From

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From Monotonic Transition Systems to Monotonic
Games
Parosh Aziz Abdulla Uppsala University
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Outline

• Model Checking
• Infinite-State Systems
• Methodology
• Monotonicity
• Well Quasi-Orderings
• Models
• Petri Nets
• Lossy Channel Systems
• Timed Petri Nets
• Extension to Games

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Model Checking
T sat f ?
transition system
specification
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Model Checking
T sat f ?
transition system
specification
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(No Transcript)
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(No Transcript)
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Forward Reachability Analysis
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Forward Reachability Analysis
Post
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Forward Reachability Analysis
Post
Forward Reachability Analysis computing Post
Fin
Init
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Backward Reachability Analysis
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Backward Reachability Analysis
Pre
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Backward Reachability Analysis
Pre
Backward Reachability Analysis computing Pre
Init
Fin
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Forward Reachability Analysis
Fin
Init
Backward Reachability Analysis
Init
Fin
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Infinite-State Systems
1. Unbounded Data Structures

• stacks
• queues
• clocks
• counters, etc.

2. Unbounded Control Structures

• Parameterized Systems
• Dynamic Systems

15
Backward Reachability Analysis
Init
Fin
infinite
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Backward Reachability Analysis
Init
Fin
infinite
effective symbolic representation
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Petri Nets
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States Markings
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Transitions
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Transitions
t
Firing t
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Transitions
t
t is disabled
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Monotonicity
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Monotonicity
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Monotonicity
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Petri Nets infinite state
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Petri Nets infinite state
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Petri Nets infinite state
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Petri Nets infinite state
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Petri Nets infinite state
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Mutual Exclusion
W
R1?
R1
R0
C
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Mutual Exclusion
W
R1?
R1
R0
C
R1?
R1?
R1?
R1
R1
R1
R0
R0
R0
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Mutual Exclusion
R1?
R1?
R1?
R1
R1
R1
R0
R0
R0
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Mutual Exclusion
R1?
R1?
R1?
R1
R1
R1
R0
R0
R0

• Initial states
• R1
• All processes in

Infinitely many
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Mutual Exclusion
R1?
R1?
R1?
R1
R1
R1
R0
R0
R0

• Initial states
• R1
• All processes in

Infinitely many
Bad states Two or more processes in
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Mutual Exclusion
R1?
R1?
R1?
R1
R1
R1
R0
R0
R0
R1
W
C
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Mutual Exclusion
Set of initial states
infinite
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Mutual Exclusion
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Mutual Exclusion
R1
W
C
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Mutual Exclusion
R1
W
C
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Mutual Exclusion
R1
W
C
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Safety Properties

• mutual exclusion
• tokens in critical section gt 1

critical section
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Safety Properties

• mutual exclusion
• tokens in critical section gt 1

Ideal Upward closed set of markings
critical section
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Safety Properties

• mutual exclusion
• tokens in critical section gt 1

Ideal Upward closed set of markings
critical section
safety reachability of ideals
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Petri Nets

• Concurrent systems
• Infinite-state symbolic representation
• Monotonic behaviour
• Safety properties reachability of ideals

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Petri Nets

• Concurrent systems
• Infinite-state symbolic representation
• Monotonic behaviour
• Safety properties reachability of ideals

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Monotonicity ideals closed under computing
Pre
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Monotonicity ideals closed under computing
Pre
I
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Monotonicity ideals closed under computing
Pre
I
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Monotonicity ideals closed under computing
Pre
I
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Monotonicity ideals closed under computing
Pre
I
Pre(I)
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Backward Reachability Analysis
Fin
Ideals
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Ideals Symbolic Representation
i index (generator)
i generator of ideal i denotes all markings
larger than i
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Ideals Symbolic Representation
index (generator)
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Ideals Symbolic Representation
index (generator)
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Ideals Symbolic Representation
index (generator)
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Ideals Symbolic Representation
index (generator)
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Ideals Symbolic Representation
C
Index for bad states
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Ideals Symbolic Representation
C
Index for bad states
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Each ideal can be characterized by a finte set of
generators
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Index is minimal element of its ideal
j
If i j then
i
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Monotonicity ideals closed under computing
Pre
C
Index for bad states
Indices of Pre
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Monotonicity ideals closed under computing
Pre
C
Index for bad states
i index Pre(i) computable
Indices of Pre
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Backward Reachability Analysis
C
Step 0
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Backward Reachability Analysis
C
Step 0
Step 1
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Backward Reachability Analysis
C
Step 0
Step 1
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Backward Reachability Analysis
C
Step 0
Step 1
Step 2
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Backward Reachability Analysis
C
Step 0
Step 1
Step 2
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Backward Reachability Analysis
C
Step 0
Step 1
Step 2
Step 3
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Backward Reachability Analysis
C
Step 0
Step 1
Step 2
Step 3
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What did we need?

