Chapter 5: Join Calculus

1. Chapter 5: Join Calculus Programming Distributed Computing Systems: A Foundational Approach Carlos Varela Rensselaer Polytechnic Institute C. Varela

2. Reflexive CHAMandJoin Calculus • Attempt to overcome “distribution” issues in π-calculus, yet keeping formal investigation results. • Bridge gap between theory and practice, and form the basics for a programming language. • Imposes locality and adds reflexion to generic CHAM of Boudol and Berry. • Combines restriction, reception, and replication (of π-calculus) into a single (joint) receptor definition. C. Varela

3. Reflexive Chemical Abstract Machine x<y> atom M | M molecule e.g. ready<laser> ready<laser> | job<1> D or J P reaction e.g. ready<printer> | job<file> printer<file> C. Varela

4. Printer Example 0 ├ ready<laser>, job<1>, job<2> 0 ├ ready<laser> | job<1> , job<2> Let D = ready<printer> | job<file> printer<file> D ├ ready<laser> | job<1>, job<2> D ├ laser<1>, job<2> C. Varela

5. Reflection def D in P molecule 0 ├ def D in ready<laser> | job<1> | job<2> D ├ ready<laser> | job<1> | job<2> C. Varela

6. Names, Processes, Definitions ~ def P = x<v> asynchronous polyadic message def D in P definition of new names P | P parallel composition J = x<v> join patterns J | J D = J P matching join D D patterns to processes ~ def def C. Varela

7. Variable Types Received variables (join patterns) rv(x<v>) = {u in v} rv(J | J’) = rv(J) U rv(J’) Defined variables (join patterns and definitions) dv(x<v>) = {x} dv(J | J’) = dv(J) U dv(J’) dv(J P) = dv(J) dv(D D’) = dv(D) U dv(D’) ~ ~ def def ~ def def def def C. Varela

8. Free Variables (processes and definitions) fv(J P) = dv(J) U (fv(P) – rv(J)) fv(D D’) = fv(D) U fv(D’) fv(x<v>) = {x} U {u in v} fv(def D in P) = (fv(P) U fv(D)) – dv(D) fv(P | P’) = fv(P) U fv(P’) def def ~ ~ def def def C. Varela

9. Operational SemanticsStructural Equivalence (str-join) ├ P | Q ├ P,Q (“ “ heating, “ “ cooling) (str-and) D E ├ D,E├ (str-def) ├ def D in P D ├ P (red) J P ├ J J P ├ P σdv instantiates port variable dv(D) to fresh names. σrv substitutes transmitted names for distinct received variables rv(J). σdv σdv σrv σrv C. Varela

10. Examples • def x<u> y<u> in P messages on local name x in P are forwarded to the outside as messages on y. • def y<u> x<u> in def x<u> y<u> in P messages on local name x are forwarded in two steps on the external name x (internal x needs renaming). • def x1<u> | x2<v> x<u,v> in P performs multiplexing of messages on x whose parts are supplied on x1 and x2. C. Varela

11. Examples (continued) • def x<v> | y<k> k<v> in P models π-calculus-like channels, as values are sent on x and requests for values are sent on y. • def s<> P s<> Q in s<> express internal non-determinism P + Q • def once<> | y<v> x<v> in y<1> | y<2> | y<3> | once<> uses the message on once as a lock for non-determinism x<1> + x<2> + x<3> C. Varela

12. Recursion Example def loop<> P | loop<> in loop<> replicates the process P (!P), starting a new copy each time the definition is used. C. Varela

13. Reference Cell def mkcell<v0, k0> in P P = def c<g,s> s<w1,k> in mkcell <w0, c> def get<k> | s<v> k<v> | s<v> set<u,k> | s<v> k<> | s<u> in s<v0> | k0<get,set> C. Varela

14. Mutual Exclusion def s<> P | s<> Q | s<> in s<> (semaphore) def in s<> (guarantees mutual exclusion) s<> def JP s<> in JP s<> def JQ s<> in JQ C. Varela

15. Dining Philosophers def in P P = Π(phil<i, ci, c(i+1)mod n> | chopstick<ci>) phil<i,l,r> def l<> | r<> eat<i> | l<> | r<> in 0 chopstick<c> c<> 0 ≤ i < n C. Varela