Suppose we declare an asynchronous molecule that will be injected by another process, and we wait for it to be injected. However, at some point we stop waiting. We need to tell the other process that it should not inject that molecule any more. However, if we try organizing this by chemistry, we might not get the right sequence of asynchronous messages: we tell the other process not to inject that molecule, but the process already has scheduled this injection, and we are not prepared for this.

Another possible use case: we need to "reset" the whole process. We need a way to drop all present molecules and wait until the soup is quiet. This may take some time, so the "reset" is an asynchronous operation.

The "actor" model does this by making an actor ignore any messages not explicitly handled by the actor process. In the actor model, the actor could be simply switched to a behavior where no message is handled, to simulate the "deactivated" state. A way to implement the "deactivated" behavior for a join definition is to have a special molecule that reacts with all other molecules and does nothing. But it would be cumbersome to define a molecule that absorbs all other molecules - we would have to define, by hand, as many reactions as other molecules. It would be good if this functionality were automatic, and if such a molecule were automatically defined by the library, for any join definition.

We would have to define also a method of removing this molecule, and/or of testing its presence. An alternative is to provide a fast "join control" molecule that imperatively changes the internal state of the join object.

The asynchronous reply ("all is quiet") needs another molecule, from another join. A "join control" molecule could take such a molecule as a decoration value.

• Define special global methods for controlling the "chemical machine" as a whole, or for controlling specific local join definitions.

Possible functions: stats (get statistics on the number of molecules and reactions), pause (do not schedule any new reactions), resume (start scheduling reactions again), clear (remove all present molecules in all reactions, stop all reactions, reply immediately to all fast molecules, and ignore requests to inject any new molecules).

These global operations, as well as the corresponding local operations, can be implemented most easily via special predefined fast molecules that already have predefined reactions.

• Start a "chemical library" encapsulating useful reactions and functions. Add tests for "chemical library" functions.

• Document the internals of CocoaJoin, trying to be language-agnostic, since this kind of "dynamic" implementation might be quite useful and simple to do in other languages.

• Fix possible memory leaks and prevent bugs:

1. If we are using "reply", the join object will be circle-referenced inside the reaction block, which is always retained by the join object. Either pass the join object as a parameter to the reaction block (curried blocks will crash?), or make it weak (iOS version dependence?).

2. Whenever a join object receives its first injected molecule, there should be no more possibility to define new reactions. Otherwise - fatal exception at runtime.

3. Also at that time we should check that all molecules declared as new input molecules are actually being used as input. Otherwise - fatal exception at runtime.

4. Add a test: check that incorrect molecule injections of any kind (undeclared molecule; declared but not used; define new reactions after injecting molecules) do actually cause a runtime exception.

5. Add a test for the situation that several reactions are constantly scheduled, with time-outs, while no more molecule names are visible. Need to make sure that no memory leaks and no crashes happen in that case.

6. When no more reactions can be run and no more molecules can be injected (all molecule names go out of scope), the join and its reactions should be released. Does this happen? Is this even possible with the current architecture? (Molecule references must be weak inside reaction blocks...? Can we declare all molecule names __weak to begin with? It seems we cannot declare a join object __weak?)

• More types (BOOL, typedef NSString *, ...?) to be defined in macros

• Refactor the API so that the types for molecules are introspected, and the reaction block receives correct arguments via currying and helper blocks. (Not sure if this will even work in Objective-C but worth investigating.) Investigated. - No, it won't work.

The host language (Objective-C/Cocoa) provides the following features:

• anonymous functions (closures) with automatic retention of local scope

• object-oriented programming with automatic (reference counted) memory management

• a C-level library (dispatch_*) for scheduling on foreground and background threads

• CPP macros (simple textual substitution with parameters)

Molecule names (counter, all_done) are values in the program. Syntactically, they are of functional type. (They are Objective-C "blocks", i.e. anonymous closures.) The user can store them for later use, or pass as arguments to functions. There are typedefs for the available types of molecules, such as CjM_empty, CjM_int, CjM_empty_int and so on.

In order to inject a molecule, the user simply calls the molecule name. For slow molecules, this call injects the molecule and returns nothing. For example, counter(0) injects the counter molecule with value 0. For fast molecules, the call injects the molecule, waits synchronously for a reaction to start, and returns a value when the reaction executes the reply operation.

The result of putting a value onto a molecule name can be imagined as a "fully constructed molecule" that exists for some time in the soup. For example, counter(10) is such a fully constructed molecule. The user's program cannot directly manipulate fully constructed molecules, they are not available to the user as values. But internally the implementation of join calculus must operate with "fully constructed" molecule objects. This is so because fast molecules need to store the calling thread information: when the user says reply x to m, the value x must be sent to a particular thread that injected the particular instance of the molecule m. The information about this thread is currently stored as a semaphore value in the "fully constructed molecule" object.

The classes CjR_empty, CjR_int, CjR_empty_int etc. represent fully constructed molecules. For slow molecules, these classes hold the value on the molecule as well as a reference to the CJoin object where the molecules were defined. For fast molecules, these classes hold also the semaphore information and the return value.

