SLIDE 1 Theoretical Foundations of the UML
Lecture 7: Communicating Finite-State Machines Joost-Pieter Katoen
Lehrstuhl für Informatik 2 Software Modeling and Verification Group
moves.rwth-aachen.de/teaching/ss-20/fuml/
May 11, 2020
Joost-Pieter Katoen Theoretical Foundations of the UML 1/21
SLIDE 2 Outline
1 Introduction 2 Communicating Finite-State Machines 3 Semantics of Communicating Finite-State Machines 4 Emptiness Problem for CFMs
Joost-Pieter Katoen Theoretical Foundations of the UML 2/21
SLIDE 3 Overview
1 Introduction 2 Communicating Finite-State Machines 3 Semantics of Communicating Finite-State Machines 4 Emptiness Problem for CFMs
Joost-Pieter Katoen Theoretical Foundations of the UML 3/21
SLIDE 4 Specification to implementation
Joost-Pieter Katoen Theoretical Foundations of the UML 4/21
(c)
MSGS
as complete
Specification
. . s
a
system
.
MSG
g
,
L ( g)
=
set
Mscs
←
set
possible
scenarios
countably
infinite
(
e.g
.
CMSG
for the
Hanna kakis
example
)
Central
question
i can we
a system "
realisation
"
exhibits all
possible
scenarios in
L (g) First
question
:
how do such
system
"
realisation
, "
look like ?
the
behavior
each process
by
a
finite automaton
C
blood
"
automaton
)
can
communicate
via unbounded Fifo
channels
SLIDE 5 Specification to implementation
Consider an MSGs as complete system specifications
they describe a full set of possible system scenarios
Joost-Pieter Katoen Theoretical Foundations of the UML 4/21 L ( G )
=
set
all
possible
scenarios
SLIDE 6 Specification to implementation
Consider an MSGs as complete system specifications
they describe a full set of possible system scenarios
Can we obtain “realisations“ that exhibit precisely these scenarios?
Joost-Pieter Katoen Theoretical Foundations of the UML 4/21
question
in
the
next 3-4
lectures
SLIDE 7 Specification to implementation
Consider an MSGs as complete system specifications
they describe a full set of possible system scenarios
Can we obtain “realisations“ that exhibit precisely these scenarios? Map MSGs, i.e., scenarios onto an executable model
model each process by a finite-state automaton that communicate via unbounded directed FIFO channels
Joost-Pieter Katoen Theoretical Foundations of the UML 4/21
k
"
p→q
MSG
1-7
communicating
finite-state
a ,b
machine
(
CFM
)
p→
→
a
SLIDE 8 Specification to implementation
Consider an MSGs as complete system specifications
they describe a full set of possible system scenarios
Can we obtain “realisations“ that exhibit precisely these scenarios? Map MSGs, i.e., scenarios onto an executable model
model each process by a finite-state automaton that communicate via unbounded directed FIFO channels
⇒ This yields Communicating Finite-state Machines
Joost-Pieter Katoen Theoretical Foundations of the UML 4/21
Brand
&
Zafiropou
lov
SLIDE 9 Intuition
Joost-Pieter Katoen Theoretical Foundations of the UML 5/21 process p
"
realisation
"
process I
Example
a
! Cpi e.
a ) →
AOD
?Cqp,a )
"
local
"
automaton
p
"
local
"
automaton
I
global
initial stele
=
( n
,
A )
global
final States
=
{
(
a A ) }
CMSC p 9- CFM
possible
behavior
accepts
if
the
CFM
: a
→
①
all
channels p
a are empty
e a .
I
a -→
②
we are
in g-
→
p
used
M
state
G.
A )
SLIDE 10 Eixample
process p
process
I
→ .
I
?a
tb
③ ②
global
initial state
=
(
n
,
A )
global
Arial
states
=
{
( 2,13 ) }
p
p
I
ae
.
LE
.
b
.