1. Computable ordering
2. Monotonicity, Computability of Pre
3. Termination -- Ordering is WQO

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What did we need?

1. Computable ordering
2. Monotonicity, Computability of Pre
3. Termination -- Ordering is WQO

nice properties
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Well Quasi-Ordering (WQO)
( A , ) is WQO if
a0 a1 a2 a3 .......
i,j iltj and ai aj
WQO Simple Example
( Nat , ) is WQO
x0 x1 x2 x3 ....... natural numbers
i,j iltj and xi xj
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Properties of WQO
Finite Sets
( A , ) is WQO if A is finite
a0 a1 a2 b a3 a4 a5 b a6 ..............
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Properties of WQO
Words
if ( A , ) is WQO
w1 a0 a1
a2

w2 b0 b1 b2 b3 b4 b5
b6


then ( A , ) is WQO
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Properties of WQO
Multisets
if ( A , ) is WQO
then ( AM , M ) is WQO
M1 M M2
M2
M1
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Methodology

• Start from a finite domain
• Build more complicated data structures
• words, multisets, lists, sets, etc.

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Examples -- WQO
( A , )
A finite alphabet
w1 w2 w1 subword of w2
e.g. ab xaybz
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Examples -- WQO
Words of natural numbers
5 2 7
w1
w2
3 7 1 4 2 8
w1
w2
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Multisets over a finite alphabet
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Words of multisets over a finite alphabet
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Lossy Channel Systems
!m

• finite state process
• unbounded lossy channel
• send and receive operations

?n
m n n m

• Infinite state space
• Perfect channel Turing machine
• Motivation Link protocols

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State
!m
mpnm npn
?n
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Transitions
Send
!m
m
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Transitions
Send
!m
m
Receive
?m
m
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Transitions
Send
!m
m
Receive
?m
m
Messages may nondeterministically be lost
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Example
!m
?n
p n m p n
n m p m
m p m
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Ordering

• same colour
• subword

m n p m p n p
m n p m p n p
m n p m p
m n p m p n p
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Ordering

• same colour
• subword

m n p m p n p
Computable and WQO
m n p m p n p
m n p m p
m n p m p n p
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Monotonicity
w1
w3
w2
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Monotonicity
w1
w3
w2
Downward closed
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Ideal Index
m n p
denotes all larger states
m n m p m m n m p
m n p
m n m p m m n m p
m n p
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Each ideal can be characterized by a finite set
of generators
By WQO of
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Computing Pre
Pre ( ) contains the following
w
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Computing Pre
Pre ( ) contains the following
w
!m
and w w m
if
w
then
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Computing Pre
Pre ( ) contains the following
w
!m
and w w m
if
w
then
!m
and last(w) m
if
w
then
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Computing Pre
Pre ( ) contains the following
w
!m
and w w m
if
w
then
!m
and last(w) m
if
w
then
?m
m w
then

if
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Example
Pre ( )
a d b
!b
a d
if
!d
if
a d b
?d

d a d b
if
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Methodology (applied to LCS)

