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Self-Loop Aggregation Product A New Hybrid Approach to On-the-Fly LTL Model Checking Alexandre Duret-Lutz (LRDE/EPITA) Kais Klai (LIPN/Paris 13) Denis Poitrenaud (LIP6/Paris 6) Yann Thierry-Mieg (LIP6/Paris 6) ATVA 2011 October 2011

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SLIDE 1

Self-Loop Aggregation Product — A New Hybrid Approach to On-the-Fly LTL Model Checking

Alexandre Duret-Lutz (LRDE/EPITA) Kais Klai (LIPN/Paris 13) Denis Poitrenaud (LIP6/Paris 6) Yann Thierry-Mieg (LIP6/Paris 6) ATVA 2011 October 2011 http://move.lip6.fr/software/DDD/ltl_bench.html

ATVA’11 Self-Loop Aggregation Product 1 / 17

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SLIDE 2

Automata-Theoretic Explicit LTL Model Checking

High-level model M State-space generation State-space automaton AM LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ Synchronized product L (A¬ϕ ⊗ AM) = L (A¬ϕ)∩L (AM) Product Automaton A¬ϕ ⊗ AM Emptiness check L (A¬ϕ⊗AM) ? = ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 2 / 17

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SLIDE 3

Automata-Theoretic Explicit LTL Model Checking

High-level model M State-space generation LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ Synchronized product L (A¬ϕ ⊗ AM) = L (A¬ϕ)∩L (AM) State-space automaton AM Product Automaton A¬ϕ ⊗ AM Emptiness check L (A¬ϕ⊗AM) ? = ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 2 / 17

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SLIDE 4

Automata-Theoretic Explicit LTL Model Checking

High-level model M On-the-fly generation

  • f state-space automaton

AM LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ Synchronized product L (A¬ϕ ⊗ AM) = L (A¬ϕ)∩L (AM) Product Automaton A¬ϕ ⊗ AM Emptiness check L (A¬ϕ⊗AM) ? = ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 2 / 17

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SLIDE 5

Automata-Theoretic Explicit LTL Model Checking

High-level model M On-the-fly generation

  • f state-space automaton

AM LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ On-the-fly synchronized product L (A¬ϕ ⊗ AM) = L (A¬ϕ) ∩ L (AM) Emptiness check L (A¬ϕ⊗AM) ? = ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 2 / 17

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SLIDE 6

Explicit Approach

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤

ATVA’11 Self-Loop Aggregation Product 3 / 17

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SLIDE 7

Explicit Approach

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ A¬ϕ ⊗ AM: q0, s0

q0, s1 q0, s2 q0, s3 q0, s4 q1, s5 q1, s6 q1, s7 q1, s4

a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c a¯ bc ab¯ c a¯ bc ¯ ab¯ c ¯ abc ab¯ c Emptiness check = search for an accepting cycle in the product State explosion problem

ATVA’11 Self-Loop Aggregation Product 3 / 17

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SLIDE 8

Symbolic Observation Graph (SOG)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ For stuttering invariant properties (e.g., LTL\ X) Ignore non-observable propositions Aggregate Kripke states with homogeneous labels Represent aggregates using BDDs

  • K. Klai and D. Poitrenaud. MC-SOG: An LTL model checker based on

symbolic observation graphs. In Proc. of PN’08, vol. 5062 of LNCS, pp. 288–306. Springer

ATVA’11 Self-Loop Aggregation Product 4 / 17

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SLIDE 9

Symbolic Observation Graph (SOG)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ SOG: { s0 s1

s2 s3 }

a¯ b {s4} {s5} {s6s7} a¯ b a¯ b ab a¯ b ¯ ab

  • K. Klai and D. Poitrenaud. MC-SOG: An LTL model checker based on

symbolic observation graphs. In Proc. of PN’08, vol. 5062 of LNCS, pp. 288–306. Springer

