Tools for Automated Verification of
Concurrent Software
Tevfik Bultan
Department of Computer Science
University of California, Santa Barbara
bultan@cs.ucsb.edu
http://www.cs.ucsb.edu/~bultan/
http://www.cs.ucsb.edu/~bultan/composite/
Summary
• Goal: Reliable concurrent programming
• Sub-goals:
– Developing reliable concurrency controllers in Java
– Developing reliable concurrent linked lists
• Approach: Model Checking
– Refined Approach: Composite Model Checking
• Specification Language: Action Language
• Tools:
– Composite Symbolic Library
– Action Language Verifier
Students
Joint work with my students:
• Tuba Yavuz-Kahveci
• Constantinos Bartzis
• Xiang Fu (co-advised with Jianwen Su)
• Aysu Betin-Can
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Difficulties in Concurrent Programming
• Concurrent programming is difficult and error prone
– In sequential programming you only worry about the
states of the variables
– In concurrent programming you also have to worry about
the states of the threads
• State space increases exponentially with the number of
threads
Concurrent Programming in Java
• Java uses a variant of monitor programming
• Synchronization using locks
– Each object has a lock
synchronized(o) { ... }
• Coordination using condition variables
– Objects can be used as condition variables
synchronized (condVar){
while (!condExp) wait(condVar);
...
notifyAll(condVar);
}
Dangers in Java Concurrency
• Nested locks
synchronized m(other) {
other.m();
}
Thread1: run() { o1.m(o2); }
Thread2: run() { o2.m(o1); }
Thread1
o1
lock
Thread2
lock
o2
Dangers in Java Concurrency
• Missed notification
notify(condVar);
• Forgotten condition check
if(!condExp) wait(condVar);
• Dependency among multiple condition variables can be
complicated
– Conservative notification and condition check
Inefficient
– Optimizing the notification and condition checks
Error prone
Example: Airport Ground Traffic Control Simulation
A simplified model of Seattle Tacoma International Airport from [Zhong 97]
Control Logic
• An airplane can land using 16R only if no airplane is using
16R at the moment
• An airplane can takeoff using 16L only if no airplane is
using 16L at the moment
• An airplane taxiing on one of the exits C3-C8 can cross
runway 16L only if no airplane is taking off at the moment
• An airplane can start using 16L for taking off only if none of
the crossing exits C3-C8 is occupied at the moment
(arriving airplanes have higher priority)
• Only one airplane can use a taxiway at a time
Java Implementation
• Simulate behavior of each airplane with a thread
• Use a monitor (a Java class)
– private variables for number of airplanes on each runway
and each taxiway
– methods of the monitor enforce the control logic
• Each thread calls the methods of the monitor based on the
airport layout to move from one point to the next
Example Implementation
public synchronized void C8_To_B11A() {
while (!((numRW16L == 0) && (numB11A == 0)))
wait();
numC8 = numC8 - 1;
numB11A = numB11A + 1;
notifyAll();
}
• This code is not efficient since every thread wakes up every
other thread
• Using separate condition variables complicates the
synchronization
– nested locks
Difficulties In Implementing Concurrent Linked Lists
• Linked list manipulation is difficult and error prone
– State of the heap: unbounded
• State space:
– Sequential programming
• states of the variables
– Concurrent programming
• states of the variables
• states of the threads
– Concurrent linked lists
• states of the variables
• states of the threads
• state of the heap
Examples
• singly linked lists
n1
prev
• doubly linked lists
n2
next
next
next
n1
n2
prev
• stack
top
n1 next
n2
next
last
• queue
first
n1
next
n2
next
• single lock
• double lock
– allows concurrent inserts and deletes
next
Outline of Our Approach
1. Specify concurrency controllers and concurrent linked lists
in Action Language
2. Verify their properties using composite model checking
3. Generate Java classes from the specifications which
preserve their properties
Action Language Tool Set
Action Language
Specification
Action Language
Parser
Action Language
Verifier
Code Generator
Verified code
(Java monitor classes)
Composite Symbolic Library
Omega
Library
Presburger
Arithmetic
Manipulator
CUDD
Package
BDD
Manipulator
MONA
Automata
Manipulator
Outline
• Difficulties in concurrent programming
• A short history of model checking in 7 slides
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Idea 1: Temporal Logics for Reactive Systems
[Pnueli FOCS 77, TCS 81]
Transformational systems
get input;
compute something;
return result;
Reactive systems
while (true) {
receive some input,
send some output
}
For reactive systems
• termination is not relevant
• pre and post-conditions are not
enough
Temporal Logics
• Invariant p (G p, AG p, p)
• Eventually p (F p, AF p, p)
• Next p : (X p, AX p, p)
• p Until q : ( p U q, A(p U q) )
Branching vs.
Linear Time
p
p
p
LTL
G(p) F(p)
p
p
CTL
AF(p), EG(p)
p
p
p
p
p
.
.
.
.
.
.
.
.
.
p
.
.
.
.
.
.
Idea 2: Automated Verification of Finite State
Systems [Clarke and Emerson 81], [Queille and Sifakis 82]
Transition Systems
• S : Set of states (finite)
• I  S : Set of initial states
• R  S  S : Transition relation
Model checking problem: Given a
temporal logic property, does the
transition system satisfy the
property?
– Complexity: linear in the size
of the transition system
Verification vs. Falsification
Verification:
show: initial states  truth set of p
Falsification:
find: a state  initial states  truth
set of p
generate a counter-example
starting from that state
Idea 3: Temporal Properties  Fixpoints
[Emerson and Clarke 80]
EF(p)  states that can reach p

