Introduction to join calculus

Introduction to Join Calculus
Sergei Winitzki
Scala Study Group
November 10, 2013
Sergei Winitzki (Versal Group Inc.) Introduction to Join Calculus November 10, 2013 1 / 15

Motivation
Imperative concurrency is dicult:
callbacks, semaphores, locks, threads, shared state
testing??
Pure functional concurrency is better:
futures = async monads
Erlang's purely functional messaging; actors (Akka in Scala)
Join Calculus:
Elegant, concise model of concurrent computation
Join Calculus = more purely functionally concurrent actors
Working implementation: JoCaml (jocaml.inria.fr)

A taste of OCaml
Common features to F#, Haskell, OCaml, SML, Scala:
Expression-oriented programming:
let s = (if 1=1 then Hello, world! else Error) in print_string s
Algebraic data types, parametric polymorphism:
type 'a bTree = Leaf of 'a | Node of ('a bTree * 'a bTree)
Immutable, scoped values, with statically inferred types:
# let x = 3 in (let x = x+1 in x/2) * x;;
- : int = 6
Mutually recursive denitions:
# let rec isEven n = if n=0 then true else isOdd (n-1)
and isOdd n = if n=0 then false else isEven (n-1);;
val isEven : int - bool = fun
val isOdd : int - bool = fun
# let result = List.map (fun x - (isEven x, isOdd x)) [1; 2];;
val result : (bool * bool) list = [ (false, true); (true, false) ]

Join Calculus in a nutshell
The Reexive Chemical Abstract Machine (RChAM)
Abstract chemistry: molecules and reactions
Chemical soup contains many molecules
A group of molecules starts a chemical reaction
jocaml def a() b() = c()
and a() c() = 0;;
val a : unit Join.chan = abstr
val b : unit Join.chan = abstr
val c : unit Join.chan = abstr
A
B
C
C
A
Using the chemical machine:
We dene arbitrary chemical laws and molecules: a, b, c, ...
We inject some molecules into the soup: spawn a() a() b()
Note: a() a() b() is the syntax for molecule-valued expressions
The runtime system evolves the soup asynchronously

Concurrent computations
Sequential computation = evaluating an expression
Concurrent computation = evaluating several expressions at once
Join Calculus organizes concurrent computations through chemistry:
Each molecule carries a value
Each reaction computes a
molecule-valued expression
Results of computation are
injected back into the soup
Reactions start asynchronously
after injecting initial molecules
a(x)
b(y) c()
a(x)
print x
a(z)compute z(x,y)
def a(x) b(y) =
let z = (big_function x y) in a(z)
and a(x) c() = (print_int x; 0);;
When reaction starts: input molecules disappear, expression is computed,
output molecules are injected

More features of JoCaml
Mutual recursion:
def a(x) = a(x+1) b(x+2) and b(x) c(y) = a(x+y)
Pattern-matching on molecule's payload values:
def a(Some x) b(y) = b(x+y) or a(None) b(y) = b(y)
Instant molecules (same type as function calls):
def a(x) f() = a(x+1) reply x to f
Local molecules and reactions:
def c(n) = ( if n0 then c(n-1) else 0 ) in spawn c(10)
Injection as side-eect: let x=3 in (spawn a(x); printf %dn x)
Molecule constructors are dened as values and can be manipulated:
# def a(x) = Printf.printf %dn x; 0;;
val a : int Join.chan = abstr
# def b(m,y) = Printf.printf injecting m(%d)n y; m(y);;
val b: (int Join.chan * int) Join.chan = abstr

Join Calculus: Examples
Options, Futures, and Map/Reduce
Future with synchronous poll (get):
# def fut(f,x) = let res = f x in finished(res)
and get() finished(res) = reply res to get;;
val get : unit - '_a = fun
val finished : '_a Join.chan = abstr
val fut : (('a - '_b) * 'a) Join.chan = abstr
Future with synchronous callback:
def fut(f,x,c) = let res = f x in ( c(res); finished(res) )
and get() finished(res) = reply res to get
Future with asynchronous callback:
def fut(f,x,m) = let res = f x in ( m(res) finished(res) )
and get() finished(res) = reply res to get
Exercise: implement a future with cancellable callback