ATVA’11 Self-Loop Aggregation Product 4 / 17

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SLIDE 10

Product Sizes: Kripke vs. SOG

A¬ϕ ⊗ AM: q0, s0

q0, s1 q0, s2 q0, s3 q0, s4 q1, s5 q1, s6 q1, s7 q1, s4

a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c a¯ bc ab¯ c a¯ bc ¯ ab¯ c ¯ abc ab¯ c A¬ϕ ⊗ SOG: q0, { s0 s1

s2 s3 }

q0, a¯ b q0, {s4} q1, {s5} q1, {s6s7} q1, {s4} a¯ b a¯ b a¯ b ab a¯ b ¯ ab ab

  • K. Klai and D. Poitrenaud. MC-SOG: An LTL model checker based on

symbolic observation graphs. In Proc. of PN’08, vol. 5062 of LNCS, pp. 288–306. Springer

ATVA’11 Self-Loop Aggregation Product 5 / 17

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SLIDE 11

BCZ: Multiple-State Tableaux

Similar to SOG with only 1 step par aggregate:

supports full LTL no need to search for livelock cycles

Low aggregation power: on our example with low branching, it does not reduce the Kripke structure.

  • A. Biere, E. M. Clarke, and Y. Zhu. Multiple state and single state tableaux

for combining local and global model checking. In Correct System Design,

  • vol. 1710 of LNCS, pp. 163–179. Springer, 1999

ATVA’11 Self-Loop Aggregation Product 6 / 17

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Building a Product Directly

High-level model M On-the-fly generation

  • f state-space automaton

AM LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ On-the-fly synchronized product L (A¬ϕ ⊗ AM) = L (A¬ϕ) ∩ L (AM) Emptiness check L (A¬ϕ⊗AM) ? = ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 7 / 17

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SLIDE 13

Building a Product Directly

High-level model M LTL property ϕ LTL translation Negated property au- tomaton A¬ϕ Dynamic and on-the-fly generation

  • f an automaton D such that

L (D) = ∅ ⇐ ⇒ L (A¬ϕ ⊗AM) = ∅. Emptiness check L (D)

?

= ∅ M | = ϕ or counterexample

ATVA’11 Self-Loop Aggregation Product 7 / 17

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SLIDE 14

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 15

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 16

Self-Loop Aggregation Product (SLAP)

AM: s0 s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 17

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 18

Self-Loop Aggregation Product (SLAP)

AM: s0 s0 s1 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 19

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 20

Self-Loop Aggregation Product (SLAP)

AM: s0 s0 s1 s1 s2 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 21

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 22

Self-Loop Aggregation Product (SLAP)

AM: s0 s0 s1 s1 s2 s2 s3 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product:

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 23

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • ATVA’11

Self-Loop Aggregation Product 8 / 17

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SLIDE 24

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 25

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 26

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

q1, { s4 s5

s6 s7 }

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 27

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 28

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 29

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 30

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s6 s7 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 31

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 32

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s4 s5 s6 s6 s7 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 33

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 34

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s4 s5 s5 s6 s6 s7 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 35

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

q1, { s4 s5

s6 s7 }

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 36

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s4 s5 s5 s6 s6 s7 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

q1, { s4 s5

s6 s7 }

⊤ ⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 37

Self-Loop Aggregation Product (SLAP)

AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ Self-Loop Aggregation Product: q0, s0 s1

s2 s3 s4

  • q1, {s5}

q1, { s4 s5

s6 s7 }

⊤ ⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 8 / 17

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SLIDE 38

Fully Symbolic Approach (FS)

Encode the Kripke structure and the property automaton using BDDs. Combine the two BDDs. (Symbolic product.) Use fixpoint computations to decide whether the product contains an accepting cycle. Two symbolic emptiness checks compared:

EL OWCTY

  • K. Fisler, R. Fraer, G. Kamhi, M. Y. Vardi, and Z. Yang. Is there a best

symbolic cycle-detection algorithm? In Proc. of TACAS’01, vol. 2031 of LNCS, pp. 420–434. Springer

