Understanding distributed calculi in Haskell

Understanding Distributed Calculi
in Haskell
Pawel Szulc
@rabbitonweb

distributed-process
also known as Cloud Haskell

distributed-process
data Process a

distributed-process
data Process a
ProcessId
send :: Serializable a => ProcessId -> a -> Process ()

distributed-process
data Process a
ProcessId
expect :: forall a. Serializable a => Process a
send :: Serializable a => ProcessId -> a -> Process ()

master
ping
pong
Init(masterPid, pongPid)

master
ping
pong
Ping(pingPid)

Syntax
L, M, N :: = x variable
λx.M abstraction
M N application

Few definitions
λx.M
Bound variable

Few definitions
λx.λy.(x y x) ≡ λz.λw.(z w z)
alpha equivalence

Few definitions
λx.M ≡ λy.M {y/x} alpha conversion

Few definitions
λx.(x y)
free variable

Beta reduction
● Applying argument to a function.

Beta reduction
● Applying argument to a function.
● Consider it as a single computation step.

Beta reduction
(λx.N) M ➝ N{M/x}

Beta reduction - example
(λx.λy.x) z w

(λx.λy.x) z w ➝
(λy.z) w

(λx.λy.x) z w ➝
(λy.z) w ➝
z

Is that it?
L, M, N :: = x
λx.M
M N
(λx.N) M ➝ N{M/x}

Encodings
● support for multi argument functions
λ(x,y).M

Encodings
λ(x,y).M ≡ λx.λy.M

Encodings
○ Moses Schönfinkel

Encodings
○ Haskell Curry

Encodings
● boolean values

Encodings
● boolean values
True = λt.λf.t
False = λt.λf.f

Encodings
● boolean values
True = λt.λf.t
False = λt.λf.f
If = λl.λm.λn l m n
And = λb.λc. b c False

Encodings
● boolean values
● Church numerals

Is there a
calculus for
distributed
computing?

Is there a calculus for distributed computing?
“The inevitability of the lambda-calculus arises
from the fact that the only way to observe a
functional computation is to watch which output
values it yields when presented with different
input values” [1]

Is there a calculus for distributed computing?
“Unfortunately, the world of concurrent
computation is not so orderly. Different notions of
what can be observed may be appropriate in
different circumstances, giving rise to different
definitions of when two concurrent systems have
‘the same behavior’ ” [1]

CCS & CSP
CCS - Calculus of Communicating Systems [3]
CSP - Communicating Sequential Processes [4]

What is π-calculus
“π -calculus is a model of computation for
concurrent systems.” [2]

What is π-calculus
“In lambda-calculus everything is a function (...)
In the pi-calculus every expression denotes a
process - a free-standing computational activity,
running in parallel with other process.” [1]

What is π-calculus
“Two processes can interact by exchanging a
message on a channel” [1]

What is π-calculus
“It lets you represent processes, parallel
composition of processes, synchronous
communication between processes through
channels, creation of fresh channels, replication of
processes, and nondeterminism.” [2]

Syntax
P, Q, R :: = 0 inert process
x(y).P input prefix
xy.P output prefix
P | Q parallel composition
(νx)P restriction
!P replication

Input and output example
Model a program that runs single process P, which
sends message hello on channel x and then
receives a message msg on the same channel x.

P :: =x(hello).x(msg).0

P :: = x(hello).x(msg).0
P

P
xhello

P x
hello

P x
msg

P
xhello
Waits for message ‘msg’ on channel
‘x’, before continuing.

P
xhello
This is a synchronous call.
It waits until somebody receives it.