p  Pre(p)  Pre(Pre(p))  ...
p
• • •
Initial
states
EF(p)
initial states that satisfy EF(p)
 initial states that violate AG(p)
EG(p)  states that can avoid reaching p
 p  Pre(p)  Pre(Pre(p))  ...
• • •
EG(p)
Initial
states
initial states that satisfy EG(p)
 initial states that violate AF(p)
Idea 4: Symbolic Model Checking
[McMillan et al. LICS 90]
• Represent sets of states and the transition relation as
Boolean logic formulas
• Fixpoint computation becomes formula manipulation
– pre and post-condition computations: Existential variable
elimination
– conjunction (intersection), disjunction (union) and
negation (set difference), and equivalence check
• Use an efficient data structure
– Binary Decision Diagrams (BDDs)
Tool 1: SMV [McMillan 93]
•
•
•
•
BDD-based symbolic model checker
Finite state
Temporal logic: CTL
Focus: hardware verification
– Later applied to software specifications, protocols, etc.
• SMV has its own input specification language
– concurrency: synchronous, asynchronous
– shared variables
– boolean and enumerated variables
– bounded integer variables (binary encoding)
• SMV is not efficient for integers, can be fixed
Idea 5: LTL Properties  Büchi automata
[Vardi and Wolper LICS 86]
• Büchi automata: Finite state
automata that accept infinite
strings
• A Büchi automaton accepts a
string when the corresponding
run visits an accepting state
infinitely often
true
Gp
p
p
true
Fp
• The size of the property
automaton can be exponential in
the size of the LTL formula
p
true
G (F p)
p
p
true
Tool 2: SPIN [Holzmann
91, TSE 97]
• Explicit state, finite state
• Temporal logic: LTL
• Input language: PROMELA
– Asynchronous processes
– Shared variables
– Message passing through
(bounded) communication
channels
– Variables: boolean, char,
integer (bounded), arrays
(fixed size)
• Property automaton from the
negated LTL property
• Product of the property
automaton and the transition
system (on-the-fly)
• Show that there is no accepting
cycle in the product automaton
• Nested depth first search to look
for accepting cycles
• If there is a cycle, it corresponds
to a counterexample behavior
that demonstrates the bug
Model Checking Research
• These 5 key ideas and 2 key tools inspired a lot of research
[Clarke, Grumberg and Peled, 99]
–
–
–
–
–
–
–
–
–
efficient symbolic representations
partial order reductions
abstraction
compositional/modular verification
model checking infinite state systems (pushdown
automata)
model checking real time systems
model checking hybrid systems
model checking programs
...
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Action Language
[Bultan, ICSE 00], [Bultan, Yavuz-Kahveci, ASE 01]
• A state based language
– Actions correspond to state changes
• States correspond to valuations of variables
– boolean
– enumerated
– integer (possibly unbounded)
– heap variables (i.e., pointers)
• Parameterized constants
– specifications are verified for every possible value of the
constant
Action Language
• Transition relation is defined using actions
– Atomic actions: Predicates on current and next state
variables
– Action composition:
• asynchronous (|) or synchronous (&)
• Modular
– Modules can have submodules
– A modules is defined as asynchronous and/or
synchronous compositions of its actions and
submodules
Readers Writers Example
module main()
integer nr;
boolean busy;
restrict: nr>=0;
initial: nr=0 and !busy;
S : Cartesian product of
variable domains defines
the set of states
I : Predicates defining
the initial states
module Reader()
boolean reading;
R : Atomic actions of the
initial: !reading;
Reader
rEnter: !reading and !busy and
nr’=nr+1 and reading’;
rExit: reading and !reading’ and nr’=nr-1;
Reader: rEnter | rExit;
endmodule
R : Transition relation of
Reader defined as
module Writer()
asynchronous composition
...
of its atomic actions
endmodule
main: Reader() | Reader() | Writer() | Writer();
spec: invariant([busy => nr=0])
endmodule
R : Transition relation of main defined as asynchronous
composition of two Reader and two Writer processes
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Which Symbolic Representation to Use?
BDDs
• canonical and efficient
representation for Boolean logic
formulas
• can only encode finite sets
x  y  {(T,T), (T,F), (F,T)}
F
x
a > 0  b = a+1
T
y
F
F
Linear Arithmetic Constraints
• can encode infinite sets
• two representations
– polyhedral representation
– automata representation
• not efficient for encoding
boolean domains
T
T
 {(1,2), (2,3), (3,4),...}
Composite Model Checking
[Bultan, Gerber, League ISSTA 98, TOSEM 00]
• Map each variable type to a symbolic representation
– Map boolean and enumerated types to BDD
representation
– Map integer type to a linear arithmetic constraint
representation
• Use a disjunctive representation to combine different
symbolic representations: composite representation
• Each disjunct is a conjunction of formulas represented by
different symbolic representations
– we call each disjunct a composite atom
Composite Representation
composite atom
n
P   pi  pi  ...  pi
i 1
1
symbolic
rep. 1
2
symbolic
rep. 2
t
symbolic
rep. t
Example:
x: integer, y: boolean
x>0 and x´x-1 and y´ or x<=0 and x´x and y´y
arithmetic constraint
representation
BDD
arithmetic constraint
representation
BDD
Composite Symbolic Library
[Yavuz-Kahveci, Tuncer, Bultan TACAS01], [Yavuz-Kahveci, Bultan STTT]
• Uses a common interface for each symbolic representation
• Easy to extend with new symbolic representations
• Enables polymorphic verification
• Multiple symbolic representations:
– As a BDD library we use Colorado University Decision
Diagram Package (CUDD) [Somenzi et al]
– As an integer constraint manipulator we use Omega
Library [Pugh et al]
Composite Symbolic Library Class Diagram
Symbolic
+intersect()
+union()
+complement()
+isSatisfiable()
+isSubset()
+pre()
+post()
BoolSym
–representation: BDD
+intersect()
+union()
•
•
•
CUDD Library
CompSym
IntSym
–representation: list
of comAtom
–representation: Polyhedra
+intersect()
+ union()
•
•
•
compAtom
–atom: *Symbolic
+intersect()
+union()
•
•
•
OMEGA Library
Composite Symbolic Representation
x: integer, y:boolean
x>0 and x´x-1 and y´ or x<=0 and x´x and y´y
: CompSym
representation : List<compAtom>
: ListNode<compAtom>
data : compAtom
y´
0 b’
1 x>0 and x´=x-1
next :*ListNode<compAtom>
: ListNode<compAtom>
data : compAtom
0
1
y’=y
x<=0 and x’=x
next: *ListNode<compAtom>
Pre and Post-condition Computation
Variables:
x: integer, y: boolean
Transition relation:
R: x>0 and x´x-1 and y´ or x<=0 and x´x and y´y
Set of states:
s: x=2 and !y or x=0 and !y
Compute post(s,R)
Pre and Post-condition Distribute
R: x>0 and x´x-1 and y´ or x<=0 and x´x and y´y
s: x=2 and !y or x=0 and y
post(s,R) = post(x=2 , x>0 and x´x-1)  post(!y , y´)
x=1
y

