1 carnegie mellon universityspinflavio lerda spin an explicit state model checker
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Carnegie Mellon University SPINFlavio Lerda
SPIN
An explicit state model checker
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Carnegie Mellon University SPINFlavio Lerda
Explict State Model Checker
• Represents the system as an finite state machine
• Visits each reachable state (state space) explicitly
• Checks some property– Property is satisfied– Counterexample
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DFS
• DFS visit of the state space
procedure DFS(s)visited = visited {s};for each successor s’ of s
if s’ visited then DFS(s’);end if
end forend procedure
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Carnegie Mellon University SPINFlavio Lerda
DFS
• How do we:– Represent the transition relation– Store the visited set
• Needs fast access (hash table)• State space explosion
– Check properties
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Carnegie Mellon University SPINFlavio Lerda
Promela
• Process Algebra– An algebraic approach to the study of
concurrent processes. Its tools are algebraical languages for the specification of processes and the formulation of statements about them, together with calculi for the verification of these statements. [Van Glabbeek, 1987]
• Describes the system in a way similar to a programming language
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Promela
• Asynchronous composition of independent processes
• Communication using channels and global variables
• Non-deterministic choices and interleavings
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
NC
C
T
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
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Carnegie Mellon University SPINFlavio Lerda
An Examplemtype = { NONCRITICAL, TRYING, CRITICAL };show mtype state[2];proctype process(int id) {beginning:noncritical:
state[id] = NONCRITICAL;if:: goto noncritical;:: true;fi;
trying:state[id] = TRYING;if:: goto trying;:: true;fi;
critical:state[id] = CRITICAL;if:: goto critical;:: true;fi;goto beginning;}
init { run process(0); run process(1); }
NC
C
T
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Carnegie Mellon University SPINFlavio Lerda
Enabled Statements
• A statement needs to be enabled for the process to be scheduled.
bool a, b;
proctype p1()
{
a = true;
a & b;
a = false;
}
proctype p2()
{
b = false;
a & b;
b = true;
}
init { a = false; b = false; run p1(); run p2(); }
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Carnegie Mellon University SPINFlavio Lerda
Enabled Statements
• A statement needs to be enabled for the process to be scheduled.
bool a, b;
proctype p1()
{
a = true;
a & b;
a = false;
}
proctype p2()
{
b = false;
a & b;
b = true;
}
init { a = false; b = false; run p1(); run p2(); }
These statements are enabled only if both a and b are true.
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Carnegie Mellon University SPINFlavio Lerda
Enabled Statements
• A statement needs to be enabled for the process to be scheduled.
bool a, b;
proctype p1()
{
a = true;
a & b;
a = false;
}
proctype p2()
{
b = false;
a & b;
b = true;
}
init { a = false; b = false; run p1(); run p2(); }
These statements are enabled only if both a and b are true.
In this case b is always false and therefore there is a deadlock.
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Carnegie Mellon University SPINFlavio Lerda
Other constructs
• Do loopsdo
:: count = count + 1;
:: count = count - 1;
:: (count == 0) -> break
od
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Carnegie Mellon University SPINFlavio Lerda
Other constructs
• Do loops
• Communication over channelsproctype sender(chan out)
{
int x;
if
::x=0;
::x=1;
fi
out ! x;
}
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Other constructs
• Do loops
• Communication over channels
• Assertionsproctype receiver(chan in)
{
int value;
out ? value;
assert(value == 0 || value == 1)
}
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Carnegie Mellon University SPINFlavio Lerda
Other constructs
• Do loops
• Communication over channels
• Assertions
• Atomic Stepsint value;
proctype increment()
{ atomic {
x = value;
x = x + 1;
value = x;
} }
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On-The-Fly
• System is the asynchronous composition of processes
• The global transition relation is never build
• For each state the successor states are enumerated using the transition relation of each process
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
1 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
On-The-Fly0
1
0
1
0 0
1 1
0 11 0
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Carnegie Mellon University SPINFlavio Lerda
Visited Set
• Represents all the states that have been reached so far
• Will eventually become the set of all reachable state (state space)
• Test of presence of a state in the set must be efficient– It is performed for each reached state
procedure DFS(s)visited = visited {s};for each successor s’ of s
if s’ visited then DFS(s’);end if
end forend procedure
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Carnegie Mellon University SPINFlavio Lerda
Visited Set
• Hash table– Efficient for testing even if the number of
elements in it is very big (≥ 106)
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Carnegie Mellon University SPINFlavio Lerda
Visited Set
• Hash table– Efficient for testing even if the number of
elements in it is very big (≥ 106)
• Reduce memory usage– Compress each state When a transition is executed only a
limited part of the state is modified
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Carnegie Mellon University SPINFlavio Lerda
Visited Set
• Hash table– Efficient for testing even if the number of
elements in it is very big (≥ 106)
• Reduce memory usage– Compress each state
• Reduce the number of states– Partial Order Reduction
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State Representation
• Global variables
• Processes and local variables
• Queues
Global Variables Processes Queues
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Compression
• Each transition changes only a small part of the state
• Assign a code to each element dynamically
• Encoded states + basic elements use considerably less spaces than the uncompressed states
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Carnegie Mellon University SPINFlavio Lerda
Compression
i=0 j=0P0x=0
P0x=0
P0x=1
Q0{1}
P1y=0
i=0 j=0P0x=0
P0x=1
Q0{1}
P1y=0
0
3
2
1
0
3
2
1
3
2
1
0 0 1 0 0 2
0
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Carnegie Mellon University SPINFlavio Lerda
0 0
P0x=0
Q0{1}
Compression
i=0 j=0P0x=0
P0x=1
P0x=1
Q0{}
P1y=0
i=0 j=0P0x=0
P0x=1
Q0{1}
P1y=0
0
3
2
1
0
3
2
1
3
2
1
0 0 1 2
0
Q0{}
1 1
q ? x
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Hash Compaction
• Uses a hashing function to store each state using only 2 bits
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Carnegie Mellon University SPINFlavio Lerda
Hash Compaction
• Uses a hashing function to store each state using only 2 bits
• There is an non-zero probability that two states are mapped into the same bits
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Carnegie Mellon University SPINFlavio Lerda
Hash Compaction
• Uses a hashing function to store each state using only 2 bits
• There is an non-zero probability that two states are mapped into the same bits
• If the number of states is quite smaller than the number of bits available there is a pretty good chance of not having conflicts
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Carnegie Mellon University SPINFlavio Lerda
Hash Compaction
• Uses a hashing function to store each state using only 2 bits
• There is an non-zero probability that two states are mapped into the same bits
• If the number of states is quite smaller than the number of bits available there is a pretty good chance of not having conflicts
• The result is not (always) 100% correct!
