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A practical and complete approach to predicate abstraction

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### A practical and complete approach to predicate abstraction

Ranjit Jhala

UCSD

Ken McMillan

Cadence Berkeley Labs

Completeness of abstraction

- An abstraction is a restricted language L
- Example: predicate abstraction
- L is the language of Boolean combinations of predicates in P

- We try to compute the strongest inductive invariant of a program in L

- Example: predicate abstraction
- An abstraction refinement heuristic chooses a sequence of sublangauges L0µ L1,... from a broader langauge L.
- Example: predicate abstraction
- L is the set of quantifier-free FO formulas (QF)
- Li is characterized by a set of atomic predictes Pi

- Example: predicate abstraction

An abstraction refinement heuristic is complete for language L, iff it always eventually chooses a sublanguage Liµ L containing a safety invariant whenever L contains a safety invariant.

Predicates

Bad

Predicates

Program

x=i; y=i;

while(x!=0)

{x--; y--;}

assert y==0;

x=y

x=0, y=0

x=1, y=1

x=2, y=2

...

Divergence- Existing refinement heuristics for predicate abstraction are incomplete.
- They can produce an infinite sequence of refinements even when a saftey invariant exists in L.

Cause: CEGAR loop

- CounterExample Guided Abstraction Refinement (CEGAR)
- Abstract counterexample is sequence of minterms in Li.
- Refinement adds predicates sufficient to refute counterexample.
- Not complete, since predicates can diverge as number of loop iterations in counterexample increases.

Verify

unsafe

abstract

counterexample

Li

Refine

Abstraction

safe

Example: refinement using weakest procondition (WP)

x=0,y=0

x=i,y=i

x=i; y=i;

while(x!=0)

{x--; y--;}

assert y==0;

x¹0,y¹0

[x!=0]

x--, y--

Add these

predicate

x=0,y¹0

[x=0]

[y!=0]

x=0,y¹0

Error!

Divergence example- Most heuristics derive predicates in some way from the refutation of the counterexample.

i=1 ) i=1

x=1 ) y=1

x=0 ) y=0

False

1. Stratify L into finite languages L0µL1µL

...

L2

x=2

L1

L

L

L

L

L

x=1

L0

x=0

Enforcing completenessLattice of sublanguages

L

2. Refute counterexample

at lowest possible stratum

x=y

If a saftey invariant exists in Lk, then we never exit Lk. Since

this is f finite language, abstraction refinement must converge.

Split prover approach

- To restrict the refutation of counterexamples, we use a "split prover"
- Each prover component knows just one time frame
- Components can only communicate facts in Lk

R1

R2

R3

Rn

L

By restricting the language of communication between time

frames, we prevent the prover from using larger constants as

the number of loop iterations increase, and force it to generalize.

Programs

- A program is a pair (T, P) where
- T is a set of symbolic transition relations (program statements)
- PµT * is a regular language defining the unsafe runs of the program

x=y

False

True

Example: let L be {x=y}:

Error!

[x!=0]

x--, y--

[x=0]

[y!=0]

x=i,y=i

Predicate abstraction- Definitions:
- Given a set of fmlas L, let B(L) be the set of Boolean combinations of L
- Let spL(s)(f), where s2T be the strongest -postcondition of f in B(L)
- Let spL(p1Lpn) = spL(p1) ±L± spL(pn)

- A program path pis L-refutable when spL(p)(True)=False

Fact: Predicate abstraction with predicates b proves a program safe exactly when every unsafe path p2P is b-refutable. A counterexample for predicate abstraction is a non-b-refutable path in P.

Consequence finders

- A consequence finder takes as input a set of hypothese and returns a set of consequences of .
- Consequence finder R is complete for L-generation iff, for any f2 L
G²fimpliesR(G)Å L²f

That is, the consequence finder need not generate all consequences

of in L, but the generated L-consequences must imply all others.

R1

R2

R3

Rn

L

- Each Ri knows just i(*)
- Ri and Ri+1 exchange only facts in L(i)ÅL(i+1)
- iRi is the composition of the Ri’s

Divide the prover into a sequence of communicating consequence finders...

Theorem: If each Ri is complete for L(i+1)-generation, then

iRi is complete for refutation [McIlraith & Amir, 2001].

*Actually, here we mean i instantiated at time i, as in BMC

L-restricted split prover

- In the L-restricted composition, LRi, the provers can exchange only formulas in L.

R1

R2

R3

Rn

L

L

L

L

L

Theorem: If each Ri is complete for LÅL(Ti+1)-generation, then path is L-refutable exactly when is refuted by LRi.

Corrolary: Let b’ be the set of AP’s exchanged by LRi in refuting .

is ’-refutable

Complete heuristic

- Given finite languages L0µ L1, µL where [ Li = QF...

bÃ{}

Pred

Abs

safe

p not b-refutable

k Ã 0

’ µ Lk

s.t. is

’-refutable?

yes

k Ã k+1

no

b Ã[’

Theorem: This procedure is complete for QF. That is, if a

safety invariant exists in QF, we conclude "safe".

An efficient split prover

- Complete consequence generation could be expensive!
- We will consider QF formulas with
- integer difference bound constraints (e.g., x · y + c)
- equality and uninterpreted functions
- restricted use of array operations "select" and "store"

- Our restriction language Lk will be determined by
- The finite set CD of allowed constants c in x · y + c
- The finite set CB of allowed constants c in x · c
- The bound bf on the depth of nesting of function sybols

Note that for a finite vocabulary, Lk is finite, and as long as the

constant and depth bounds are increasing, every formula is included

in some Lk.

=,f

=,f

=,f

L

·+

·+

·+

·+

Prover architecture- SAT solver generates propositionally satisfying minterms
- Split prover refutes this minterm
- Hypotheses of split prover are thus literals, not clauses

- Convexity: theory is convex if all consequences are Horn
- In convex case, provers only exchange literals [Nelson & Oppen, 1980]
- Simple proof rules for complete unit consequence finding
- In case of a non-Horn consequence, split cases in SAT solver
- Integers and array store operations introduce non-convexities.

- Multiple theories handled by hierarchical decomposition

These and other optimizations can result in a relatively efficient prover...

Performance comparison

- Refuting counterexamples for two hardest Widows device driver examples in the Blast benchmark set.
- Compare split prover against Nelson-Oppen style, same SAT solver

Some "trivial" benchmarks

example: substring copy

main(){

char x[*], z[*];

int from,to,i,j,k;

i = from;

j = 0;

while(x[i] != 0 && i < to){

z[j] = x[i];

i++;

j++;

}

/* prove strlen(z) >= j */

assert !(k >= 0 && k < j && z[k] == 0);

}

Results

X = refine fail, = bug, = diverge, TO = timeout, = verified safe

Conclusions

- An abstraction refinement heuristic is complete for langauge L if it guarantees to find a safety invariant if one exists in L
- Existing PA heuristics are incomplete and diverge on trivial programs
- CEGAR can be made complete by...
- Stratifying L into hierarchy of finite sublanguages L0, L1, ...
- Refuting counterexamples as low as possible in hierarchy
- Using Lk-restricted split prover

- A split prover can be made efficient enough to use in practice
- (at least for some useful theories)

- Future work:
- New theories (transitive closure?)
- Quantified invariants, indexed predicate abstraction
- Interpolant-based software model checker (coming soon)

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