post(x=2 , x<=0 and x´x)  post (!y , y´y)
false
!y

post(x=0 , x>0 and x´x-1)  post(y , y´)
false
y

post (x=0 , x<=0 and x´x)  post (y, y´y )
x=0
y
= x=1 and y or x=0 and y
Polymorphic Verifier
Symbolic TranSys::check(Node *f) {
•
•
•
Symbolic s = check(f.left)
case EX:
s.pre(transRelation)
case EF:
do
sold = s
s.pre(transRelation)
s.union(sold)
while not sold.isEqual(s)
•
•
•
}
 Action
Language Verifier
is polymorphic
 It becomes a BDD based model
checker when there or no integer
variables
Heuristics for Composite Representation
[Yavuz-Kahveci, Bultan FroCos 02]
• Masking
– compute operations on BDDs first
– avoid redundant computations on integer part
• Incremental subset check
– Exploit the disjunctive structure by computing subset
checks incrementally
• Interleaving pre-condition computation with the subset
check in least-fixpoint computations
• Simplification
– Reduce the number of disjuncts in the composite
representation by iteratively merging matching disjuncts
Some Experiments
Problem
Instance
All Heuristics
Time (sec)
Memory (MB)
No Heuristics
Time (sec)
Memory (MB)
Barber2-2
0.27
8.80
1327.82
464.14
Barber3-2
0.35
9.50