8.
c
>
is
"
accepted
"
by
MSC
the
example
CFM
(
Yannakakis
example
)
SLIDE 11 The need for synchronisation messages
Joost-Pieter Katoen Theoretical Foundations of the UML 6/21
Suppose
we
want
to
realise
:-p
I
p I
# µ
find
states
{ ( 2. B) ( 3.
c))
I
I
D
A
Cfm
:
Process
p
y
apgcessqcE7ieh@oToatoE.o
Otto
7 3
B A C
SLIDE 12 ⇐
p
I p 9-
left
n
→
←
b
b
Process
p
informs
process
q
whether
to go
"
left
"
" right "
Automaton
for
process
p
:
Fe
{ (
0,0 )
, ( 0,0 ))
⇐
⇐
do
For
process I
L
A
deadlock
like
in
the
previous example
cannot
Occur
SLIDE 13 Overview
1 Introduction 2 Communicating Finite-State Machines 3 Semantics of Communicating Finite-State Machines 4 Emptiness Problem for CFMs
Joost-Pieter Katoen Theoretical Foundations of the UML 7/21
SLIDE 14 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
(
a
SLIDE 15 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Definition (communication actions, channels)
Act!
p := {!(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of send actions by process p
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
SLIDE 16 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Definition (communication actions, channels)
Act!
p := {!(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of send actions by process p Act?
p := {?(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of receive actions by process p
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
SLIDE 17 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Definition (communication actions, channels)
Act!
p := {!(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of send actions by process p Act?
p := {?(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of receive actions by process p Actp := Act!
p ∪ Act? p
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
SLIDE 18 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Definition (communication actions, channels)
Act!
p := {!(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of send actions by process p Act?
p := {?(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of receive actions by process p Actp := Act!
p ∪ Act? p
Act :=
p∈P Actp
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
SLIDE 19 Preliminaries
Definition
Let P be a finite set of at least two (sequential) processes C be a finite set of message contents
Definition (communication actions, channels)
Act!
p := {!(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of send actions by process p Act?
p := {?(p, q, a) | q ∈ P \ {p}, a ∈ C}
the set of receive actions by process p Actp := Act!
p ∪ Act? p
Act :=
p∈P Actp
Ch := {(p, q) | p, q ∈ P, p = q} “channels“
Joost-Pieter Katoen Theoretical Foundations of the UML 8/21
( P
.
I )
( a
, p)
SLIDE 20 Communicating finite-state machines
Definition
A communicating finite-state machine (CFM) over P and C is a structure A = (((Sp, ∆p))p∈P, D, sinit, F) where
Joost-Pieter Katoen Theoretical Foundations of the UML 9/21 Sef
→
global
Anil
steles
local
"
↳
global
initial
automaton
state
£719
" (
synchronisation
messages ( e.g
.
left
,
right )
SLIDE 21 Communicating finite-state machines
Definition
A communicating finite-state machine (CFM) over P and C is a structure A = (((Sp, ∆p))p∈P, D, sinit, F) where D is a nonempty finite set of synchronization messages (or data) We often write s
σ,m
− →p s′ instead of (s, σ, m, s′) ∈ ∆p
Joost-Pieter Katoen Theoretical Foundations of the UML 9/21
e- g
.
, right
SLIDE 22 Communicating finite-state machines
Definition
A communicating finite-state machine (CFM) over P and C is a structure A = (((Sp, ∆p))p∈P, D, sinit, F) where D is a nonempty finite set of synchronization messages (or data) for each p ∈ P:
Sp is a non-empty finite set of local states (the Sp are disjoint) ∆p ⊆ Sp × Actp × D × Sp is a set of local transitions
We often write s
σ,m
− →p s′ instead of (s, σ, m, s′) ∈ ∆p
Joost-Pieter Katoen Theoretical Foundations of the UML 9/21
s
,
'
. Cp
. e.
