Overview of IC3/PDR

© Anvesh KomuravelliIC3/PDR

Overview of IC3/PDR

Anvesh KomuravelliCarnegie Mellon University


(<k)-reachable?

SAT-Based Bounded Model Checking (BMC)

Is there an execution leading to error in < k steps, beginning in I ?


(<k)-reachable?

BMC + k-Induction

k-inductive?

Is there an execution leading to error in < k steps, beginning in I ?Is there an execution leading to error in k steps, beginning in P ?


(<k)-reachable?

Downside of k-Induction

k-inductive?

P may not be inductive for any k !


Strengthen P !

reachable states


BMC + Interpolation (McMillan ‘03)

k-reachable?

(k-1)

k-reachable?

(k-1)

…

If abstract counterexample found, start all over again with a bigger k

Many improvements followed

Approximate Forward-Reachability!


Forward-Reachability in a nutshell

Initial States

Over-approximations

Also, w.l.o.g., assume that :

No counterexample of length (k-1) from Fn

No counterexample of length (k-1)+1 from Fn-1

…


Formalizing BMC + Interpolation

Rule Condition Transition

Init −

Unfold

Refine

Unsafe return UNSAFE

Safe return SAFE

Abstract Transition System




Init −

Unfold

Refine


Safe return SAFE

Abstract Transition SystemState triple




Init −

Unfold

Refine


Safe return SAFE

Downsides

• Blow-up in SAT formula size as k gets big• Resolution proof of UNSAT is non-trivial to obtain

Abstract Transition System


…

??

A different search strategy

Let us restrict to 1-reachable queries


? …






…


Formalizing the new search strategy


Init −

UnfoldCandidate

Decide

Conflict


Safe return SAFE




Init −

UnfoldCandidate

Decide

Conflict


Safe return SAFE

Checks k-reachability by explicit state backward search !


CDCL – Local Interpolants

Given

Find

Use algorithms to minimize cores (MUS)


…


Given

Find

t


…Strengthen

Fi+1


Given

Find


…Strengthen

Fi+1


Hence,


…Strengthen

Fi


Hence,



Init −

UnfoldCandidate

Decide

Conflict


Safe return SAFE



…

Forward Propagation

t is bad for Fi+2 as well!Can we reuse φ?

t


Forward Inductive Propagation

Given

Find


Forward Inductive Propagation

Given

Find

unsat



Init −

UnfoldCandidate

Decide

Conflict

Induction


Safe return SAFE



…

Forward Propagation

Block φ or s at Fi+2, Fi+3, …


…

Long Counterexamples!

k

m

…

Block φ or s at Fi+2, Fi+3, …



Init −

UnfoldCandidate

Decide

Conflict

Induction


Safe return SAFE



cube

Generalizing Predecessors

Given Find


cube


Given Find

cube



Given

Find

T

−−−

…

−−−

−−−

…−−

−

Ternary Simulation

…


…

?



… ?



…

…




Init −

UnfoldCandidate

Decide

Conflict

Induction


Safe return SAFE




Init −

UnfoldCandidate

Decide

Conflict

Induction


Safe return SAFE

IC3/PDR !


To summarize…

1-step reachability queriesGeneralizing PredecessorsLocal InterpolantsForward Inductive PropagationReusing Counterexamples

Fi is in CNF


To summarize…

• Competitive with variants of McMillan’s Interpolation• 3rd place in HWMCC’10 – competing with well-established tools• Well received by hardware industry• Implemented in Berkeley’s ABC tool• Extensions to progress and CTL properties• Extensions to LRA – implemented in Z3


Fi is in CNF

Efficient Implementation of IC3/PDR

SAT Context C

…


Decide/Conflict Rules

Assumptions A

Y N

Ternary Simulation


least j ≥ i such that

If none, add to F∞

MUS extraction to get

Conflict Rule

✗

Additionally,pushing the clause to

higher levels


Induction Rule

Similar to Conflict Rule, with repeated checks!


Extending to First-order Theories

can do sometheory-generalizationLocal Interpolants ?

LRA : Linear combination of literals (Hoder and Bjorner, 2012)

∞state



References

1. SAT-Based Model Checking without Unrolling, Bradley, VMCAI 20112. Efficient Implementation of Property Directed Reachability, Een,

Mishchenko and Brayton, FMCAD 20113. An Incremental Approach to checking Progress Properties, Bradley et al.,

FMCAD 20114. Understanding IC3, Bradley, SAT 20125. Generalized Property Directed Reachability, Hoder and Bjorner, SAT 20126. Incremental, Inductive CTL Model Checking, Hassan et al., CAV 2012

Overview of IC3/PDR

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