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Concurrent Object-Oriented
Programming
Arnaud Bailly, Bertrand Meyer and
Volkan Arslan
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Lecture 7:
Modeling with the Pi-Calculus.
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The Pi-calculus





Super-set of CCS.
Allows monadic name communication.
Scope extrusion is fundamental.
Well-developed theory.
Programming languages: PICT, Join, TyCO, …
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Pi-Calculus Syntax
P ::
0
|
( x ) P
| [ x  y ]P
|
 .P
|
|
|
PP
P|P
!P
 We shall omit trailing
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 :: a | a | a (b) | ab | 
0’s, as usual.
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Free and Bound Names
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bn ( x ( y ).P ) { y} bn ( P )
bn (( x ) P ) {x} bn ( P )
bn (0)

bn ( xy.P )
bn ( x .P )
bn ( P  Q )
bn ( P | Q )
bn (! P )
bn ( x.P )
bn ( .P )
bn ( P ) bn (Q )
bn ( P ) bn (Q )
bn ( P )
 Free names are names that are not bound.
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bn ( P )
Substitution
 Behaves similarly to the Lambda-Calculus
substitution:
 replaces the free occurrences of names,
 many names can be substituted simultaneously,
 substitution may involve renaming of bound
names to avoid incidental capture.
 Examples:
(c.( c)ac.c.P )[a / c]  a.( c)ac.c. P
(c.( c)ac.c.P )[c / a ]  a.( d )cd .d .P[d , c / c, a ]
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Pi-Calculus Semantics (1)
( P  a (b).Q ) | ( ac.R  S )  Q[c / b] | R
P  P'
P |Q  P' |Q
P  P'
( x ) P  ( x ) P '
P  Q  Q'  P'
P  P'
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Pi-Calculus Semantics (2)
P0  P
P Q  Q  P
( P  Q )  R  P  (Q  R )
( x)0  0
P |0  P
P |Q  Q | P
( P | Q ) | R  P | (Q | R)
( x )( y ) P  ( y )( x ) P
( x ) P | Q  ( x )( P | Q ) if x  fn(Q )
! P  P |! P
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Example: Scope Extrusion
P  ( c)ac.c.P ', Q  a( x ).bx.Q ', R  b( y ). y .R '
a
P
Q
{c}
{c}
c
b
{c}
R
P | Q | R  ( c)(c.P ' | bc.Q '[c / x ]) | R
 ( c)( c.P ' | Q '[c / x ] | c . R '[c / y ])
 ( c)( P ' | Q '[c / x ] | R '[c / y ])
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Encoding Polyadic Interactions
c (a1,..., an ).P
( x )(cx. xa1. ... . xan.P)
with x  fv ( P, a1,..., an )
c(a1,..., an ).P
c( x ). x(a1). ... . x(an ).Q
with x  fv (Q, a1,..., an )
 We can use these shortcuts transparently.
 We introduce notations:
x
( x1,..., xm) P
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x1,..., xm
( x1)...( xm) P
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Encoding Named Processes
 Defining named processes by adding a context
to (plain) terms.
def
and
and
and
in
K 1( x1)  P1
K 2( x 2 )  P 2
...
Kn ( xn )  Pn
Q
( yK 1,..., yKn )(! yK 1( x1).P1 | ... |! yKn( xn ).Pn | Q ')
where Q ' Q[ yK 1 x1,..., yKn xn / K 1( x1),..., Kn( xn)]
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Named Process Example
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def Get (i, o)  i ( x ).Put (i, o, x )
and Put (i, o, x )  ox.Get (i, o)
in
Get ( p, q)
( xGet , xPut )(
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! xGet (i, o).i ( x ). xPut i o x
|
! xPut (i, o, x ).ox. xGet i o
|
xGet p q
)
The Mobile Phone Example
 Borrowed from Milner(1991).
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The Mobile Phone Example (2)
System
Car(t, s)
( talki, switchi , givei, alerti : i  1, 2)
( Car (talk 1, switch1)
| Base1
| IdleBase2
| Centre1
)
t.Car(t, s)  s(t ', s ').Car(t ', s ')
Centre1
give1talk 2 switch 2.alert 2.Centre 2
Centre2
give2talk 1switch1.alert1.Centre1
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The Mobile Phone Example (3)
Base(t , s, g , a )
Basei
Base(talki, switchi, givei, alerti )
Base(t , g , s, a )
Basei
t.Base(t , s, g , a )
 g (t ' s ').st ' s '. IdleBase(t, s, g , a )
a.Base(t , g , s, a )
IdleBase(talki, switchi, givei, alerti )
 What happens when the CENTRE give the
switching order?
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Encoding Booleans
 Now a process has to provide a port, in order to
receive an answer!
True( r )
False( r )
if r then P else Q
And (b1, b2, r )
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( b)( rb |!b(t , f ).t )
( b)( rb |!b(t , f ). f )
( t , f )(r (b).b (t , f ).(t.P | f .Q ))
if b1 then r b2 else False(r)
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Encoding Numerals
n ( x, z )
x . x . ... . x . z
n times
 Define Copy so that:
( x z )( n ( x, z ) | Copy ( xz, yw))  n ( y, w)
 Solution:
Copy ( xz, yw)
x. y.Copy ( xz, yw)  z.w
 Define Successor so that:
( x z)(n( x, z) | Succ( xz, yw))  n  1( y, w)
 Solution:
Succ( xz, yw)
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y.Copy ( xz, yw)
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Addition and Multiplication
Add ( x1 z1 , x2 z2 , yw)
x1. y . Add ( x1 z1 , x2 z2 , yw)
 z1.Copy ( x2 z2 , yw)
Mult ( x1 z1 , x2 z2 , yw)
( x2 ' z2 ', x2 '' z2 '', xr zr )(
Double( x2 z2 , x2 ' z2 ', x2 '' z2 '') |
Add ( x2 ' z2 ', xr zr , yw) |
Mult ( x1 z1 , x2 '' z2 '', yw)
)
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Duplication & Zero Test
Double( xz, y1w1 , y2 w2 )
( y1 ' w1 ', y2 ' w2 ')(
x.( Succ( y1 ' w1 ', y1w1 ) | Succ( y2 ' w2 ', y2 w2 ) |
Double( xz, y1 ' w1 ', y2 ' w2 '))
 z.( w1 | w2))
Iszero( r, xz, yw)
 Double( xz, yw, y ' w ') | 
( y ' w ') 