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Minimized Automata Reduction
• Turns the state in a sequence of integers
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Carnegie Mellon University SPINFlavio Lerda
Minimized Automata Reduction
• Turns the state in a sequence of integers
• Constructs an automata which accepts the states in the visited set
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Carnegie Mellon University SPINFlavio Lerda
Minimized Automata Reduction
• Turns the state in a sequence of integers
• Constructs an automata which accepts the states in the visited set
• Works like a BDD but on non-binary variables (MDD)
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Carnegie Mellon University SPINFlavio Lerda
Minimized Automata Reduction
• Turns the state in a sequence of integers
• Constructs an automata which accepts the states in the visited set
• Works like a BDD but on non-binary variables (MDD)– The variables are the components of the state
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Carnegie Mellon University SPINFlavio Lerda
Minimized Automata Reduction
• Turns the state in a sequence of integers
• Constructs an automata which accepts the states in the visited set
• Works like a BDD but on non-binary variables (MDD)– The variables are the components of the state– The automata is the minimal automata
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Carnegie Mellon University SPINFlavio Lerda
Minimized Automata Reduction
• Turns the state in a sequence of integers
• Constructs an automata which accepts the states in the visited set
• Works like a BDD but on non-binary variables (MDD)– The variables are the components of the state– The automata is the minimal automata– The automata is updated efficiently
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Carnegie Mellon University SPINFlavio Lerda
Partial Order Reduction
• Some interleavings of processes are equivalent
x=0y=0
x=1y=0
x=0y=1
x=1y=1
x++y++
y++x++
x=1y=0
x=1y=0
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Carnegie Mellon University SPINFlavio Lerda
Partial Order Reduction
• Some interleavings of processes are equivalent
• Computing such interleavings and storing the intermediate states is expensive
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Carnegie Mellon University SPINFlavio Lerda
Partial Order Reduction
• Some interleavings of processes are equivalent
• Computing such interleavings and storing the intermediate states is expensive
• Partial order reduction defines a reduced system which is equivalent to the original system but contains less states and transitions
Defines an equivalent relation between states and computes the quotient of the state transition graph to obtain a reduced state transition graph.
Properties are true of the reduced state transition graph if and only if are true of the original graph.
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Partial Order Reduction
• Optimal partial order reduction is as difficult as model checking!
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Carnegie Mellon University SPINFlavio Lerda
Partial Order Reduction
• Optimal partial order reduction is as difficult as model checking!
• Compute an approximation based on syntactical information
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Carnegie Mellon University SPINFlavio Lerda
Partial Order Reduction
• Optimal partial order reduction is as difficult as model checking!
• Compute an approximation based on syntactical information– Independent– Invisible– Check (at run-time) for actions postponed at
infinitum
Access to local variablesReceive on exclusive receive-access queues
Send on exclusive send-access queues
Not mentioned in the property
So called stack proviso
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Properties
• Safety properties– Something bad never happens– Properties of states
• Liveness properties– Something good eventually happens– Properties of paths
Reachability is sufficient
We need something more complex to check liveness properties
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Carnegie Mellon University SPINFlavio Lerda
LTL Model Checking
• Liveness properties are expressed in LTL– Subset of CTL* of the form:
• A f
where f is a path formula with does not contain any quantifiers
• The quantifier A is usually omitted.• G is substituted by (always or box)• F is substituted by (eventually or diamond)• X is substituted by (next)
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Carnegie Mellon University SPINFlavio Lerda
LTL Formulae
• Always eventually p: p AGFp in CTL*
AG(pFq) in CTL*
• Fairness:
( p )
AG(p AFq) in CTL
AG AF p in CTL
(AGF p) in CTL*
Can’t express it in CTL
• Always after p there is eventually q: ( p ( q ) )
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References• http://spinroot.com/ • Design and Validation of Computer Protocols by Gerard
Holzmann• The Spin Model Checker by Gerard Holzmann• An automata-theoretic approach to automatic program
verification, by Moshe Y. Vardi, and Pierre Wolper • An analysis of bitstate hashing, by G.J. Holzmann • An Improvement in Formal Verification, by G.J. Holzmann
and D. Peled • Simple on-the-fly automatic verification of linear temporal
logic, by Rob Gerth, Doron Peled, Moshe Vardi, and Pierre Wolper
• A Minimized automaton representation of reachable states, by A. Puri and G.J. Holzmann