DistributedAlgorithms computa@onandmessagedelivery Asynchronous - PDF document

3/4/09
 Distributed Asynchronous
Systems
 Bertinoro, March 2009 Algorithms Part II 2
 • Synchronous
 – lock‐step
synchrony,
rounds
 Distributed
Algorithms
 • computa@on
and
message
delivery
 • Asynchronous
 PART II 
 – we
lose
the
no@on
of
“@me”
 ASYNCHRONOUS
 • processes
can
take
any
amount
of
@me
in
performing
 computa@on
steps
 SYSTEMS 
 • channels
can
take
any
amount
of
@me
in
delivering
a
 message
 • Difficult
to
deal
with
 – many
impossibility
results
 1
 Distributed Asynchronous
Model
 Bertinoro, March 2009 Distributed I/O
Automata
Model
 Bertinoro, March 2009 Algorithms Algorithms Part II 3
 Part II 4
 • Every
component
of
a
distributed
system
is
 • I/O
automaton
model
 described
by
means
of
an
automaton
 – a
very
general
model
for
describing
asynchronous
 systems
 State
(variables)
 a=1
 send( m ) i,j init( v ) i msg=“Hello”
 • Modular
descrip@on
 recv( m ) j,i – composi@on
opera@on
 Transi:ons
(ac:ons)
 decide( v ) i Send(msg)
 • can
combine
automata
to
create
larger
automaton
 
Precondi@on:
a<5
 
Effect:
a:=a+1
 • Invariant
asser@ons
proofs
 • I/O
 – automaton
has
input
and
output
ac@ons
 – but
also
internal
ac@ons
 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Algorithms Algorithms Part II 5
 Part II 6
 • Formally,
an
IOA
 A 
consists
of
5
components
 Channel i,j 
automaton
 – sig ( A ),
set
of
ac@ons
(input,
output,
internal)
 Signature:
 – states ( A )
 Input: 
 
 
 
 
 
 
 
Output:
 
 send ( m ) i,j m ∈ M receive ( m ) i,j m ∈ 
 M • a
(not
necessarily
finite)
set
of
states
 – start ( A )
 States:
 queue, a
FIFO
queue
of
elements
of M ,
ini@ally
empty . • a
nonempty
subset
of
 states ( A )
 – trans ( A ),
a
state‐transi@on
func@on
 Transi:ons: 

 • trans( A )
 ⊆ 
states( A )
 × 
 acts ( sig ( A ))
 × 
 states ( A )
 send ( m ) i,j receive ( m ) i,j Ef Effect: fect: add
 m 
to
 queue Pr Precondition econdition: 
 m 
is
first
on
 queue • ∀ 
state
 s ,
input
ac@on
 π ,
 ∃ 
state
 s ’
s.t.
( s ,
 π ,
 s ’)
 ∈ 
 Ef Effect: fect: 
 remove
first
element
of
 queue trans ( A )
 Tasks:
 – tasks ( A )
 { receive(m) i,j : m ∈ 
 M } • equivalence
rela@on
on
“locally
controlled”
ac@ons
of
A
 1