Bakery2i
0.21
7.80
5.52
94.66
Bakery3i
8.26
19.60


Lightcontrol
0.12
7.90
81.05
48.40
Without the simplification for 15 out of 39 problem
instances the verifier ran out of memory
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Application to Concurrency Controllers
[Yavuz-Kahveci, Bultan ISTTA 02] [Betin-Can, Bultan SoftMC 03]
Outline of our approach:
1. Specify concurrency controllers and concurrent linked lists
in Action Language
2. Verify their properties using composite model checking
3. Generate Java classes from the specifications which
preserve their properties
Readers-Writers Controller
module main()
integer nr;
boolean busy;
restrict: nr>=0;
initial: nr=0 and !busy;
module Reader()
boolean reading;
initial: !reading;
rEnter: !reading and !busy and
nr’=nr+1 and reading’;
rExit: reading and !reading’ and nr’=nr-1;
Reader: rEnter | rExit;
endmodule
module Writer()
boolean writing;
initial: !writing;
wEnter: !writing and nr=0 and !busy and
busy’ and writing’;
wExit: writing and !writing’ and !busy’;
Writer: wEnter | wExit;
endmodule
main: Reader() | Reader() | Writer() | Writer();
spec: invariant([busy => nr=0])
endmodule
Arbitrary Number of Threads
• Counting abstraction
– Create an integer variable for each local state of a
thread
– Each variable will count the number of threads in a
particular state
• Local states of the threads have to be finite
– Specify only the thread behavior that relates to the
correctness of the controller
– Shared variables of the controller can be unbounded
• Counting abstraction can be automated
Readers-Writers After Counting Abstraction
Parameterized constants
module main()
introduced by the counting
integer nr;
abstractions
boolean busy;
parameterized integer numReader, numWriter;
restrict: nr>=0 and numReader>=0 and numWriter>=0;
initial: nr=0 and !busy;
Variables introduced by the
module Reader()
counting abstractions
integer readingF, readingT;
initial: readingF=numReader and readingT=0;
rEnter: readingF>0 and !busy and
nr’=nr+1 and readingF’=readingF-1 and
readingT’=readingT+1;
rExit: readingT>0 and nr’=nr-1 readingT’=readingT-1
and readingF’=readingF+1;
Reader: rEnter | rExit;
endmodule
module Writer()
...
endmodule
main: Reader() | Writer();
spec: invariant([busy => nr=0])
endmodule
Verification of Readers-Writers Controller
Integers
Booleans
Cons. Time
(secs.)
Ver. Time
(secs.)
Memory
(Mbytes)
RW-4
1
5
0.04
0.01
6.6
RW-8
1
9
0.08
0.01
7
RW-16
1
17
0.19
0.02
8
RW-32
1
33
0.53
0.03
10.8
RW-64
1
65
1.71
0.06
20.6
RW-P
7
1
0.05
0.01
9.1
SUN ULTRA 10 (768 Mbyte main memory)
What about the Java Implementation?
• We can automatically generate code from the controller
specification
– Generate a Java class
– Make shared variables private variables
– Use synchronization to restrict access
• Is the generated code efficient?
– Yes!
– We can synthesize the condition variables automatically
– There is no unnecessary thread notification
Specific Notification Pattern
[Cargill 96]
public class ReadersWriters{
private int nr;
private boolean busy;
private Object rEnterCond, wEnterCond;
private synchronized boolean Guard_rEnter() {
if (!busy) {
nr++;
return true;
}
All condition variables and
else return false;
}
wait and signal operations are
public void rEnter() {
generated automatically
synchronized(rEnterCond) {
while(!Guard_rEnter())
rEnterCond.wait();
}
public void rExit() {
synchronized(this) { nr--; }
synchronized(wEnterCond) { wEnterCond.notify(); }
}
...
}
rEnter: !reading and !busy and nr’=nr+1 and reading’;
Example: Airport Ground Traffic Control
A simplified model of Seattle Tacoma International Airport from [Zhong 97]
Action Language Specification
module main()
integer numRW16R, numRW16L, numC3, ...;
initial: numRW16R=0 and numRW16L=0 and ...;
module Airplane()