How to pronounce ν?
lower-case: ν
sound: Nee
greek name: Νι

(νx)P restriction bounds a variable
P ::= yx.0

P ::= yx.0
Q ::= (νx)P

P ::= yx.0
Q ::= (νx)P →
(νx)(yx.0)

(νx)yx.0 == (νz)yz.0

Input and output example revisited
P :: = x(hello).0
Q :: =x(msg).0
R :: = νx(P | Q).0

P :: = x(hello)
Q :: =x(msg)
R :: = νx(P | Q)

R
P :: = x(hello)
Q :: =x(msg)
R :: = νx(P | Q)

R
P :: = x(hello)
Q :: =x(msg)
R :: = νx(P | Q)
P Q

R
P :: = x(hello)
Q :: =x(msg)
R :: = νx(P | Q)
P Qx

R
P :: = x(hello)
Q :: =x(msg)
R :: = νx(P | Q)
P Qx
hello

R
P :: = x(hello)
Q :: = x(msg)
R :: = νx(P | Q)
Q
msg

R
P :: = x(hello)
Q :: = x(msg)
R :: = νx(P | Q)

P :: = x(hello)
Q :: = x(msg)
R :: = νx(P | Q)

Structural Congruence
“Two processes are structurally congruent, if they
are identical up to structure.” [5]

P | Q ≡ Q | P commutativity of parallel composition

(P | Q) | R ≡ P | (Q | R) associativity of parallel composition

((νx)P) | Q ≡ (νx)(P | Q) “scope extrusion”

!P ≡ P | !P replication

(νx)(νy)P ≡ (νy)(νx)P restriction

Reduction rules ⟶
Think of it as a operational semantics.
P ⟶ P’ represents a single computation step

Reduction rules ⟶
xy.P | x(z).Q communication

Reduction rules ⟶
xy.P | x(z).Q → P | [y/z]Q communication

Reduction rules ⟶
P | R → if P → Q reduction under |

Reduction rules ⟶
P | R → Q | R if P → Q reduction under |

Reduction rules ⟶
(νx)P → if P → Q reduction under ν

Reduction rules ⟶
(νx)P → (νx)Q if P → Q reduction under ν

Reduction rules ⟶
P → if P ≡ P’ Q’ ≡ Q structural congruence

Reduction rules ⟶
P → if P ≡ P’ → Q’ ≡ Q structural congruence

Reduction rules ⟶
P → Q if P ≡ P’ → Q’ ≡ Q structural congruence

Some examples(1) - PingPong
PING ::= x(ping).x(pong)
PONG ::= x(ping).x(pong)
P ::= PING | PONG

P ::= x(ping).x(pong) | x(ping).x(pong)

P ::= x(pong) | x(pong)

P ::= 0 | 0

P ::= 0 | 0
P | 0 ≡ P missing equivalence?

P ::= 0 | 0
P | 0 ≡ P wikipedia equivalence

P ::= 0

Some examples(0) - Sending channels
P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= (νx)(P | Q)

P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= (νx)((νy)xy.yw.yz | x(y).y(h).y(h))

P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= (νx)((νy)yw.yz | y(h).y(h))

P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= (νx)((νy)yz | y(h))

P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= (νx)((νy) 0 | 0)

P ::= (νy)xy.yw.yz
Q ::= x(y).y(h).y(h)
R ::= 0

P ::= PING | PONG | PONG

P ::= x(ping).x(pong) | x(ping).x(pong) | x(ping).x(pong)

P ::=x(pong) | x(ping).x(pong) |x(pong)

P ::= 0 | x(ping).x(pong) | 0

P ::= PING | PONG

P ::= !PING | !PONG

P ::= PING | !PING | PONG | !PONG

P ::= 0 | !PING | 0 | !PONG

P ::= 0 | 0 | !PING | 0 | 0 | !PONG

Some examples(4) - Race Conditions
P ::= xy | xz | x(w).w(v)

P ::= xy | xz | x(w).w(v)
P ::= 0 | xz | y(v)

P ::= xy | xz | x(w).w(v)
P ::= 0 | xz | y(v) P ::= xy | 0 | z(v)

P ::= xy | xz | x(w).w(v)
P ::= xz | y(v) P ::= xy | z(v)

Two processes P and Q are bisimilar if ...
“Two processes P and Q are bisimilar if every
action of one can be matched by a corresponding
action of the other to reach bisimilar state” [1]

Benefits
● reason about computation
● detect dead-locks
● detect non-determinism
● reason about structure
● good to describe protocols
● a formal framework for providing semantics for
a high-level language

Typed channels
data SendPort a
data ReceivePort a

Typed channels
data SendPort a
data ReceivePort a
newChan :: Serializable a => Process (SendPort a, ReceivePort a)