a )
,
d
,
s
' )
e
Dp Og
SLIDE 23 Communicating finite-state machines
Definition
A communicating finite-state machine (CFM) over P and C is a structure A = (((Sp, ∆p))p∈P, D, sinit, F) where D is a nonempty finite set of synchronization messages (or data) for each p ∈ P:
Sp is a non-empty finite set of local states (the Sp are disjoint) ∆p ⊆ Sp × Actp × D × Sp is a set of local transitions
sinit ∈ SA is the global initial state
where SA :=
p∈P Sp is the set of global states of A
We often write s
σ,m
− →p s′ instead of (s, σ, m, s′) ∈ ∆p
Joost-Pieter Katoen Theoretical Foundations of the UML 9/21 P ,
I
,
r
( p
,
I
,
r )
SLIDE 24 Communicating finite-state machines
Definition
A communicating finite-state machine (CFM) over P and C is a structure A = (((Sp, ∆p))p∈P, D, sinit, F) where D is a nonempty finite set of synchronization messages (or data) for each p ∈ P:
Sp is a non-empty finite set of local states (the Sp are disjoint) ∆p ⊆ Sp × Actp × D × Sp is a set of local transitions
sinit ∈ SA is the global initial state
where SA :=
p∈P Sp is the set of global states of A
F ⊆ SA is the set of global final states We often write s
σ,m
− →p s′ instead of (s, σ, m, s′) ∈ ∆p
Joost-Pieter Katoen Theoretical Foundations of the UML 9/21
SLIDE 25 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over P = {1, 2} and C = {req, ack}
D = { , , } S1 = {s0, s1, s2} S2 = {t0, t1, t2} ∆1: s0
!(1,2, req )
− − − − − − →1 s0 ... ∆2: t0
?(2,1, req )
− − − − − − →2 t1 ... sinit = (s0, t0) F = {(s2, t2)}
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
e
2
SLIDE 26 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
. . . . . .
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21 1
2
←
←
1-32
2-21
SLIDE 27 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req
. . . . . .
!(1, 2, req)
!(1, 2, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21 CMS
C
C
O
linearis
atsuis
O
SLIDE 28 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req req
. . . . . .
!(1, 2, req) !(1, 2, req)
!(1, 2, req) !(1, 2, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
c
'
:
.
O
←
SLIDE 29 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
so
x
SLIDE 30 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req ack
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
j
←
O
SLIDE 31 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
ack
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
f
O
X
SLIDE 32 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
ack ack
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
Oo
I
°
SLIDE 33 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
ack
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
j
O
→
SLIDE 34 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req ack
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
x
SLIDE 35 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
T
SLIDE 36 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req req
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req)
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
SLIDE 37 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
req
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req) ?(2, 1, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req) ?(2
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
±
x
=
SLIDE 38 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
. . . . . .
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req) ?(2, 1, req) ?(2, 1, req)
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req) ?(2
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
.
X
SLIDE 39 Communicating finite-state machines
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
. . . . . . 1 2
req req ack ack req req
!(1, 2, req) !(1, 2, req) ?(2, 1, req) !(2, 1, ack) ?(2, 1, req) !(2, 1, ack) ?(1, 2, ack) !(1, 2, req) ?(1, 2, ack) !(1, 2, req) ?(2
Joost-Pieter Katoen Theoretical Foundations of the UML 10/21
M M
is
accepted
by
this
CFM
.
SLIDE 40 Overview
1 Introduction 2 Communicating Finite-State Machines 3 Semantics of Communicating Finite-State Machines 4 Emptiness Problem for CFMs
Joost-Pieter Katoen Theoretical Foundations of the UML 11/21
SLIDE 41 Formal semantics of CFMs
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (configurations)
Configurations of A: ConfA := SA × {η | η : Ch → (C × D)∗}
Joost-Pieter Katoen Theoretical Foundations of the UML 12/21
=
[
"
the
content
all
global
state channels
the
Local
(
= a
state for
Cfm
"
every
process
p )
2
:
Ch
→
(
c
×
D)
*
2 ( Crs ))
=
E
r ( Cp , s ) )
=
( a
,
, )
SLIDE 42 Formal semantics of CFMs
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (configurations)
Configurations of A: ConfA := SA × {η | η : Ch → (C × D)∗}
Definition (global step)
= ⇒
A ⊆ ConfA × Act × D × ConfA is defined as follows:
Joost-Pieter Katoen Theoretical Foundations of the UML 12/21
(
Tpp
the
Csi
,
.
she)
,
e )
(
Cs
,
!