enumerated pc {arFlow, touchDown, parked, depFlow,
taxiTo16LC3, ..., taxiFr16LB2, ..., takeoff};
initial: pc=arFlow or pc=parked;
reqLand: pc=arFlow and numRW16R=0 and pc’=touchDown
and numRW16R’=numRW16R+1;
exitRW3: pc =touchDown and numC3=0 and
numC3’=numC3+1 and numRW16R’=numRW16R-1 and
pc’=taxiTo16LC3;
...
Airplane: reqLand | exitRW3 | ...;
endmodule
main: AirPlane() | Airplane() | Airplane() | ....;
spec: AG(numRW16R1 and numRW16L 1)
spec: AG(numC3 1)
spec: AG((numRW16L=0 and numC3+numC4+...+numC8>0) =>
AX(numRW16L=0))
endmodule
Airport Ground Traffic Control
• Action Language specification
– Has 13 integer variables
– Has 6 Boolean variables per airplane process to keep
the local state of each airplane
– 20 actions
• Automatically generated Java monitor class
– Has 13 integer variables
– Has 14 condition variables
– Has 34 methods
Experiments
A: Arriving Airplane
D: Departing Airplane
P: Arbitrary number of threads
Processes Construction(sec)
Verify-P1(sec)
Verify-P2(sec)
Verify-P3(sec)
2
0.81
0.42
0.28
0.69
4
1.50
0.78
0.50
1.13
8
3.03
1.53
0.99
2.22
16
6.86
3.02
2.03
5.07
2A,PD
1.02
0.64
0.43
0.83
4A,PD
1.94
1.19
0.81
1.39
8A,PD
3.95
2.28
1.54
2.59
16A,PD
8.74
4.6
3.15
5.35
PA,2D
1.67
1.31
0.88
3.94
PA,4D
3.15
2.42
1.71
5.09
PA,8D
6.40
4.64
3.32
7.35
PA,16D
13.66
9.21
7.02
12.01
PA,PD
2.65
0.99
0.57
0.43
Efficient Java Implementation
public class airport {
private int numRW16R;
private int numRW16L;
private int numC3;
....
private Object CondreqLand;
private Object CondexitRW3;
...
public airport() {
numRW16R = 0 ;
numRW16L = 0 ;
...
}
private synchronized boolean
Guarded_reqLand(){
if(numRW16R == 0) {
numRW16R = numRW16R + 1;
return true;
}else return false ;
}
public void reqLand(){
synchronized(CondreqLand){
while (! Guarded_reqLand()){
try{
CondreqLand.wait();
}
catch(InterruptedException e){;}
}
}
}
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Heap Type
[Yavuz-Kahveci, Bultan SAS 02]
• Heap type in Action Language
heap {next} top;
• Heap type represents dynamically allocated storage
top’=new;
• We need to add a symbolic representation for the heap type
to the Composite Symbolic Library
numItems > 2 => top.next != null
Concurrent Stack
module main()
heap {next} top, add, get, newTop; boolean mutex; integer numItems;
initial: top=null and mutex and numItems=0;
module push()
enumerated pc {l1, l2, l3, l4};
initial: pc=l1 and add=null;
push1: pc=l1 and mutex and !mutex’ and add’=new and pc’=l2;
push2: pc=l2 and numItems=0 and top’=add and numItems’=1 and pc’=l3;
push3: pc=l3 and top’.next =null and mutex’ and pc’=l1;
push4: pc=l2 and numItems!=0 and add’.next=top and pc’=l4;
push5: pc=l4 and top’=add and numItems’=numItems+1 and
mutex’ and pc’=l1;
push: push1 | push2 | push3 | push4 | push5;
endmodule
module pop()
...
endmodule
main: pop() | pop() | push() | push() ;
spec:AG(mutex =>(numItems=0 <=> top=null))
spec: AG(mutex => (numItems>2 => top->next!=null))
endmodule
Shape Graphs
• Shape graphs represent the states of the heap
heap variables add and top
point to node n1
add
top
n1
next
n2
next
add.next is node n2
top.next is also node n2
add.next.next is null
• Each node in the shape graph represents a dynamically
allocated memory location
• Heap variables point to nodes of the shape graph
• The edges between the nodes show the locations pointed
by the fields of the nodes
Composite Symbolic Library
Symbolic
+union()
+isSatisfiable()
+isSubset()
+forwardImage()
HeapSym
IntSym
CompSym
–representation:
BDD
–representation:
list of ShapeGraph
–representation:
list of Polyhedra
–representation:
list of comAtom
+union()
+union()
+union()
+ union()
BoolSym
•
•
•
CUDD Library
•
•
•
ShapeGraph
–atom: *Symbolic
•
•
•
OMEGA Library
•
•
•
compAtom
–atom: *Symbolic
Forward Fixpoint
arithmetic constraint
representation
BDD
pc=l1  mutex

