Interpolant Compaction In Unbounded Model Checking Danilo - - PowerPoint PPT Presentation

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Interpolant Compaction In Unbounded Model Checking Danilo - - PowerPoint PPT Presentation

Feb 11, 2024 460 likes •884 views

Alpine Verification Meeting 2013, FBK, Trento, Italy. Optimization Techniques For Craig Interpolant Compaction In Unbounded Model Checking Danilo Vendraminetto PhD student at Formal Methods Group, Politecnico di Torino, Torino, Italy. Before

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Optimization Techniques For Craig Interpolant Compaction In Unbounded Model Checking

Danilo Vendraminetto

PhD student at Formal Methods Group, Politecnico di Torino, Torino, Italy. Alpine Verification Meeting 2013, FBK, Trento, Italy.

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Before starting..

 Talk in part based on a paper we presented at

DATE 2013 Conference:

Gianpiero Cabodi, C. Loiacono, D. Vendraminetto.

Optimization techniques for craig interpolant compaction in unbounded model checking. DATE 2013: 1417-1422

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Outline

 Motivations & background

 Hardware designs verification  Craig Interpolants in MC  ITP size compaction & scalability

 Contributions

 Redundancy removal and reduction of

 UNSAT proofs  Craig interpolants

 Heuristic procedure for scalable ITP compaction

 Experimental results & Conclusions

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Motivations

 Can ITPs compete with IC3 ?  Main limitations of ITP

 BMC-based model (vs. cube/clause-based reachability)  ITPs are highly redundant

IC3 ITP 2-level (AND-OR) characteristic functions Multiple level circuits Single instance of TR TR unrollings

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Motivations

 Can ITPs compete with IC3 ?  Main limitations of ITP

 BMC-based model (vs. cube/clause-based reachability)  ITPs are highly redundant

IC3 ITP 2-level (AND-OR) characteristic functions Multiple level circuits Single instance of TR TR unrollings

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Bounded Model Checking

 Trading off completeness for productivity

find BUGs !!!

I F T T T CNF clauses

SAT solver

Buggy states Initial states

Gianpiero Cabodi - IBM2011

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Interpolation [Craig’57]

 Given AB = 0  A’ = interpolant(A,B)

 A A’  A’B = 0  A’ refers only to common variables of A,B

 Interpolants from proofs

 Given a resolution refutation of AB  A' is derived in linear time and space

[Pudlak,Krajicek’97]

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Interpolation [McMillan’03]

 Interpolant as over-approx. image operator

 Over-approximation  Variable quantification

 Works whenever a representation of

backward reachable space is given

 A: From T (FWD)  B: paths to failure states (BWD)  A’: over-approx image

 Approx image is called adequate w.r.t. B

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ITP from refutation proof

A B CNF clauses

UNSAT problem (AB = 0)

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ITP from refutation proof

A B

Resolution graph

Null clause

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ITP from refutation proof

A B

Resolution graph

Null clause

Unsatisfiable core Resolution rule (A  p) (p  B) (A  B)

pivot variable

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ITP from refutation proof

A B Null clause

Unsatisfiable core Resolution rule (A  p) (p  B) (A  B)

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Interpolant from refutation proof

A B

Resolution graph

Null clause

AND-OR circuit

1 A’ = Interpolant (A,B)

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Interpolant from refutation proof

A B

Resolution graph

Null clause

AND-OR circuit

1 A’ = Interpolant (A,B) A gate for each resolution node

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Interpolant rules

 Interpolation is a circuit that follows the

structure of the proof

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A = (p)(p  q) B = (q  r)(r)

(p) (p  q) (q) (q  r) (r) (r) ^ ^ ^ ^ q =q

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W V V’

Image+

From

T

T To+ To

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W V V’

Image+

From

T

T To+ To To+(V’) = IMG+(From,T) = Approx((V,W)From(v)T(V,W,V’))

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Adequate Image+

From

T

T To+ B To

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Adequate Image+

From

T

T To+ B To To+ adequate w.r.t. B

  • if To outside B
  • then To+ outside B
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Adequate Image+ by Interpolant

From

T

T To+ B To To+ = interpolant (From T,B)

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ITP

Rk,bwd F Fromi T T T T Toi T To+i

Standard ITP: to+i computed from appr. Fromi

A B

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Why use adequate IMG+ ?

