Formal Methods in Software

Lecture 5. Pi-Calculus

Vlad PatryshevSCU2014 Prerequisites

λ-1, λ-2

This is Tier 2 of Modern Comp Sci

History: Hoare, CSP, 1978

The trouble is: it’s static in structure - no process creation allowed

What is π-calculus

• Describes processes developing in time

• Is conceptually similar to λ-calculus

• Is used to describe o gameso security protocols (spi-calculus)o biological and chemical processeso business processes

• Is the base of modern concurrency libraries, like Scala Actors

Scala Actors, an implementation of

πclass CustomerSupport extends Actor {

def act() {

while (true) {

receive {

case Stop =>

println("End of my sufferings")

exit()

case question:String => sender ! s"What do you mean, $question?!"

}

}

}

}

Elements of PiNotion Simplified Notation Milner Notation Meaning

variable x,y,... x,y,... Stores (points to) data; typeless

channel a,b,... a,b,... Used for communication

process (agent) P,Q,... P,Q,... Denotes a combination of elementary processes

write a!(x1,...xn) a x⟨ 1,...xn⟩ A process in which channel a outputs the values of x (sends a message)

read a?(x1,...xn) a(x1,...xn) A process in which channel a reads a message with values x1,...xn

new channel new(x1,...xn) ν(x1,...xn) Creates new names to be used within the scope

identifier A(x1,...)=P A(x1,...)=P given a process, give it a name

Operations on ProcessesOperation Simplified Notation Milner Notation Meaning

sequence P.Q P.Q run process P, then process Q

parallel P|Q P|Q run processes P and Q in parallel

choice P+Q P+Q nondeterministically run either P or Q

match if x=y then P if x=y then P obvious (this op is optional)

mismatch if x≠y then P if x≠y then P obvious (this op is optional)

null process 0 0 does nothing, and ends

replication *(P)=P|*(P) !P unlimited replication of process P

reduction P → Q P → Q evaluation: P performs a step and becomes Q

Example

Printer = b?doc . Println(doc) . Printer

Server = a!b . Server

Client = a?p . p!doc

Life = Client | Server | Printer

Laws and Rules

First, a free variable is something that’s exposed to external world, while a bound variable is hidden in a scope - so we are free to rename bound variables, but we better not touch free variables.

Defining Free and Bound Names

• 0 has no bound or free names

• a?x - a is free, x is bound (it’s a new name to be used further down)

• new(x) - x is bound

• a!x - a and x are free

• A(x1,...xn)=P - x1,...xn are bound

• P.Q, P|Q, P+Q - free(P) free(Q)∪ , bound(P) bound(Q) ∪ (where’s the

rest?!)

Laws and Rules for Reduction

• fundamental rule (channels communication)

• alpha-conversion (like in lambda), and unfolding law

• monoid laws

• choice rule

• parallelization rule

• replication rule

• name binding rule

• scope extension laws

Laws and Rules for Reduction

• fundamental rule (channels communication, beta-reduction)

(x!z . P) | (x?y . Q) → P|(Q[y/z])

• alpha-conversion (like in lambda), and unfolding lawo P[x/y] ≡ Po if A(x) = P, then A(y) = P[x/y]

• monoid laws

• choice rule

• parallelization rule

• replication rule

• name binding rule

• scope extension laws

Laws and Rules for Reduction• fundamental rule (channels communication)

• alpha-conversion (like in lambda), and unfolding law

• monoid laws

| is a commutative monoid, 0 is a neutral element

+ is commutative and associative

• choice ruleo If P→Q, then (P+R) → (Q+R) (?)

• parallelism ruleo If P→Q, then (P|R) → (Q|R)

• replication ruleo If P→Q, then *(P) → Q|*(P) (this follows from definition)

• name binding rule

• scope extension laws

Laws and Rules for Reduction• fundamental rule (channels communication)

• alpha-conversion (like in lambda), and unfolding law

• monoid laws

• choice rule

• parallelism rule

• replication rule

• name binding rule

If P→Q, then new(x).P → new(x).Q (new names don’t break it)

• scope extension lawso new(x).0 = 0o new(x).(P|Q)=P|(new(x).Q) if x fn(P)∉o new(x).(P+Q)=P+(new(x).Q) if x fn(P)∉o new(x).new(y).P = new(y).new(x).P

Can Model Lambda-Calculusλ-expression π-agent on port p Meaning

M [M](p) Build an agent with a communication channel p

λx M p?x.p?q.[M](q) Given a channel p, obtain the value of x and the communication channel q; call agent [M] on channel q

x x!p A variable, when used, just publishes its channel

M N new(a,b).(([M](a))|(a!b.a!f)|*((b?c).[N](c))

Applying M to N means: create control channels a and b, and launch in parallel:

● [M] on a (it will wait)● pass b and f to M via channel a● read channel b on y, start [N] which will work with channel c;

run it forever, we may need result more than once.

Modeling Lambda: ExampleFor M=λx x how will [M N] look? Can we reduce it to just [N]?

[(λx x) N](f) = new(a).new(b) . ([M](a) | (a!b.a!f) | *(b?c.[N](c)) ) = new(a).new(b) . (a?x.a?y.x!y | (a!b.a!f) | *(b?c.[N](c)) ) → new(a).new(b) . (a?y.b!y | (a!f) | *(b?c.[N](c)) ) → new(a).new(b) . (b!f | 0 | *(b?c.[N](c)) ) → new(a).new(b) . (b!f | *(b?c.[N](c)) ) → new(a).new(b) . (b!f | b?c.[N](c)| *(b?c.[N](c)) ) → new(a).new(b) . (0 | [N](f) | *(b?c.[N](c)) ) → new(a).new(b) . ([N](f) | *(b?c.[N](c)) ) → new(b) . ([N](f) | *(b?c.[N](c)) ) → ([N](f) | new(b). *(b@c.[N](c)) ) → [N](f)

Modeling Lambda: Example

M=λa a, N=b // using (x!z).P | (x?y).Q ) → P|(Q[z/y])[(λa a) b](p) = (new(q) . [M](q)) | (new(y) . q!(y,p)) | *((y?r) . [N](r)) = (new(q) . q?(x,q’) . x!q’ | (new(y) . q!(y,p)) | *(y?r . z!r))) → /* reduces to */ (new(q) . (new(y) . y!p) | *(y?r . z!r)) = (new(q) . (new(y) . y!p) | (y?r . z!r)) | *(y?r . !z(r))) → /* reduces to */ (new(q) . (new(y) . z!p) | *(?y(r) . z!r)) “=” /* nobody’s calling local channel y*/ (new(q) . (new(y) . z!p) “=” /* nobody uses q and y */ z!p = [z](p)

Referenceshttp://scienceblogs.com/goodmath/2007/04/16/back-to-calculus-a-better-intr-1/

http://users.soe.ucsc.edu/~abadi/Papers/isss02.pdf

https://www.doc.ic.ac.uk/~pg/Concurrency/4pi.pdf

drona.csa.iisc.ernet.in/~deepakd/pav/crchandbook.ps

https://github.com/leithaus/SpecialK/blob/master/docs/presentations/Agents%20and%20agency%20in%20the%20Internet.pdf?raw=true

http://www.scala-lang.org/old/node/242

http://basics.sjtu.edu.cn/~yuxi/papers/lambda_in_pi.pdf

http://scala-programming-language.1934581.n4.nabble.com/scala-Actors-versus-processes-td1993744.html

- Greg Meredith discussing Pi and Scala Actors with Martin Odersky. Not much.

Wikipedia