verification of java programs using symbolic execution and loop invariant generation corina...
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Verification of Java Programs using Symbolic Execution and Loop Invariant Generation
Corina Pasareanu (Kestrel Technology LLC)Willem Visser (RIACS/USRA)
Automated Software Engineering GroupNASA Ames
Outline
• Motivation and Overview
• Examples
• Symbolic Execution and Java PathFinder
• Program Verification and Invariant Generation
• Experiments
• Related Work and Conclusions
Motivation
Ariane 501
Mars Polar Lander
Software errors can be very costly. Software verification is recognized as an important and difficult problem.
Spirit
More recently …
Java PathFinder with Symbolic Execution
Previous work:• Java PathFinder (JPF) - explicit-state model checker for Java• Extended with symbolic execution [TACAS’03]
– Motivation• Open systems, large input data domains• Complex data structures
– Applications: test-input generation, error detection– Shortcoming: cannot prove properties of looping programs
New:• Invariant generation to deal with loops
Verification Framework Overview
• Uses symbolic execution
• Requires annotations– method preconditions – loop invariants
• Novel technique for invariant generation– uses invariant strengthening, approximation, and
refinement– handles boolean and numeric constraints,
dynamically allocated structures, arrays
Array Example 1
// precondition: a!=null;public static void set(int a[]) { int i = 0; while (i < a.length) { a[i] = 0; i++; } assert a[0] == 0;}
Loop invariants: 0≤i ¬(a[0] 0 0<i)
Array Example 2
// precondition: a!=null;public static void set(int a[]) { int i = 0; while (i < a.length) { a[i] = 0; i++; } assert forall int j: a[j] == 0;}
Loop invariant:¬(a[j] 0 a.length ≤ i 0 ≤ j < a.length) ¬(a[j] 0 j < i 0 ≤ i,j < a.length)
Symbolic Execution
• Execute a program on symbolic input values• For each path, build a path condition
– condition on inputs in order for the execution to follow that path– check satisfiability of path condition
• Symbolic state– symbolic values/expressions for variables– path condition– program counter
• Various applications– test case generation– program verification
• Traditionally: sequential programs with fixed number of integers
x = 1, y = 0
1 > 0 ? true
x = 1 + 0 = 1
y = 1 – 0 = 1
x = 1 – 1 = 0
0 > 1 ? false
Swap Example
int x, y;
if (x > y) {
x = x + y;
y = x – y;
x = x – y;
if (x > y)
assert false;
}
Concrete Execution Path:Code that swaps 2 integers:
Swap Example
[PC:true] x = X, y = Y
[PC:true] X > Y ?
[PC:X>Y] y = X + Y – Y = X
[PC:X>Y] x = X + Y – X = Y
[PC:X>Y] Y > X ?
int x, y;
if (x > y) {
x = x + y;
y = x – y;
x = x – y;
if (x > y)
assert false;
}
Code that swaps 2 integers: Symbolic Execution Tree:
[PC:X≤Y] END [PC:X>Y] x = X+Y
false true
[PC:X>YY≤X ] END [PC:X>YY>X] END
false true
path condition
Generalized Symbolic Execution
• Handles– dynamically allocated data structures, arrays
– preconditions, concurrency
• Uses JPF – to generate and explore the symbolic execution tree
• Implementation via instrumentation– programs instrumented to enable JPF to perform
symbolic execution
– Omega library used to check satisfiability of numeric path conditions (for linear integer constraints)
– lazy initialization for arrays and structures
Instrumentation
void set (int a[]) { int i = 0;
while (i < a.length) { a[i] = 0; i++; } assert a[0] == 0;}
void set() { IntArrayStruct a = new IntArrayStruct(); Expression i = new IntConstant(0); while (i._LT(a.length)) { a._set(i,0); i = i._plus(1); } assert a._get(0)._EQ(0);}
Instrumented code
Library classes
class Expression { … static PathCondition pc; Expression _plus(Expression e) {…} boolean _LT(Expression e) { return pc._update_LT(this,e); } }
class PathCondition { … Constraints c; boolean _update_LT(Expression l, Expression r) { boolean result = Verify.choose_boolean(); if(result) c.add_constraint_LT(l, r); else c.add_constraint_GE(l, r); Verify.ignoreIf(!c.is_satisfiable()); return result; } }
class ArrayCell { Expression elem; Expression idx; }class IntArrayStruct { … Vector _v; Expression length;
public Expression _get(Expression idx) { Verify.ignoreIf !inbounds; //assert inbounds ArrayCell cell = _ArrayCell(idx); return cell.elem;}
ArrayCell _ArrayCell(Expression idx) { for(int i=0; i<_v.size(); i++) { ArrayCell cell=(ArrayCell)_v.elementAt(i); if(cell.idx._EQ(idx)) return cell; } ArrayCell ac = new ArrayCell(…); _v.add(ac); return ac; } }
Induction Step
Base Case
Proving Properties of Programs
X = init;while (C(X)) X = B(X);assert P(X);
Looping program:
Program execution:
while …true
while …true
while …true
…May be infinite …
How to reason about infinite executions?