 FWD approximate reachable states

 computed by adequate IMG+  do not intersect BWD reachable states

Rbwd R+ I F IMG+ adequate

…

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ITP compaction

Proof reduction ITP circuit compaction

Alternative proofs

  • different resolution schemes

BDD/SAT sweeping Const propagation Equivalent proofs

  • redundancy removal

ODC

Refactor rewrite

Resolution

graph

AND-OR

circuit

1

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ITP compaction

Proof reduction ITP circuit compaction

Alternative proofs

  • different resolution schemes

BDD/SAT sweeping Const propagation Equivalent proofs

  • redundancy removal

ODC

Refactor rewrite

Resolution

graph

AND-OR

circuit

1

Problem #1:

  • SCALABILITY
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Proof reduction

 Recycle-pivots [Bar-Inal & al. HVC08]

C1 (1 2 3) C2 (-2 4) C3 (1 3 4) C4 (-1 -2 5) C6 (2 6) C5 (-2 3 4 5) C7 (3 4 5 6) C1 (1 2 3) C2 (-2 4) C3 (1 3 4) C4 (-1 -2 5) C6 (2 6) C5 (-2 3 4 5) C7 (3 4 5 6)

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C1 (1 2 3) C2 (-2 4) C3 (1 3 4) C4 (-1 -2 5) C6 (2 6) C5 (-2 3 4 5) C7 (3 4 5 6) C2 (-2 4) C3 (-2 4) C6 (2 6) C5 (-2 4) C7 (4 6)

RL = {-2 1} RL = {-2}

RL denotes the Removable-Literals

Proof reduction

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 Recycle-pivots + restruct proof [Bar-Inal & al.

HVC08]

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Our Contribution: exploit proof topology

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Our Contribution: exploit proof topology

Proof node chain

 Simpler data structure for proof reduction algorithms

and further techniques

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ITP Circuit Compaction

 Logic synthesis manipulations on the proof

 Constant propagation  BDD-based sweeping (for equivalences)  Observability Don’t Care (lightweight)

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 Proof into AIG

 ODC (lightweight)  Logic synthesis

 rewrite / refactor, using ABC tool  AIG balance

 ITE-based decomposition (iff necessary)

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Observability don’t care

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 If A == 0  out = 0 ; no matters f(.) or g(.)

 don’t-care set

A f(x, .. , A) g(x, .. , A)

  • ut
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Observability don’t care

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 If A == 1  f(.) and g(.) can be simplified

 care set

A f(x, .. , 1) g(x, .. , 1)

  • ut
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ITP ITE decomposition

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x1 xN ITP

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ITP ITE decomposition

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x1 xN ITP 1

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ITP ITE decomposition

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x1 xN ITP 1 Ni

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ITP0

ITP ITE decomposition

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x1 xN ITP1 1 1

MUX 1 X Ni ITP1 ITP0 ITP

1 0 x1 xN

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Ad-Hoc ITP compaction

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AigIteDecomp (ITP)

if (max recursions || |ITP| < th) standardLogicSynth (ITP) do search node Ni with highest FO

ITE(Ni,ITP1,ITP0) //compute cofactors; equals to ITP

if (accept (ITE decomp)) //size-based heuristic AigIteDecomp (Ni) AigIteDecomp (ITP1) AigIteDecomp (ITP0) ITP = ITE(Ni,ITP1,ITP0) while max try reached

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Experimental results

 Framework: PdTrav

 State-of-the-art academic Model Checker  HWMCC ’07 to ‘12

  • Ranked 1st at 2010 Model Checking Competition – UNSAT

category

 ITP compaction => better MC runs  Experience on IBM & Intel benchmarks

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200 400 600 800 1000 1200 Time [s] Circuit name Best Opt Time Std Itp Time

Experimental results

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Conclusions

 ITP-based MC heavily relies on scalability,

i.e. ability to compact ITPs

 We developed effective techniques to

compact ITPs.

 Scalable techniques, applied incrementally

 Best suited as a second engine

 Hard-to-prove properties (hard for IC3)  Explosion of standard interpolation  Can afford extra time (for memory)

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Thank you!