3/4/09
 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Algorithms Algorithms Part II 7
 Part II 8
 • Execu@on
 • Opera@ons
on
automata
 – s 0 ,
 π 1 , s 1 ,
 π 2 ,…, s r ,
 π r ,…
 – Composi@on
 • ( s k ,
 π k , s k+1 )
is
a
transi@on
of
A
 – Hiding
 • s 0
 is
a
start
state
of
A
 • Composi@on
 • Examples
of
execu@ons
for
Channel i,j – A
complex
automaton
described
by
smaller
automata
 • each
describing
a
simpler
piece
of
the
system
 – Smaller
automata
combined
together
 α 1 
=
[ λ ],
send(1) i,j ,[1],
receive(1) i,j ,[ λ ],
send(2) i,j ,[2],
receive(2) i,j ,[ λ ]
 – Ac@ons
with
same
name
executed
together
 α 2 
=
[ λ ],
send(1) i,j ,[1],
receive(1) i,j ,[ λ ],
send(2) i,j ,[2] 
 • only
one
automaton
has
“control”
of
the
ac@on
 • INPUT
ac@on
of
one
automaton
gets
executed
when
the
 OUTPUT
ac@on
with
the
same
name
of
another
automaton
 α 3 
=
[ λ ],
send(1) i,j ,[1],
send(1) i,j ,[11],
send(1) i,j ,[111],
… 
 gets
executed
 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Algorithms Algorithms Part II 9
 Part II 10
 • Now
we
can
compose
Channel i,j and p i 
for
every
 i,j .
 Process i 
automaton
 – when
 p 1 
executes
send( m ) 1,2 ,
channel
C 1,2 
executes
send( m ) 1,2.
 – when
channel
C 1,2 
executes
receive( m ) 1,2 


 p 2 
executes
 Signature:
 Input: 
 
 
 
 
 
 
 
Output:
 receive( m ) 1,2 

 
 init ( v ) i v ∈ V decide ( v ) i, v ∈ 
 V receive ( m ) j,i m ∈ 
 M, j ≠ i send ( m ) i,j m ∈ 
 M, j ≠ i States:
 • Internal
ac@ons
 val, a
vector
indexed
by
{1,2,…, n } of
elements
in
V
 ∪ 
{ ⊥ },
all
ini@ally
 ⊥ – are
unobservable
externally,
so
they
are
local
and
cannot
be
 involved
in
the
composi@on
 Transi:ons: 

 init ( v ) i, , v ∈ V receive ( v ) J,i, , v ∈ V • to
compose
two
automata,
the
two
sets
of
internal
ac@ons
must
be
 fect: val [ i ]= v Ef fect: 
 val [ j ]= v disjoint
 Ef Effect: Effect: • Output
ac@ons
 send ( v ) i,j , v ∈ V decide ( v ) i , v ∈ V – also
disjoint
 econd: 
 val [ i ]= v econd: ∀ j =1,…,n,
 val [ j ]≠ ⊥ Pr Precond Pr Precond Ef Effect: fect: 
 none 
 
 
 
 
 
 




 
 



 v = f ( val [1],…, val [ n ])
 • otherwise
two
automata
“control”
the
same
ac@on;
an
output
ac@on
 
 
 
 
 
 
 
 
 




 Ef fect: none 
 Effect: can
correspond
to
one
(or
more)
input
ac@ons
is
other
automata.
 • Input
ac@ons
 Tasks:
 – can
be
executed
at
any
@me
 ∀ j ≠ i :
 { send ( v ) i,j, : v ∈ V }






{ receive ( v ) i,j, : v ∈ V } Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Distributed I/O
Automaton
Model
 Bertinoro, March 2009 Algorithms Algorithms Part II 11
 Part II 12
 init ( v ) 1
 decide ( v ) 1
 p 1
 send ( m ) 1,2
 receive ( m ) 3,1
 receive ( m ) 2,1
 send ( m ) 1,3
 p 1
 C 3,1
 C 1,2
 C 1,2
 C 1,3
 C 3,1
 C 2,1
 C 2,1
 1,3
 C init ( v ) 3
 decide ( v ) 2
 receive ( m ) 1,2
 send ( m ) 3,1
 C 3,2
 p 3
 p 2
 receive ( m ) 1,3
 init ( v ) 2
 C 2,3
 decide ( v ) 3
 send ( m ) 2,1
 • Hiding
 receive ( m ) 3,2
 C 3,2
 send ( m ) 3,2
 p 3
 p 2
 – we
are
“hiding”
the
“output”
ac@ons
send
and
receive
 C 2,3
 – those
ac@ons
are
reclassified
as
internal
to
A
 send ( m ) 2,3
 receive ( m ) 2,3
 2