Has finite execution.Easy to reason about!
Problem: How do we come up with Inv?Requires great user ingenuity.
X = init;assert Inv(X);X = new symbolic values;assume Inv(X);if (C(X)) { X = B(X); assert Inv(X); } else assert P(X);
Non-looping program:
Find loop invariant Inv
Iterative Invariant Strengthening
Model check the program:
Start with Inv0 = ¬(¬C ¬P)
Base case violation: error in the program!
No errors: done, found loop invariant!
Induction step violation: apply strengthening
- counterexample path conditions:
PC1, PC2 … PCn
- strengthen invariant:
Inv1 = Inv0 ¬ PCi
- repeat
X = init;assert Inv(X);X = new symbolic values;assume Inv(X);if (C(X)) { X = B(X); assert Inv(X); } else assert P(X);
Iterative Strengthening
Inv0 Inv1 …
More precise invariants
State space:
Inv
May result in an infinite sequence of exact invariants:
Inv0, Inv1, Inv2 …
(we may get infinitely many generated constraints)
Heuristic for Termination
At each step k, apply heuristic for current candidate Invk
-it is also iterative strengthening, but … -use oldPC instead of PC - oldPC is weaker than PC (PC → oldPC)- obtains a stronger invariant:
Invkj+1 = Invk
j¬ oldPCi
Iterative approximation
X = init;assert Inv(X);X = new symbolic values;assume Inv_k(X);if (C(X)) { X = B(X); assert Inv_k(X); } else assert P(X);
Check the inductive step:
oldPC= q r
PC= q r v
new constraint(encodes the effect of the loop)
X = init;assert Inv(X);X = new symbolic values;assume Inv_k(X);if (C(X)) { X = B(X); assert Inv_k(X); } else assert P(X);
Check the inductive step:
Iterative Approximation
Symbolic execution results in finite universe of constraints Uk
New constraints from Inv(B(X))
Refinement:•if base case fails for Invk
j
•backtrack•compute Invk+1
•apply approximation
State space at step k:Approximation
too coarse Invk
InvInvk+1
Invk1
PColdPC
Results in finite sequence of approximate invariants: Invk
1, Invk2 … Invk
m
Invariant Generation Method
Inv0 Inv1 … Invk Invk+1 …
Invk1 Invk
2 … Invkm
Refinement - backtrack on base case violation
Iterative strengthening
Iterative approximation
•If there is an error in the program, the method is guaranteed to terminate•If the program is correct wrt. the property, the method might not terminate
Array Example 1
// precondition: a!=null;public static void set(int a[]) { int i = 0; while (i < a.length) { a[i] = 0; i++; } assert a[0] == 0;}
[PC: I<a.length a[0]0] i=I
public static void set(int [] a) { int i = 0; assert Inv; //i,a = new symbolic values; assume Inv; if (i < a.length) { a[i]=0; i++; // oldPC if (!Inv) { // PC assert false; } } else assert a[0]==0;}
Proof + tree
Error
Inv0 = ¬(i ≥ a.length a[0] 0)=
(i<a.length a[0] 0) (i<a.length a[0] = 0) (i ≥ a.length a[0] = 0)
[PC: 0<I<a.length a[0]0] a[I]=0
[PC: I<a.length a[0]0 I0 0≤I<a.length][PC: I<a.length a[0]0 I=0] …
[PC: 0<I<a.length a[0]0] i=I+1
[PC: I<a.length a[0]0] i=I
public static void set(int [] a) { int i = 0; assert Inv; //i,a = new symbolic values; assume Inv; if (i < a.length) { a[i]=0; i++; // oldPC if (!Inv) { // PC assert false; } } else assert a[0]==0;}
Proof + tree
Error
[PC: 0<I<a.length a[0]0] a[I]=0
[PC: I<a.length a[0]0 I0 0≤I<a.length][PC: I<a.length a[0]0 I=0] …
Inv0 = ¬(i ≥ a.length a[0] 0)=
(i<a.length a[0] 0) (i<a.length a[0] = 0) (i ≥ a.length a[0] = 0)
[PC: 0<I<a.length a[0]0] I+1≥a.length a[0]0 ?
[PC: 0<I<a.length a[0]0] i=I+1
[PC: I<a.length a[0]0] i=I
public static void set(int [] a) { int i = 0; assert Inv; //i,a = new symbolic values; assume Inv; if (i < a.length) { a[i]=0; i++; // oldPC if (!Inv) { // PC assert false; } } else assert a[0]==0;}
Proof + tree
Error
[PC: 0<I<a.length a[0]0] a[I]=0
[PC: I<a.length a[0]0 I0 0≤I<a.length][PC: I<a.length a[0]0 I=0] …
Inv0 = ¬(i ≥ a.length a[0] 0)=
(i<a.length a[0] 0) (i<a.length a[0] = 0) (i ≥ a.length a[0] = 0)
[PC: 0<I<a.length a[0]0 I+1≥a.length]
true
Error
…
[PC: 0<I<a.length a[0]0] I+1≥a.length a[0]0 ?