,
ri )
(
ch
→
C Cx D)£
? Csi , sj
,
a )
,
m
SLIDE 43 Formal semantics of CFMs
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (configurations)
Configurations of A: ConfA := SA × {η | η : Ch → (C × D)∗}
Definition (global step)
= ⇒
A ⊆ ConfA × Act × D × ConfA is defined as follows:
sending a message: ((s, η), !(p, q, a), m, (s′, η′)) ∈ = ⇒
A if
(s[p], !(p, q, a), m, s′[p]) ∈ ∆p η′ = η[(p, q) := (a, m) · η((p, q))] s[r] = s′[r] for all r ∈ P \ {p}
Joost-Pieter Katoen Theoretical Foundations of the UML 12/21
1PI=k
✓
✓
[
]
(
5=6
, ,
. . . . . .
.sk )
,
r )
::i÷:÷÷l⇐÷÷÷ii
:
SLIDE 44 Formal semantics of CFMs
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (configurations)
Configurations of A: ConfA := SA × {η | η : Ch → (C × D)∗}
Definition (global step)
= ⇒
A ⊆ ConfA × Act × D × ConfA is defined as follows:
sending a message: ((s, η), !(p, q, a), m, (s′, η′)) ∈ = ⇒
A if
(s[p], !(p, q, a), m, s′[p]) ∈ ∆p η′ = η[(p, q) := (a, m) · η((p, q))] s[r] = s′[r] for all r ∈ P \ {p}
receipt of a message: ((s, η), ?(p, q, a), m, (s′, η′)) ∈ = ⇒
A if
(s[p], ?(p, q, a), m, s′[p]) ∈ ∆p η((q, p)) = w · (a, m) = and η′ = η[(q, p) := w] s[r] = s′[r] for all r ∈ P \ {p}
Joost-Pieter Katoen Theoretical Foundations of the UML 12/21
SLIDE 45 ¥
¢
,
n )
anent
configuration
|
? Cp
. e.
a ) ,
m w
I
((
k )
,
hi )
rice ,pD=w
for all
channels
C
⑧
n' Cd
.
red
SLIDE 46 Example
Joost-Pieter Katoen Theoretical Foundations of the UML 13/21
SLIDE 47 E
←
shiny
:
(
cnn.s.ca#ta99-D--
e
=
to
I
, p )) =
E
[/enpty
(( r
, c)
,
le
,
b ))
=
t
,
I empty
, (
a.
b ))
=
tz
II
Vcrs
D=
a
a
2 I (
Ep
))
. .
. .
( I 2,4
, ( age ))
⇐
I C
s
,
c )
,
Caa
, b ))
=L
,
÷
SLIDE 48 Linearizations of a CFM
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (accepting runs)
A run ρ of CFM A on word w = σ1 . . . σn ∈ Act∗ is an alternating sequence ρ = γ0 m1 γ1 . . . γn−1 mn γn such that
1 γ0 = (sinit, ηε) with ηε mapping any channel to ε 2 γi−1
σi,mi
⇒A γi for any i ∈ {1, . . . , n}
Joost-Pieter Katoen Theoretical Foundations of the UML 14/21
y
! C Pie
, a) ,
? Ce
, p , b )
( empty
contest )
SLIDE 49 Linearizations of a CFM
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (accepting runs)
A run ρ of CFM A on word w = σ1 . . . σn ∈ Act∗ is an alternating sequence ρ = γ0 m1 γ1 . . . γn−1 mn γn such that
1 γ0 = (sinit, ηε) with ηε mapping any channel to ε 2 γi−1
σi,mi
⇒A γi for any i ∈ {1, . . . , n} The run ρ is accepting if γn ∈ F × {ηε}.