Options, Futures, and Map/Reduce
Asynchronous counter:
# def inc() c(n) = c(n+1)
or get() c(n) = reply n to get c(n);;
val inc : unit Join.chan = abstr
val get : unit - int = fun
val c : int Join.chan = abstr
# spawn c(0) inc() inc() inc();;
- : unit = ()
# get();;
- : int = 3
Map/Reduce:
def res(list) c(s) = res (s::list) or get() res(list) = reply list to get;;
spawn res([]);;
List.map (fun x- spawn c(x*2)) [1; 2; 3];;
get();; (* this returned [4; 6; 2] in one test *)
Exercise: implement a concurrent fold (e.g. sum of int list)

Five Dining Philosophers
Philosophers A, B, C, D, E; forks fAB, fBC, fCD, fDE, fEA.
let report(message) = Printf.printf %sn message;
Unix.sleep (Random.int 3600) ;;
def hA() fEA() fAB() = report(A is eating); tA() fEA() fAB()
or hB() fAB() fBC() = report(B is eating); tB() fAB() fBC()
or hC() fBC() fCD() = report(C is eating); tC() fBC() fCD()
or hD() fCD() fDE() = report(D is eating); tD() fCD() fDE()
or hE() fDE() fEA() = report(E is eating); tE() fDE() fEA()
and tA() = report(A is thinking); hA()
and tB() = report(B is thinking); hB()
and tC() = report(C is thinking); hC()
and tD() = report(D is thinking); hD()
and tE() = report(E is thinking); hE() ;;
spawn fAB() fBC() fCD() fDE() fEA()
tA() tB() tC() tD() tE() ;;

Limitations and restrictions of Join Calculus
Less is more!
Reactions are dened statically and with local scope:
no molecules with computed names:
a(x) molecule_named(b)(x) = (not allowed!)
cannot dynamically add a new reaction to a previously dened
molecule:
def a(x) b(y) = ... ;;
def b(y) c(z) = ... shadows the old definition of b()!
No guard conditions for reactions:
def a(x) b(y) start_if (x==y) = ... (not allowed!)
No duplicated input values: a(x) b(x) = (not allowed!)
No duplicated input molecules: a(x) a(y) = (not allowed!)
No way to test dynamically for the presence/absence of a molecule!
def a(x) b(y) = if have_molecules(c d) then ... else ... (not allowed!)

Limitations and restrictions of Join Calculus
It seems they do not limit the expressive power!
What if we need a reaction with pairs of molecules?
a(x) a(y) = a(x+y)
Solution: use two or-coupled reactions with new molecules a' and b:
def a(x) b() = a'(x) or a(x) a'(y) = whatever(x,y)
Make sure that one b() is injected together with each a(x)
Questions:
Can we prevent the error of not injecting b()?
Can we do a reaction with n molecules, where n is dynamic?
Can we do n dining philosophers?

Local scope and recursion
Skeleton code for concurrent merge-sort
The mergesort molecule:
receives the upper-level sorted_result molecule
denes its own sorted molecule in local scope
emits upper-level sorted_result when done
def mergesort(arr, sorted_result) =
if Array.length arr = 1 then sorted_result(arr)
else
let (part1, part2) = array_split arr
in
def sorted(x) sorted(y) = sorted_result(array_merge x y)
in
mergesort(part1, sorted) mergesort(part2, sorted)
Note: sorted(x) sorted(y) is pseudocode, easy to rewrite.
See tutorial for complete working JoCaml code.

Comparison: Join Calculus vs. Actor model
Reaction ≈ actor; molecule ≈ message to actor.
Actors:
receive messages asynchronously
process one message at a time (one actor = one thread)
must hold mutable state (e.g. for a thread-safe counter)
explicitly create and congure other actors
Reactions:
start when several molecules are available
many reactions can start at once, automatically
do not need mutable state
all reactions are dened statically, but locally scoped
simulate actors: def message(m) actor(state) = actor(compute_new state m)

Implementation of Join Calculus
JC = a DSL + run-time library, or just DSL?
Implement Join Calculus using Actors (Akka)?
Each reaction has 1 monitor actor and ≥ 1 worker actors
Monitor receives messages for each spawn, keeps track of molecules
Monitor starts a worker actor when all molecules are present
Monitors have to talk to competing monitors - use up molecules
but all competitions are statically dened!
Monitors / workers need to be locally scoped!
No globally shared state of the soup is needed!
Discuss

Conclusions and outlook
Join Calculus is concurrent programming in pure functional style
Similar to Actors, but more concurrent and more pure
Very little known, and very little used in practice
Existing literature is not suitable as introduction to practical
programming
My tutorial text is in progress (search Google for tutorial on join
calculus)

Introduction to join calculus

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