[PC: 0<I<a.length a[0]0] i=I+1
[PC: I<a.length a[0]0] i=I
public static void set(int [] a) { int i = 0; assert Inv; //i,a = new symbolic values; assume Inv; if (i < a.length) { a[i]=0; i++; // oldPC if (!Inv) { // PC assert false; } } else assert a[0]==0;}
Proof + tree
Error
[PC: 0<I<a.length a[0]0] a[I]=0
[PC: 0<I<a.length a[0]0 I+1≥a.length]
true
[PC: I<a.length a[0]0 I0 0≤I<a.length][PC: I<a.length a[0]0 I=0] …
Iterative approximation:Inv0
1 =Inv0 ¬oldPC = ¬(i ≥ a.length a[0] 0) ¬(0<i <a.length a[0] 0) = ¬(i > 0 a[0] 0)
oldPC: 0<i <a.length a[0] 0
PC: 0<i <a.length a[0] 0 (i + 1) ≥ a.length
oldPC:
PC:
Inv0 = ¬(i ≥ a.length a[0] 0)=
(i<a.length a[0] 0) (i<a.length a[0] = 0) (i ≥ a.length a[0] = 0)
Error
…
Array Example 2
// precondition: a!=null;public static void set(int a[]) { int i = 0; while (i < a.length) { a[i] = 0; i++; } assert forall int j: a[j] == 0;}
Proof
Inv0 = ¬(i ≥ a.length a[j] 0 0 ≤ j< a.length)
oldPC: a[j] 0 j < i 0 ≤ i,j < a.length
Iterative approximation:Inv0
1 =Inv0 ¬oldPC = ¬(i ≥ a.length a[j] 0 0 ≤ j< a.length) ¬(a[j] 0 j < i 0 ≤ i,j < a.length)
PC: a[j] 0 j<i 0 ≤ i,j <a.length (i + 1) ≥ a.length
public static void set(int [] a) { int i = 0; assert Inv; //i,a = new symbolic values; //j = new symbolic value; assume Inv; if (i < a.length) { a[i]=0; i++; // oldPC if (!Inv) { // PC assert false; } } else assert a[j]==0;}
Partition Example
class Cell {int val;Cell next;
Cell partition (Cell l, int v) { Cell curr = l, prev = null; Cell nextCurr, newl = null; while (curr != null) { nextCurr = curr.next; if (curr.val > v) { if (prev != null) prev.next = nextCurr; if (curr == l) l = nextCurr; curr.next = newl; assert curr != prev; newl = curr; } else prev = curr; curr = nextCurr; } return newl; }}
Loop invariant: ¬(curr=prev curr≠null curr.elem>v) ¬(curr≠ prev prev≠ null curr ≠ null prev.elem>v curr.elem>v prev≠curr.next)
Pathological Example
void m (int n) { int x = 0; int y = 0; while (x <n) {/* loop 1 */ x++; y++; } /* hint: x == y */ while (x!=0) {/* loop 2 */ x--; y--; } assert y==0;}
•First, attempt computation of invariant for loop 2
•Iterative invariant generation does not terminate
•Constraint x=y is important, but not discovered
•Using x=y as a hint we get two invariants:
Loop 2:
¬(y 0 x=0) ¬(y ≤ 0 x>0) ¬(y>0 x y)
Loop 1:
¬(x y x ≥ n) ¬(x<0) ¬(x≥0 x<n x y)
Related Work
• Invariant generation:– INVEST (Verimag), STEP (Stanford)– Graf & Saidi (CAV 1996), Havelund & Shankar (FME 1996), Tiwari et al. (TACAS
2001), Wegbreit (CACM 1974)– … (a lot of work)– Iterative forward/backward computations– Domain specific; focus on numeric invariants– Heuristics for termination, e.g. using auxiliary invariants
• Abstract interpretation– Cousot & Cousot (CAV 2002), Cousot & Halbwachs (POPL 1978)– Widening operator to compute fixpoints systematically
• Flanagan & Qadeer (POPL 2002)– Loop invariant generation for Java programs– Uses predicate abstraction– Predicates need to be provided by the user
• Extended Static Checker (ESC)– Uses theorem proving to check partial correctness specifications of Java programs– Rely heavily on user provided specifications, such as loop invariants
Conclusion and Future Work
• Framework for verification of light-weight specifications of Java programs: new use of JPF
• Iterative technique for discovering (some) loop invariants automatically– Uses invariant strengthening, approximation, and refinement– Handles different types of constraints– Allows checking universally quantified formulas
• … Very preliminary work
• Future work:– Instead of dropping newly generated constraints, replace them with an
appropriate boolean combination of exiting constraints from Uk
• Similar to predicate abstraction– Use more powerful abstraction techniques in conjunction with our framework– Use heuristics/dynamic methods to discover useful constraints/hints (e.g. Daikon)– Study relationship to widening and predicate abstraction– Extend to multithreading and richer properties– …