  • F. Somenzi, K. Ravi, and R. Bloem. Analysis of symbolic SCC hull
  • algorithms. In Proc. of FMCAD’02, vol. 2517 of LNCS, pp. 88–105. Springer

ATVA’11 Self-Loop Aggregation Product 9 / 17

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SLIDE 39

SLAP-FST

FST = Fully Symbolic search in Terminal automata states When reaching a state of A¬ϕ which is terminal (and accepting) use a fully symbolic search. AM: s0 s1 s2 s3 s4 s5 s6 s7 a¯ bc a¯ b¯ c a¯ bc a¯ b¯ c ab¯ c a¯ bc ¯ ab¯ c ¯ abc A¬ϕ: q0 q1 a¯ b b ⊤ SLAP-FST: q0, s0 s1

s2 s3 s4

  • q1, {s5}

⊤ ⊤

ATVA’11 Self-Loop Aggregation Product 10 / 17

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SLIDE 40

Summary of methods

Explicit BCZ SOG SLAP PDP FS logic LTL LTL\ X LTL

  • prop. aut.

explicit symb./expl. data str. graph BDD+graph BDD[] BDD

  • empt. chk

explicit symbolic (OWCTY/EL)

PDP = Property Driven Partitioning,

  • R. Sebastiani, S. Tonetta, and M. Y. Vardi. Symbolic systems, explicit

properties: on hybrid approches for LTL symbolic model checking. In Proc.

  • f CAV’05, vol. 3576 of LNCS, pp. 350–363. Springer

ATVA’11 Self-Loop Aggregation Product 11 / 17

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SLIDE 41

Experimental Framework

Implemented algorithms BCZ, SOG, SLAP, SLAP-FST, EL, OWCTY Tools used SDD Hierarchical Set Decision Diagrams (http://ddd.lip6.fr/) Hierarchy (memory gain) Automatic saturation (improves fixpoint computations) Spot Model checking library (http://spot.lip6.fr/) Good LTL-to-TGBA translation Explicit emptiness-check algorithms Data Models Scalable toy models with 106 . . . 1066 states. Formulas Random formulas (filtered), plus a set of (random) weak fairness properties.

ATVA’11 Self-Loop Aggregation Product 12 / 17

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SLIDE 42

Cumulative Plot: Violated Formulas

500 1000 1500 2000 2500 3000 3500 4000 4500 20 40 60 80 100 120 SLAP SLAP-FST SOG BCZ EL OWCTY ATVA’11 Self-Loop Aggregation Product 13 / 17

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SLIDE 43

Cumulative Plot: Verified Formulas

500 1000 1500 2000 2500 3000 3500 20 40 60 80 100 120 SLAP SLAP-FST SOG BCZ EL OWCTY ATVA’11 Self-Loop Aggregation Product 14 / 17

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SLIDE 44

Scatter Plots: SLAP-FST vs. Hybrid

Runtime in seconds. Timeout at 120s.

0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 SOG SLAP-FST empty non-empty unknown 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 BCZ SLAP-FST empty non-empty unknown

ATVA’11 Self-Loop Aggregation Product 15 / 17

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SLIDE 45

Scatter Plots: SLAP-FST vs. Fully Symbolic

Runtime in seconds. Timeout at 120s.

0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 EL SLAP-FST empty non-empty unknown 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 OWCTY SLAP-FST empty non-empty unknown

ATVA’11 Self-Loop Aggregation Product 16 / 17

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SLIDE 46

Conclusion

SLAP, a new hybrid approach:

Aggregates states at the product level, not in the Kripke structure Can therefore adjust dynamically to the features (self-loops and acceptance conditions) present in the property automaton. SLAP-FST: a variant using a fully-symbolic algorithm for terminal states.

SLAP-FST most competitive algorithm of those we compared,

  • n this benchmark

On-the-fly computations have strong impact when counterexamples exist. Hybrid approaches outperform fully symbolic approaches in these cases. http://move.lip6.fr/software/DDD/ltl_bench.html

ATVA’11 Self-Loop Aggregation Product 17 / 17