numItems=2
A set of shape graphs
add

top

pc=l2  mutex

numItems=2

add
top

pc=l4  mutex

numItems=2

add
top

pc=l1  mutex

numItems=3

add
top
Post-condition Computation: Example
set of
states

pc=l4  mutex
transition
relation
pc=l4 and mutex’
pc’=l1
pc=l1  mutex

numItems=2

add
numItems’=numItems+1

numItems=3


add
top
top’=add
top
Fixpoints Do Not Converge
• We have two reasons for non-termination
– integer variables can increase without a bound
– the number of nodes in the shape graphs can increase
without a bound
• The state space is infinite
• Even if we ignore the heap variables, reachability is
undecidable when we have unbounded integer variables
• So, we use conservative approximations
Conservative Approximations
• Compute a lower ( p ) or an upper ( p+ ) approximation to
the truth set of the property ( p )
• Model checker can give three answers:
I
p
I
p
“The property is satisfied”
sates which
violate the
property
p
p
“I don’t know”
I
p
p+
“The property is false and here is a counter-example”
 p
Conservative Approximations
• Truncated fixpoint computations
– To compute a lower bound for a least-fixpoint
computation
– Stop after a fixed number of iterations
• Widening
– To compute an upper bound for the least-fixpoint
computation
– We use a generalization of the polyhedra widening
operator by
• [Cousot and Halbwachs POPL’77]
• Summarization
– Generate summary nodes in the shape graphs which
represent more than one concrete node
Summarization
• The nodes that form a chain are mapped to a summary
node
• No heap variable points to any concrete node that is
mapped to a summary node
• Each concrete node mapped to a summary node is only
pointed by a concrete node which is also mapped to the
same summary node
• During summarization, we also introduce an integer
variable which counts the number of concrete nodes
mapped to a summary node
Summarization Example
pc=l1  mutex

numItems=3

add
top
summarized nodes
After summarization, it becomes:
add
pc=l1  mutex

numItems=3  summarycount=2
a new integer variable
representing the number
of concrete nodes encoded
by the summary node
top

summary node
Simplification
pc=l1  mutex

numItems=3
 summaryCount=2

add
top

pc=l1  mutex

numItems=4

(numItems=4
 summaryCount=3

add
top
=
pc=l1  mutex
 summaryCount=3
 numItems=3
 summarycount=2)

add
top
Simplification On the Integer Part

pc=l1  mutex
(numItems=4
 summaryCount=3

add
top
 numItems=3
 summaryCount=2)
=
pc=l1  mutex

numItems=summaryCount+1
 3  numItems
 numItems  4

add
top
Widening
• Fixpoint computation still will not converge since numItems
and summaryCount keep increasing without a bound
• We use the widening operation:
– Given two composite atoms c1 and c2 in consecutive
fixpoint iterates, assume that
c1 = b1  i1  h1
c2 = b2  i2  h2
where b1 = b2 and h1 = h2 and i1  i2
Assume that i1 is a single polyhedron and i2 is also a
single polyhedron
Widening
• Then
– i1  i2 is defined as: all the constraints in i1 which are also
satisfied by i2
• Replace i2 with i1  i2 in c2
• This generates an upper approximation to the forwardfixpoint computation
Widening Example
pc=l1  mutex