Joost-Pieter Katoen Theoretical Foundations of the UML 14/21
fn
=
global
final starlet
all
channels
are
.
SLIDE 50 Linearizations of a CFM
Let A = (((Sp, ∆p))p∈P, D, sinit, F) be a CFM over P and C.
Definition (accepting runs)
A run ρ of CFM A on word w = σ1 . . . σn ∈ Act∗ is an alternating sequence ρ = γ0 m1 γ1 . . . γn−1 mn γn such that
1 γ0 = (sinit, ηε) with ηε mapping any channel to ε 2 γi−1
σi,mi
⇒A γi for any i ∈ {1, . . . , n} The run ρ is accepting if γn ∈ F × {ηε}.
Definition (linearization of a CFM)
The (word) language of CFM A is defined by: Lin(A) := {w ∈ Act∗ | there is an accepting run of A on w}
Joost-Pieter Katoen Theoretical Foundations of the UML 14/21
SLIDE 51 Linearizations of an example CFM
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over {1, 2} and {req, ack}
Joost-Pieter Katoen Theoretical Foundations of the UML 15/21
E
I
Jux
SLIDE 52 Linearizations of an example CFM
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over {1, 2} and {req, ack}
Lin(A) =
- w ∈ Act∗ | there is n 1 such that:
w1 = !(1, 2, req))n (?(1, 2, ack) !(1, 2, req))n w2 = (?(2, 1, req) !(2, 1, ack))n (?(2, 1, req))n for any u ∈ Pref (w) and (p, q) ∈ Ch:
|u|!(p,q,a) −
|u|?(q,p,a) 0
Theoretical Foundations of the UML 15/21
n
h
WM
=
the
sequence
actions
in
w
that
n
7
at
process
1 n
w T2 =
.
for
process
2
. .
Cta)
C
tart )
SLIDE 53 Linearizations of an example CFM
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over {1, 2} and {req, ack}
!(1, 2, req) and !(2, 1, ack) are always independent. !(1, 2, req) and ?(1, 2, ack) are always dependent. !(1, 2, req) and ?(2, 1, req) are sometimes independent. non-regular (word) languages
Joost-Pieter Katoen Theoretical Foundations of the UML 15/21
→ more
expressive
than
finite
automata
!
SLIDE 54 Linearizations and MSCs of an example CFM
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over {1, 2} and {req, ack}
Lin(A) =
- w ∈ Act∗ | there is n 1 such that:
w1 = (!(1, 2, req))n (?(1, 2, ack) !(1, 2, req))n w2 = (?(2, 1, req) !(2, 1, ack))n (?(2, 1, req))n for any u ∈ Pref (w) and (p, q) ∈ Ch:
|u|!(p,q,a) −
|u|?(q,p,a) 0
Theoretical Foundations of the UML 16/21
SLIDE 55 Linearizations and MSCs of an example CFM
Example
s0 s1 s2 t0 t1 t2 !(1, 2, req ) ?(2, 1, req ) ?(1, 2, ack) !(2, 1, ack) !(1, 2, req ) ?(1, 2, ack) ?(2, 1, req ) !(2, 1, ack)
CFM A over {1, 2} and {req, ack}
L(A) =
- M ∈ M | there is n ≥ 1 such that:
M 1 = (!(1, 2, req))n (?(1, 2, ack) !(1, 2, req))n M 2 = (?(2, 1, req) !(2, 1, ack))n (?(2, 1, req))n
Joost-Pieter Katoen Theoretical Foundations of the UML 16/21
r
t
MS
Cs
accepted
by
CFM
A
.
SLIDE 56 Overview
1 Introduction 2 Communicating Finite-State Machines 3 Semantics of Communicating Finite-State Machines 4 Emptiness Problem for CFMs
Joost-Pieter Katoen Theoretical Foundations of the UML 17/21
#
CFMS
are more
expressive
than
finite
automata
I
does
a
CFM
accept
at
least
word
?
undecidable
SLIDE 57 Elementary questions are undecidable for CFMs
Emptiness of CFMs is undecidable
[Brand & Zafiropulo 1983]
The following problem is undecidable (even if C is a singleton): Input: CFM A over processes P and message contents C Question: Is L(A) empty?