The programmer should not manipulate the fully constructed molecules directly. Due to the limitations of Objective-C, it is not possible to make the CjR_* classes hidden.

The implementation of join calculus proceeds like this:

• the user creates a join definition in a local scope

• the user injects some molecules

In order to create a join definition, the user writes some macros that expand to the following code:

• create a CJoin object in the local scope; the object is held in a local variable with a fixed name _cj_LocalJoin.

The CPP macro system does not allow us to create unique names or to mark names as "used", which would be desirable to avoid name conflicts! For this reason, it is currently impossible to create two join definitions in the same local scope.

• define, separately, each input molecule name and its value type; report the new molecule names to the join object. The join object marks these names as possible input types.

Each input molecule name is a closure that takes an argument of the appropriate type. When evaluated, the closure constructs and injects a full molecule. The closure returns void (for slow molecules) or the appropriate value type (for fast molecules).

Thus, the code for defining a counter molecule looks like this,

CjM_int counter = ^(int n){ ... create and inject 
      an object of class CjR_int that represents
      the fully constructed molecule counter(n) ... };

There is no separate inject operator: calling the molecule name with arguments is the same as injecting the molecule, both for slow and for fast molecules.

The molecule name, counter, is therefore a new local variable in the local scope.

• define, separately, each reaction in the join definition. A reaction is an object of class CjReaction that contains the following values: a list of input molecule names, a boolean flag "schedule on UI thread", and a closure (the reaction body). The reaction body takes an array of input molecule objects, and returns nothing. The join object stores all reaction objects in an array.

It is an error to define a reaction object with an input molecule that has not been defined in the join definition. It is also an error to define a molecule in the join definition but to fail to use that molecule as an input molecule to some reaction.

The standard formulation avoids these errors by making the join definition and the input molecule declarations at the same time. We cannot do this in Objective-C, so we have to separately declare the input molecules and the reactions.

Each locally created join object maintains its own "soup". By the semantics of join calculus, a molecule can be injected only into the join definition where the molecule was defined as an input molecule. (Each new join definition in each local scope introduces its own molecule names, valid only within that local scope, even if the molecule names are the same.)

The join object holds a dictionary that maps each molecule name to a set of fully constructed molecule instances currently present in the soup.

Whenever a new molecule is injected, the join object checks whether some reaction is now possible. For example, if we define a reaction

consume counter(n) & inc() => ...

then the join object checks whether any counter molecules are present. (Many molecule instances with the same name may be present.) If so, the join object checks whether any inc molecules are present. If so, the join object will randomly select one molecule instance of counter and one molecule instance of inc from the soup. These two instances will be then removed from the soup and put into the input molecule array for the reaction. The input array contains fully constructed molecule instances in the order they were defined when defining the reaction. By the linearity restriction of join calculus, the input molecule names cannot repeat.

A method is then called that schedules a reaction. In this method, the array with the input molecules is given to the reaction block, which is run on a background thread.

After scheduling one reaction and removing the consumed molecules, the join object tries finding a further possible reaction, and also schedules it if possible. The process is repeated until no further reactions can be started.

This mechanism allows us to inject several molecules at once (although this is not implemented now). If only one molecule is ever injected at once, it is not possible that more than two reaction can start as a result of this injection.

The random shuffling of the list of available reactions is needed to make sure that available reactions are chosen randomly.

A fast molecule, when injected, constructs a molecule object and sets a semaphore in the "closed" state, then injects the molecule object into the join object. The join object tries to run some reaction as usual. When a reaction consumes the fast molecule object, the semaphore is available inside the molecule object and thus is made available to the reaction body (since the reaction body is a block that takes an array of molecule objects as argument). If during the reaction body there is ia reply operation for the fast molecule, the value given to reply is stored in the fast molecule object, and then the semaphore is opened. This allows the calling thread to continue. By the time the semaphore is opened, the "reply value" is now stored in the molecule object, and can be returned to the caller.

A reaction object has a boolean field that says whether it is designated for the UI thread. Every iOS application runs its visual operations on the UI thread, and the programmer needs to make sure that no UI manipulations occur on any background threads. For this reason, CocoaJoin implements this extension of the join calculus.

Reactions designated for the UI thread are always scheduled to run on that thread, regardless of the thread they were called from, and regardless of the thread the join definition's code is running.

The join definition's code is a relatively quick routine that searches through the declared reactions (reshuffled to random order), finds possible reactions, removes their input molecules, and asynchronously schedules the reaction in a background queue. This routine can be run on the UI thread or on a background thread, according to whether the join definition is designated for the UI thread or not.

Again, this minor extension of join calculus is necessary for practical Cocoa programming. By designating a join definition for the UI thread, and by injecting some molecules also on the UI thread, the programmer can make sure that certain UI-thread reactions will be run immediately after injecting the molecules, without any delay caused by thread switching or task scheduling. This may be important for performance reasons when reacting to UI signals. Of course, in this case the reaction should be kept simple and short (say, changing the UI state).