Supervisory Control (4CM30)
Networks of Automata – Formal definitions & Tool support
Michel Reniers
[email protected]
2016-2017
Summary previous lecture
Networks of automata – Introduction
I
Synchronization of events
I
Shared variables
Open automata – motivation
I
reconsider automaton definition:
M = (L, V, E, −→, Lm , l0 , v0 )
I
only caters for local variables
I
extend with external variables (only to be read)
I
disambiguation of variables by prefixing with automaton name
queue1.count < 2
q1enter
queue1.count := queue1.count + 1
queue1.count = 0
queue1.count > 0
q1leave
queue1.count := queue1.count − 1
queue2.count < 2
q2enter
queue2.count := queue2.count + 1
queue2.count = 0
queue2.count > 0
q2leave
queue2.count := queue2.count − 1
Open automata – definition
An open automaton is a 8-tuple (L, V, X, E, −→, Lm , l, v) where
I L is a finite set of locations;
I V is a finite set of local variables;
I X is a finite set of variables (V ⊆ X);
I E is a finite set of events;
I −→ ⊆ L × G(X) × E × U (V, X) × L is the transition relation;
I Lm ⊆ L is a set of marked states;
I l ∈ L is the initial location;
I v : V −→ Λ is the initial valuation of the local variables.
I
I
I
U (V, X) : set of all updates of variables from V where
variables from X may be used in the right-hand sides of the
updates
X \ V : (set of) external variables
open automaton with X = V is just an
automaton (from Chapter 2)
Exercise
Provide the formal notation for the following open automata:
queue1.count < 2
q1enter
queue1.count ≤ queue2.count
queue1.count := queue1.count + 1
q1enter
queue1.count = 0
queue1.count > 0
queue2.count ≤ queue1.count
q2enter
q1leave
queue1.count := queue1.count − 1
Network of automata
Let I be an arbitrary index set.
A network of automata is a collection of open automata
Mi = (Li , Vi , Xi , Ei , −→i , Lmi , li , vi ) for i ∈ I, with
I
pairwise
disjoint
sets of local variables Vi such that
S
S
Vi =
Xi
I
each global / external variable of an open automaton is local
to exactly one of the contained open automata
i∈I
i∈I
Example
count < 2
q1enter
count := count + 1
count = 0
count > 0
q1leave
count := count − 1
count < 2
q2enter
count := count + 1
queue1.count ≤ queue2.count
q1enter
count = 0
count > 0
q2leave
count := count − 1
queue2.count ≤ queue1.count
q2enter
I
automaton queue1 has local variables {queue1 .count} and no
external variables
I
automaton queue2 has local variables {queue2 .count} and no
external variables
I
automaton customer has no local variables and has variables
{queue1 .count, queue2 .count}
I
each external variable is declared by exactly one automaton
Meaning of networks of automata
I
networks of automata for convenience of modelling
I
in principle a single automaton is represented
I
how to obtain a single automaton representing a network of
automata?
I
define synchronous product of two open automata
Synchronous product
Given Mi = (Li , Vi , Xi , Ei , −→i , Lmi , li , vi ): M1 k M2 =
(L, V, X, E, −→, Lm , l, v) where
I
I
L = L1 × L2 V = V1 ∪ V2
−→ is defined as:
I
I
I
X = X1 ∪ X2
E = E1 ∪ E2
if σ ∈ E1 ∩ E2 , then ((s1 , s2 ), g1 ∧ g2 , σ, u1 ⊕ u2 , (t1 , t2 )) ∈−→
for each (s1 , g1 , σ, u1 , t1 ) ∈−→1 and (s2 , g2 , σ, u2 , t2 ) ∈−→2
such that u1 (x) = u2 (x) for all x ∈ dom(v1 ) ∩ dom(v2 )
if σ ∈ E1 \ E2 , then ((s1 , s2 ), g1 , σ, u1 , (t1 , s2 )) ∈−→ for each
(s1 , g1 , σ, u1 , t1 ) ∈−→1
if σ ∈ E2 \ E1 , then ((s1 , s2 ), g2 , σ, u2 , (s1 , t2 )) ∈−→ for each
(s2 , g2 , σ, u2 , t2 ) ∈−→2
I
Lm = Lm1 × Lm2
I
l = (l1 , l2 )
I
v : V →(Λ is defined as follows:
v1 (x), for x ∈ dom(v1 )
v(x) =
v2 (x) otherwise
u1 ⊕ u2 : combination of updates
Example
produce
Producing
provide
Idle
Consuming
Idle
provide
consume
Synchronous product:
Producing
Idle
consume
Producing
Consuming
produce
Idle
Idle
provide
produce
consume
Idle
Consuming
Locations formally: {(Producing, Idle), (Idle, Idle),
(Producing, Consuming), (Idle, Consuming)}
Example with variables
count < 2
q1enter
count := count + 1
queue1.count ≤ queue2.count
q1enter
count = 0
count > 0
q1leave
count := count − 1
queue2.count ≤ queue1.count
q2enter
Synchronous product:
queue1.count < 2 ∧ queue1.count ≤ queue2.count
q1enter
queue1.count := queue1.count + 1
queue1.count = 0
queue2.count ≤ queue1.count
q2enter
queue1.count > 0
q1leave
queue1.count := queue1.count − 1
Tool support in CIF
count < 2
q1enter
count := count + 1
count = 0
count > 0
q1leave
count := count − 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
count < 2
q2enter
count := count + 1
queue1.count ≤ queue2.count
q1enter
count = 0
count > 0
q2leave
count := count − 1
queue2.count ≤ queue1.count
q2enter
event q1enter , q1leave , q2enter , q2leave ;
automaton queue1 :
disc int count = 0;
location l0 :
initial ;
edge q1enter when count < 2 do count := count + 1;
edge q1leave when count > 0 do count := count - 1;
end
automaton queue2 :
disc int count = 0;
location l0 :
initial ;
edge q2enter when count < 2 do count := count + 1;
edge q2leave when count > 0 do count := count - 1;
end
automaton customer :
location l0 :
initial ;
edge q1enter when queue1 . count <= queue2 . count ;
edge q2enter when queue2 . count <= queue1 . count ;
end
Computing synchronous products
For automata without variables:
I
“Apply synchronous product ...” (again the keyword event
may not be used and should be replaced by controllable or
uncontrollable)
For automata with variables the CIF tool set has two functions for
replacing a network of automata with a single automaton. Neither
of these computes the synchronous product as defined in this
chapter.
Guided self-study
Open automata (Section 3.3)
I
Pick some automata from examples and exercises and practice
writing down formal representations
Synchronous composition of automata (Section 3.4)
I
Exercises 3.22, 3.24 - 3.28
Tool support for networks of automata in CIF (Section 3.5)
I
Exercises 3.29 - 3.30
Additional modelling exercises (Section 3.6)
I
Exercises 3.31 - 3.37
I
Finish exercises from previous sections