Joost-Pieter Katoen Theoretical Foundations of the UML 18/21
e.g
.
c-
{
a )
↳
the set
Mscs
accepted
by
CFM
A
€a
,
the
set
linton
's
atoms
accepted
by
Crm
A
.
SLIDE 58 Elementary questions are undecidable for CFMs
Emptiness of CFMs is undecidable
[Brand & Zafiropulo 1983]
The following problem is undecidable (even if C is a singleton): Input: CFM A over processes P and message contents C Question: Is L(A) empty?
Proof (sketch)
Reduction from the halting problem for Turing machine TM = (Q, Σ, ∆, , q0, qf) to emptiness for a CFM with two processes. Build CFM A = ((A1, A2), D, sinit, F) over {1, 2} and some singleton set C such that L(A) = ∅ iff TM can reach qf, i.e., TM accepts. Process 1 sends current configurations to process 2 Process 2 chooses successor configurations and sends them to 1 D =
- (Σ ∪ {}) × (Q ∪ {_})
- ∪ {#}
Joost-Pieter Katoen Theoretical Foundations of the UML 18/21
✓ ✓
I of
the
TM
SLIDE 59 A CFM simulating a Turing machine
Proof (contd.)
a
a q2 b
a′ b
. . .
⊢TM ⊢TM ⊢TM
γ0
γ2 γ3
. . .
γ1 γ2 γ3 ← q0 # ← q1 a # b a ← q2 # b a′ ← q3 # ← q1 a # b a ← q2 # b a′ ← q3 #
Joost-Pieter Katoen Theoretical Foundations of the UML 19/21
Turing
machine
1CFM#
initial conf
.
: : #
SLIDE 60 A CFM simulating a Turing machine
Proof (contd.)
Left or standstill transition: Process 2 may just wait for a symbol containing a state of TM and to alter it correspondingly. In the example, the left-moving transition (q2, a, a′, L, q3) is applied so that process 2
sends b unchanged back to process 1 detects (receives) a ← q2 sends a′ to process 1 entering a state indicating that the symbol to be sent next has to be equipped with q3 receives # so that the symbol ← q3 has to be inserted before returning #
Joost-Pieter Katoen Theoretical Foundations of the UML 20/21
SLIDE 61 A CFM simulating a Turing machine
Proof (contd.)
Left or standstill transition: Process 2 may just wait for a symbol containing a state of TM and to alter it correspondingly. In the example, the left-moving transition (q2, a, a′, L, q3) is applied so that process 2
sends b unchanged back to process 1 detects (receives) a ← q2 sends a′ to process 1 entering a state indicating that the symbol to be sent next has to be equipped with q3 receives # so that the symbol ← q3 has to be inserted before returning #
Right transition: Process 2 has to guess what the position right before the head is. For example, provided process 2 decided in favor of (q2, a, a′, R, q3) while reading b, it would have to
send b ← q3 instead of just b, entering some state t(a ← q2) receive a ← q2 (no other symbol can be received in state t(a ← q2)) send a′ back to process 1
Joost-Pieter Katoen Theoretical Foundations of the UML 20/21
SLIDE 62 Communicating
Finite
Machines
"
system
forming
:::::I
"
(c)
MSG
=
" requirements
"
all
scenarios
a
system
should exhibit
SLIDE 63 A CFM simulating a Turing machine
Proof (contd.)
Introduce local final states sf and tf, one for process 1 and one for process 2, respectively (i.e., F = {(sf, tf)} and A is locally accepting). At any time, process 1 may switch into sf, in which arbitrary and arbitrarily many messages can be received to empty channel (2, 1). Process 2 is allowed to move into tf and to empty the channel (1, 2) as soon as it receives a letter c ← qf for some c. As process 2 modifies a configuration of TM locally, finitely many states are sufficient in A.
Theoretical Foundations of the UML 21/21