1. Computable ordering
2. Monotonicity, Computability of Pre
3. Ordering is WQO

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LCS -- Forward vs Backward Analysis
Pre(w) is regular and computable Post(w) is
regular but not computable
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Timed Petri Nets
2.1
0.5
8.5
6.2
4,7
1,5
3,6
0,3
4, )
1,2
4.6
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States Markings
2.1
0.5
3.5
6.2
3,6
1,5
4,7
0,3
1,2
4.6
2.1 3.5 0.5 6.2 4.6
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Timed Transitions
2.1
0.5
3.5
6.2
3,6
1,5
4,7
2.1 3.5 0.5 6.2 4.6
0,3
1,2
4.6
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Timed Transitions
2.1
0.5
3.5
6.2
3,6
1,5
4,7
2.1 3.5 0.5 6.2 4.6
0,3
increase age by 1.3
1,2
4.6
3.4
1.8
4.8
7.5
1,5
4,7
3.4 4.8 1.8 7.5 5.9
0,3
1,2
5.9
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Discrete Transitions
3.1
1.5
4.5
7.2
3,6
1,5
4,7
t
3.1 4.5 1.5 7.2 5.6
0,3
1,2
5.6
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Discrete Transitions
3.1
1.5
4.5
7.2
3,6
1,5
4,7
t
3.1 4.5 1.5 7.2 5.6
0,3
1,2
5.6
Firing t
3.1
7.2
1,5
4,7
3.1 7.2 0.8 5.6
t
0,3
1,2
0.8
5.6
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Timed Petri Nets

• Concurrent timed systems
• Infinite-state symbolic representation
• Monotonic behaviour
• Safety properties reachability of ideals

107
Equivalence on Markings
3.1
7.2
3,6
1,5
4,7
t
0,3
1,2
0.8
5.6

• max 7
• ages gt max behave identically

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Equivalence on Markings
Markings equivalent if they agree on

• colours
• integral parts of clock values
• ordering on fractional parts

3.1 4.8 1.5 6.2 5.6
3.2 4.8 1.6 6.4 5.7
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Equivalence on Markings
Markings equivalent if they agree on

• colours
• integral parts of clock values
• ordering on fractional parts

3.1 4.8 1.5 6.2 5.6
3.1 1.5 4.8
3.2 4.8 1.6 6.4 5.7
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Equivalence on Markings
Markings equivalent if they agree on

• colours
• integral parts of clock values
• ordering on fractional parts

3.1 4.8 1.5 6.2 5.6
3.1 1.5 4.8
3.2 4.8 1.6 6.4 5.7
3.2 1.6 4.7
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Equivalence on Markings
Markings equivalent if they agree on

• colours
• integral parts of clock values
• ordering on fractional parts

3.1 4.8 1.5 6.2 5.6
3 6 1 5 4
3.2 4.8 1.6 6.4 5.7
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Equivalence on Markings
Markings equivalent if they agree on

• colours
• integral parts of clock values
• ordering on fractional parts

3.1 4.8 4.8 1.1 5.4
3 1
4 4
5
3.2 4.7 4.7 1.2 5.5
words over multisets over a finite alphabet
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Ordering on Markings
M1 M2 iff M3

• M1 M3
• M3 M2

4.8 6.4 5.7
3.1 4.8 1.5 6.2 5.6
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Ordering on Markings
M1 M2 iff M3

• M1 M3
• M3 M2

4.8 6.4 5.7
4.8 6.2 5.6
3.1 4.8 1.5 6.2 5.6
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4.8 6.4 5.7
4.8 6.2 5.6
3.1 4.8 1.5 6.2 5.6
116
4.8 6.4 5.7
4.8 6.2 5.6
3.1 4.8 1.5 6.2 5.6
6 5 4

subword
6 5 4
subword
3 6 1 5 4
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Ordering on Markings
M1 M2 iff M3

• M1 M3
• M3 M2

3.2 1.2 4.7
3.1 4.8 4.8 1.1 5.4
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Ordering on Markings
M1 M2 iff M3

• M1 M3
• M3 M2

3.2 1.2 4.7
3.1 4.8 1.1
3.1 4.8 4.8 1.1 5.4
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3.2 1.2 4.7
3.1 4.8 1.1
3.1 4.8 4.8 1.1 5.4
3 1
4

subword
3 1
4
subword
3 1
4 4
5
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Properties of
subword ordering on multisets over a
finite alphabet
is a well quasi-ordering

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Properties of -- Monotonicity
M3
M1
M2
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Properties of -- Monotonicity
M3
M1
M4
M2
123
Properties of -- Monotonicity
M3
M1
M5
M4
M2
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Properties of -- Monotonicity
M3
M1
M5
M4
M2
M6
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Properties of -- Monotonicity
M3
M1
M5
M4
M2
M6
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Methodology (applied to TPN)