(
y
'.
False
(
r
)

w
'.
True
(
r
))


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Encoding Lists
Nil ( r )
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( l )( rl |!l (n, c).n )
Cons(hd , tl , r )
( l )( rl |!l (n, c).c (hd , tl ))
 Define “Pattern Matching”:
match l with Nil  P and Cons(hd , tl )  Q
( n, c)(l (l0 ).l0 (n, c).(n.P  c(hd , tl )Q)
 Define Concatenation:
Concat (l1 , l2 , r )
match l1 with Nil  r l2
and Cons(hd , tl )  ( r ')(Concat (tl , l2 , r ') | r '(l ).Cons(hd , l , r ))
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Encoding The Lambda-Calculus (!)
 x.t
x
tu
p ( x, q). t
p
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q
xp
p
p
( x. x ) z
( q)( t
p
q
| ( y )( qyp |! y ( r ). u r ))
 ( q)(  x. x
q
| ( y )( qyp |! y ( r ). z r ))
 ( q)( q( x, q '). xq ' | ( y )( qyp |! y ( r ). zr ))
 ( q, y )( yp |! y ( r ). zr )
 ( q, y )(! y ( r ). zr ) | zp  zp  z
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p
Encoding State
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 A cell with read and write methods can be
implemented in many ways. First one:
( l )( l i | Servread (l , r ) | Servwrite(l , w))
Cell ( r, w, i )
Servread (l , r ) ! r (c).l (v ).(cv | l v )
Servwrite(l , w) ! w(c, v ').l (v ).(cv | l v ')
 Second solution with exclusive access:
Cell ( r, w, i )
( l )( l i |!( Servread (l , r )  Servwrite(l , w)))
Servread (l , r )
Servwrite(l , w)
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r (c).l (v ).(cv | l v )
w(c, v ').l (v ).(cv | l v ')
Synchrony & Asynchrony
 Pi-Calculus communication is synchronous:
( x )( x ( y ).P | z.Q ) is blocked if x  fn(Q )
 But a message can be represented by:
x ( y ).0
 Example:
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( z.( x ( y ) | Q ) | z . x ( y '). P )
 ( x ( y ) | Q | x ( y '). P )
 ( x ( y ) | Q ' | x ( y '). P )
 (Q ' | P )
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Encoding Objects and Classes
 Both objects and classes correspond to processes.
 A class is an object with method new:
(...  new( r ).( self , m)( r (m).BodyObj )  ...)
 On the self name is first sent the list of method
names m .
 An object is very much like a mutable cell.
 Example:
CCell
new( p ).( cell , r, w)( p ( r, w).Cell ( r, w,0))
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BBuffer using ACT++ Primitives
class BBUFFER is
public interface: … // as before
behavior:
empty
= {put}
partial
= {put, get}
full
= {get}
implementation:
Boolean isFull, isEmpty;
put (t: OBJECT) is …
if (isFull) then become full;
else become partial;
end;
OBJECT: get () is …
if (isEmpty) then become empty;
else become partial;
end;
end BBUFFER;
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Example: The Bounded Buffer (1)
BBuffer ! newBBuffer( r ).( put, get ).r ( put, get ). Init
Init
( l , x, z )( Nil (l ) | n( x, z ) | Empty (l , x, z))
Empty (l , x, z )
Partial (l , x, z )
Put (l , x, z )
Put (l , x, z )  Get (l , x, z )
Full (l , x, z ) Get (l , x, z )
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Example: The Bounded Buffer (2)
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Get (l , x, z )
get ( r ).l (l 0).( n, c)l 0( n, c ).c( hd , tl ).
( match tl with Nil  r (hd ).Empty (tl , x, z )
and Cons( hd ', tl ')  r (hd ).Partial (tl , x, z ))
put ( r ).l (l 0).( n, c)l 0( n, c).c( hd , tl ). x.
Iszero(b, xz, yw) |


( b, yw) if b then r ( hd ).Full (l , y , w)


else r (hd ).Partial (l , y , w)

Put (l , x, z )
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




Problems
 “Faithful” encodings, preserving types.
 Complexity of checking properties.
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Next
 Some specific encodings.
 Verifying properties…
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