Typed channels
data SendPort a
data ReceivePort a
newChan :: Serializable a => Process (SendPort a, ReceivePort a)
sendChan :: Serializable a => SendPort a -> a -> Process ()
receiveChan :: Serializable a => ReceivePort a -> Process a

Example
MASTER ::= (νx)(cx.cx.(PING | PONG).c(done)
PING ::= c(x).x(ping).x(pong).x(done)
PONG::= c(x).x(ping).x(pong)

Implementation issue...
“This seems quite natural.(...).But there’s a big problem here.
ReceivePorts are not Serializable, which prevents us passing
the ReceivePort r1 to the spawned process. GHC will reject the
program with a type error.” [8]

“Why are ReceivePorts not Serializable? If you think about it a
bit, this makes a lot of sense. If a process were allowed to send
a ReceivePort somewhere else, the implementation would have
to deal with two things: routing messages to the correct desti‐
nation when a ReceivePort has been forwarded (possibly
multiple times), and routing messages to multiple destinations,
because sending a ReceivePort would create a new copy.” [8]

“This would introduce a vast amount of complexity to the
implementation, and it is not at all clear that it is a good feature
to allow. So the remote framework explicitly disallows it,
which fortunately can be done using Haskell’s type system.”
[8]

Issues
● synchronous by nature
● receiving on send channel - implementation
dilemmas
● notion of creating a named channel

Chemical State Machine
“The chemical abstract machine” G. Berry, G. Boudl

Sync π syntax
x(y).P input prefix
xy.P output prefix
(νx)P restriction
!P replication

Async π syntax
x(y).P input prefix
xy.P output prefix
(νx)P restriction
!P replication

Async π syntax
x(y).P input prefix
xy output prefix
(νx)P restriction
!P replication

Benefits
● almost all expressive power of classical
pi-calculus
● models asynchronous computation

Issues
● Is not a closed theory
● Heating / cooling rules seem like an overkill
● Not as powerful as synchronous version
○ “Comparing the expressive power of the synchronous
and asynchronous pi-calculus” Catuscia Palamidessi

ρ-calculus
“The π-calculus is not a closed theory, but rather
a theory dependent upon some theory of names.
(...) names may be tcp/ip ports or urls or object
references, etc. But, foundationally, one might ask
if there is a closed theory of processes, i.e. one in
which the theory of names arises from and is
wholly determined by the theory of processes.” [7]

Quoting
“Here we present a theory of an asynchronous
message-passing calculus built on a notion of
quoting. Names are quoted processes, and as such
represent the code of a process.(...) Name-passing,
then becomes a way of passing the code of a
process as a message.” [7]

Syntax
P, Q ::= 0 inert process
x(y).P input
⌝x⌜ drop
x⦉P⦊ lift
P | Q parallel
x, y ::= ⌜P⌝ quote

x⦉P⦊
“Process P will be packaged up as its code, ⌜P⌝,
and ultimately made available as an output at the
port x” [7]

x⦉P⦊
“Process P will be packaged up as its code, ⌜P⌝,
and ultimately made available as an output at the
port x” [7]
“The lift operator turns out to play a role
analogous to (νx)”

⌝x⌜
“The ⌝x⌜ operator (...) eventually extracts the
process from a name. We say ‘eventually’ because
this extraction only happens when quoted process
is substituted into this expression.” [7]

⌝x⌜
“A consequence of this behaviour is that the ⌝x⌜ is
inert, except under an input prefix” [7]

x(y) - syntactic sugar
x(y) ≜ x⦉⌝y⌜⦊

How to create a single name even?
P, Q ::= 0 inert process
x(y).P input
⌝x⌜ drop
x⦉P⦊ lift
P | Q parallel
x, y ::= ⌜P⌝ quote

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
x(y) output

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝ // ⌜x(x)⌝
x(y) output

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
z ::= ⌜⌜0⌝(⌜0⌝).0⌝
x(y).P input

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
z ::= ⌜⌜0⌝(⌜0⌝).0⌝ // ⌜x(x).0⌝
x(y).P input