1. Computable ordering
2. Monotonicity, Computability of Pre
3. Ordering is WQO

127
Infinite-State Games
Player A
Player B
Can B take game to ?
128
Backward Reachability Analysis
Characterize losing states for A
A-states
B-states
Pre( )
129
Backward Reachability Analysis
Characterize losing states for A
B-states
A-states
Pre( )
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Backward Reachability Analysis
Characterize losing states for A
Pre
Pre
Pre
Pre
131
Vector Addition Systems with States (VASS)
x
y --
x--

• Finite-state automaton operating on variables
• Variables range over natural numbers
• Operations increment or decrement variable

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VASS Petri nets
y--
x
x--
VASS
y
Petri net
x
133
x
VASS Games
x
x--
x--
x
Player A
Player B
Can B take game to ?
134
x
0
x
0
x--
1
x--
2
x
3
4
135
x
0
x
0
x--
1
x--
2
x
3
A cannot avoid
4
136
x
1
x
1
0
x--
2
x--
3
x
4
5
137
x
1
x
1
0
x--
2
x--
3
x
4
A can avoid
5
138
x
2
x
2
1
x--
3
0
x--
4
1
x
5
2
6
3
139
x
2
x
2
1
x--
3
0
x--
4
1
x
5
2
A cannot avoid
6
3
140
Player A 0 -- lose 1 -- win
gt1 -- lose
Monotonicity does not imply upward closedness
141
Backward Reachability Analysis
Characterize losing states for A
Pre
Pre
Pre
Pre
Why scheme does not work for VASS?
Monotonicity does not imply that ideals are
closed under
Pre
142
2-Counter Machines
x
y--
x--
x0?
Is reachable?
Problem undecidable
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Simulation of 2-Counter Machines by VASS Games
x
Counter machine
x
VASS game
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Simulation of 2-Counter Machines by VASS Games
x--
Counter machine
x--
VASS game
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Simulation of 2-Counter Machines by VASS Games
x0?
Counter machine
x--
VASS game
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Safety undecidable for Monotonic Games
Safety undecidable for VASS Games
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B-Downward Closed Games
s1
s3
s2
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B-Downward Closed Games
s1
s3
s2
Pre
any set
ideal
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Backward Reachability Analysis
B-Downward closed games
Pre
Pre
Pre
Pre
ideal
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Backward Reachability Analysis
B-Downward closed games
Pre
Pre
Pre
Pre
ideal
nice ordering
characterization of A-losing states
decidability of safety
151
Backward Reachability Analysis
B-LCS Games
!m
Player B can lose messages
!n
?m
?n
!m
B-LCS characterization of A-losing states
Safety decidable for B-LCS games
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A-Downward Closed Games
153
A-Downward Closed Games
Post
154
A-Downward Closed Games
Post
155
A-Downward Closed Games
156
A-Downward Closed Games
157
A-Downward Closed Games
F
158
A-Downward Closed Games
F
159
A-Downward Closed Games
F
T
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A-Downward Closed Games
F
T
F
T

• Termination
• all leaves closed
• Evaluate tree OR
• AND

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A-Downward Closed Games
F
T
F
T
Termination guaranteed if is WQO
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A-Downward Closed Games
F
T
F
T
Safety decidable for A-LCS Games
Can we characterize winning states ?
163
!m
A Problem for LCS
?n
characterize
sf
w w
sf

• Set regular
• But Not computable

164
A-LCS Games

• Winning set regular
• But not computable

!m
LCS
!m
A-LCS game
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A-LCS Games

• Winning set regular
• But not computable

?m
LCS
?m
A-LCS game
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A-LCS Games

• Winning set regular
• But not computable

For each
A-LCS game
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Conclusions and Planned Work

• Define a WQO on state space
• Safety properties reachability of ideals
• Examples
• Timed Petri nets
• Parameterized systems
• Broadcast protocols
• Cache coherence protocols
• Lossy channel systems, etc.

168

• Extension to Games
• Regular Model Checking
• Stochastic behaviours