numItems=summaryCount+1
add

top
 3  numItems
 numItems  4
pc=l1  mutex


numItems=summaryCount+1

add
top
 3  numItems
 numItems  5
=
pc=l1  mutex

numItems=summaryCount+1
 3  numItems
Now, fixpoint converges

add
top
Verified Properties
Specification
Verified Invariants
Stack
top=null  numItems=0
topnull  numItems0
numItems=2  top.nextnull
Single Lock Queue
head=null  numItems=0
headnull  numItems0
(head=tail  head null)  numItems=1
headtail  numItems0
Two Lock Queue
numItems>1  headtail
numItems>2  head.nexttail
Experimental Results
Verification times in secs
Number of Queue
Threads
HC
Queue
Stack
Stack
IC
2Lock
Queue
HC
2Lock
Queue
IC
IC
HC
1P-1C
10.19
12.95
4.57
5.21
60.5
58.13
2P-2C
15.74
21.64
6.73
8.24
88.26
122.47
4P-4C
31.55
46.5
12.71
15.11


1P-PC
12.85
13.62
5.61
5.73


PP-1C
18.24
19.43
6.48
6.82


HC : heap control
IC : integer control
Verifying Linked Lists with Multiple Fields
• Pattern-based summarization
– User provides a graph grammar rule to describe the
summarization pattern
L x = next x y, prev y x, L y
• Represent any maximal sub-graph that matches the pattern
with a summary node
– no node in the sub-graph pointed by a heap variable
Summarization Pattern Examples
L x  x.n = y, L y
n
n
...
L x  x.n = y, y.p = x, L y
n
n
...
p
p
n
L x  x.n = y, x.d = z, L y
d
n
d
n
n
p
...
n
d
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Shape Analysis
• There is a lot of work on Shape analysis, I will just mention
the ones which directly influenced us:
– [Sagiv,Reps, Wilhelm TOPLAS’98] , [Dor, Rodeh, Sagiv SAS’00]
• Verification of concurrent linked lists with arbitrary number
of processes in
– [Yahav POPL’01]
• 3-valued logic and instrumentation predicates
– [Sagiv,Reps, Wilhelm TOPLAS], [Lev-Ami, Reps, Sagiv, Wilhelm
ISSTA 00]
• Automatically generating instrumentation predicates
– [Sagiv,Reps, Wilhelm ESOP 03]
Shape Analysis
• Deutch used integer constraint lattices to compute aliasing
information using symbolic access paths
– [Deutch PLDI’94]
• The idea of summarization patterns is based on the shape
types introduced in
– [Fradet and Metayer POPL 97]
Model Checking Software Specifications
• [Atlee, Gannon 93]
– Translating SCR mode transition tables to input
language of explicit state model checker EMC [Clarke,
Emerson, Sistla 86]
• [Chan et al. 98,00]
– Translating RSML specifications to input language of
SMV
• [Bharadwaj, Heitmeyer 99]
– Translating SCR specifications to Promela, input
language of automata-theoretic explicit state model
checker SPIN
Specification Languages
• Specification languages for verification
– [Milner 80] CCS
– [Chandy and Misra 88] Unity
– [Lamport 94] Temporal Logic of Actions (TLA)
• Specification languages for model checking
– [Holzmann 98] Promela
– [McMillan 93] SMV
– [Alur and Henzinger 96, 99] Reactive Modules
Action Language TLA Connection
• Similarities:
– Transition relation is defined using predicates on current
(unprimed) and next state (primed) variables
– Each predicate is defined using integer arithmetic,
boolean logic, etc.
• Differences: In Action Language
– Temporal operators are not used in defining the
transition relation
• Dual language approach: temporal properties (in
CTL) are redundant, they are used to check
correctness
– Synchronous and asynchronous composition operators
are not equivalent to logical operators
Constraint-Based Verification
• [Cooper 71]
– Used a decision procedure for Presburger arithmetic to
verify sequential programs represented in a block form
• [Cousot and Halbwachs 78]
– Used real arithmetic constraints to discover invariants of
sequential programs
• [Halbwachs 93]
– Constraint based delay analysis in synchronous
programs
• [Halbwachs et al. 94]
– Verification of linear hybrid systems using constraint
representations
• [Alur et al. 96]
– HyTech, a model checker for hybrid systems
Constraint-Based Verification
• [Boigelot and Wolper 94]
– Verification with periodic sets
• [Boigelot et al.]
– Meta-transitions, accelerations
• [Delzanno and Podelski 99]