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
z ::= ⌜⌜0⌝(⌜0⌝).0⌝

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
z ::= ⌜⌜0⌝(⌜0⌝).0⌝
q ::= ⌜0 | 0 ⌝

Name game!
x ::= ⌜0⌝
y ::= ⌜⌜0⌝(⌜0⌝)⌝
z ::= ⌜⌜0⌝(⌜0⌝).0⌝
q ::= ⌜0 | 0 ⌝
p ::= ⌜0 | 0 | 0 ⌝

Are those different names?
x ::= ⌜0⌝
q ::= ⌜0 | 0 ⌝
p ::= ⌜0 | 0 | 0 ⌝

Are those different names?
“This question leads to several intriguing and
apparently fundamental questions. Firstly, if
names have structure, what is a reasonable notion
of equality on names? How much computation, and
of what kind, should go into ascertaining equality
on names?” [7]

Structural congruence
P | 0 ≡ P ≡ 0 | P
P | Q ≡ Q | P
(P | Q) | R ≡ P | (Q | R)

alpha-equivalence
x(z).w⦉y(z)⦊
x(v).w⦉y(v)⦊

alpha-equivalence
x(z).w⦉y(z)⦊
x(v).w⦉y(v)⦊
x(v).w⦉y(v)⦊ {z/v}

Name equivalence
⌜⌝x⌜⌝ ≡ x quote-drop
P ≡ Q ➝ ⌝P⌜ ≡ ⌝Q⌜ struct-equiv

Wait, what?
⌜⌝x⌜⌝ ≡ x
P ≡ Q ➝ ⌝P⌜ ≡ ⌝Q⌜
P | 0 ≡ P ≡ 0 | P
P | Q ≡ Q | P
(P | Q) | R ≡ P | (Q | R)

“If you made them and they made you...”
⌜⌝x⌜⌝ ≡ x
P ≡ Q ➝ ⌝P⌜ ≡ ⌝Q⌜
P | 0 ≡ P ≡ 0 | P
P | Q ≡ Q | P
(P | Q) | R ≡ P | (Q | R)

It all works out...
⌜⌝x⌜⌝ ≡ x
P ≡ Q ➝ ⌝P⌜ ≡ ⌝Q⌜
P | 0 ≡ P ≡ 0 | P
P | Q ≡ Q | P
(P | Q) | R ≡ P | (Q | R)

Operational Semantics
x0
≡ x1
then x0
⦉(Q)⦊ | x1
(y).P ➝ P {⌝Q⌜/y}

x0
≡ x1
then x0
⦉(Q)⦊ | x1
(y).P ➝ P {⌝Q⌜/y}
P ➝ P’ then P | Q ➝ P’ | Q

x0
≡ x1
then x0
⦉(Q)⦊ | x1
(y).P ➝ P {⌝Q⌜/y}
P ➝ P’ then P | Q ➝ P’ | Q
P ≡ P’ and Q ≡ Q’ and Q’ ➝ P’ then P ➝ Q

x(y) - syntactic sugar proof!
x(z) | x(y).P

x(z) | x(y).P
P {z/y}

x(z) | x(y).P ➝
x⦉⌝z⌜⦊ | x(y).P

x(z) | x(y).P ➝
x⦉⌝z⌜⦊ | x(y).P ➝
P {⌜⌝z⌜⌝/y}

x(z) | x(y).P ➝
x⦉⌝z⌜⦊ | x(y).P ➝
P {⌜⌝z⌜⌝/y} ≡
P {z/y}

● Bisimulation
What we have not covered?

References
1 “Foundational Calculi for Programming Languages” Benjamin C. Pierce
2 "FAQ on π-Calculus" Jeannette M. Wing
3 “A Calculus of Communicating Systems” Robin Milner
4 “Communicating Sequential Processes” C.A.R Hoare
5 https://en.wikipedia.org/wiki/%CE%A0-calculus
6 “The Polyadic Pi-Calculus: a Tutorial” Robin Milner
7 “A Reflective Higher-Ordered Calculus” L.G. Meredith, Matthias Radestock
8 “Parallel and Concurrent Programming in Haskell: Techniques for Multicore”

Pawel Szulc@rabbitonweb
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Understanding distributed calculi in Haskell

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