– Built a model checker using constraint logic
programming framework
• [Boudet Comon], [Wolper and Boigelot ‘00]
– Translating linear arithmetic constraints to automata
Automata-Based Representations
• [Klarlund et al.]
– MONA, an automata manipulation tool for verification
• [Boudet Comon]
– Translating linear arithmetic constraints to automata
• [Wolper and Boigelot ‘00]
– verification using automata as a symbolic representation
• [Kukula et al. 98]
– application of automata based verification to hardware
verification
Combining Symbolic Representations
• [Chan et al. CAV’97]
– both linear and non-linear constraints are mapped to
BDDs
– Only data-memoryless and data-invariant transitions are
supported
• [Bharadwaj and Sims TACAS’00]
– Combines automata based representations (for linear
arithmetic constraints) with BDDs
– Specialized for inductive invariant checking
• [Bensalem et al. 00]
– Symbolic Analysis Laboratory
– Designed a specification language that allows
integration of different verification tools
Model Checking Programs
• Verisoft from Bell Labs [Godefroid POPL 97]
– C programs, handles concurrency, bounded search,
bounded recursion, stateless search
• Java Path Finder (JPF) at NASA Ames [Havelund, Visser]
– Explicit state model checking for Java programs,
bounded search, bounded recursion, handles
concurrency
• SLAM project at Microsoft Research
[Ball, Rajamani et al. SPIN 00, PLDI 01]
– Symbolic model checking for C programs, unbounded
recursion, no concurrency
– Uses predicate abstraction [Saidi, Graf 97] and BDDs
• BANDERA: A tool for extracting finite state models from
programs [Dwyer, Hatcliff et al ICSE 00, 01]
Outline
• Difficulties in concurrent programming
• A short history of model checking
– 5 key ideas + 2 key tools
• Action Language
• Composite Symbolic Library
• Application to concurrency controllers
• Application to concurrent linked lists
• Related work
• Current and future work
Current and Future Work
• Automata representation for linear arithmetic constraints
• Interface based specification and verification of
concurrency controllers
• Specification and verification of web services
Automata Representation for Arithmetic Constraints
[Bartzis, Bultan, CIAA 02], [Bartzis, Bultan, IJFCS]
[Bartzis, Bultan TACAS 03], [Bartzis, Bultan CAV 03]
• Given a linear arithmetic formula construct a deterministic
finite automaton that accepts the integers that satisfy the
formula.
• Used MONA package
1
0
0
1
1
0
0
• Complexity results
0
0
0
1
-2
0
1
0 1
1, 1
A finite automaton for
2x - 3y = 2
-1
0 1
0, 0
1
1
0 1
0, 0
1
1
0 1
1, 1
sink
0 0 1 1
0, 1, 0, 1
0
Concurrency Controllers and Interfaces
[Betin-Can, Bultan SoftMC 03]
•
•
•
Concurrency Controller
– Behavior: How do the shared variables change
– Interface: In which order are the methods invoked
Separate Verification
– Behavior verification
• Action Language Verifier
– Interface verification
• Java PathFinder
A modular approach
– Build complex concurrency controllers by composing
interfaces
Example Interface
park2
reqLand
leave
park7
crossRW3
exitRW3
crossRW4
exitRW4
crossRW5
exitRW5
crossRW6
exitRW6
park9
park10
crossRW7
park11
crossRW8
exitRW7
exitRW8
reqTakeOff
Verification of Web Services
[Fu, Bultan, Hull, Su TACAS 01, WES 02], [Bultan,Fu,Hull, Su WWW 03],
[Fu, Bultan, Su CIAA 03]
• Verification of Vortex workflows using SMV and Action
Language Verifier
• A top-down approach to specification and verification of
composite web services
– Specify the composite web service as a conversation
protocol
– Generate peer specifications from the conversation
protocol
• Realizability conditions
• Working on the application of this framework to BPEL
msg1
Conversation
Schema
Peer A
msg2,
msg6
msg4
Peer B
msg3,
msg5
Peer C
BA:msg2 BC:msg5
Conversation
Protocol
AB:msg1
?
BA:msg6
BC:msg3
LTL property
G(msg1 F(msg3  msg5))
Model
Checking
C B:msg4
Peer
Synthesis
Peer A
Peer B
!msg1
Peer C
?msg1
!msg3
Input
Queue
?msg3
!msg2
?msg2
!msg5
?msg6
Virtual Watcher
?msg5
?msg4
!msg4
!msg6
...
The End