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Jan 12, 2024 366 likes •748 views

Compositional Algorithms for Succinct Safety Games Romain Brenguier, Guillermo A. P erez, Jean-Franc ois Raskin, Ocan Sankur SYNT15 Reactive Synthesis for circuits AbsSynthe https://github.com/gaperez64/AbsSynthe Specification: G( (

SLIDE 1

Compositional Algorithms for Succinct Safety Games

Romain Brenguier, Guillermo A. P´ erez, Jean-Franc ¸ois Raskin, Ocan Sankur SYNT’15

SLIDE 2

Reactive Synthesis for circuits

AbsSynthe https://github.com/gaperez64/AbsSynthe i1 i2 i3 i4 i5

1 o2 o3 o4 o5

ℓ1 ℓ2 ℓ3 Specification: G(¬(o1 ∧ o2)) ∧ G(i1 → Xo3) i1 i2 i3 i4 i5 ℓ1 ℓ2 ℓ3

1 o2 o3 o4 o5

Contr.

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 2 / 17

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Succinct Safety Games

Safety game: Stat, Actu, Actc, δ, U Succinct representation: Stat = {0, 1}L, Actu = {0, 1}Xu, Actc = {0, 1}Xc, δ and U are given by And-Inverter Graphs (AIG) → standard file format for sequential synchronous circuits → used in model checking and synthesis competitions a1 a2 ℓ1 ℓ′

1

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 3 / 17

SLIDE 4

Succinct Safety Games

Safety game: Stat, Actu, Actc, δ, U Succinct representation: Stat = {0, 1}L, Actu = {0, 1}Xu, Actc = {0, 1}Xc, δ and U are given by And-Inverter Graphs (AIG) → standard file format for sequential synchronous circuits → used in model checking and synthesis competitions a1 a2 ℓ1 ℓ′

1

∧ ∧ ∧ ¬

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 3 / 17

SLIDE 5

The classical algorithm: attractor computation

For the safety game Stat, Actu, Actc, δ, U:

1 uncontrollable predecessors: states where environment can force S in

1 step: UPRE(S) = {s | ∃au, ∀ac, δ(s, au, ac) ∈ S}

2 Compute the least fixpoint of UPRE starting from the error states U.

→ if s0 ∈ Stat \ UPRE∗(U), controller has a winning strategy s0 U

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 4 / 17

SLIDE 6

The classical algorithm: attractor computation

For the safety game Stat, Actu, Actc, δ, U:

1 uncontrollable predecessors: states where environment can force S in

1 step: UPRE(S) = {s | ∃au, ∀ac, δ(s, au, ac) ∈ S}

2 Compute the least fixpoint of UPRE starting from the error states U.

→ if s0 ∈ Stat \ UPRE∗(U), controller has a winning strategy s0 UPRE1(U) U

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 4 / 17

SLIDE 7

The classical algorithm: attractor computation

For the safety game Stat, Actu, Actc, δ, U:

1 uncontrollable predecessors: states where environment can force S in

1 step: UPRE(S) = {s | ∃au, ∀ac, δ(s, au, ac) ∈ S}

2 Compute the least fixpoint of UPRE starting from the error states U.

→ if s0 ∈ Stat \ UPRE∗(U), controller has a winning strategy s0 UPRE∗(U)

( ( (

UPRE2(U) UPRE1(U) U

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 4 / 17

SLIDE 8

Implementation with BDDs

We use Binary Decision Diagrams (BDDs): data structure to represent Boolean functions efficient Boolean operations (∧, ∨, ∀, ∃,. . . ) and equality test 2 basic approaches:

1 Compute a transition relation

T(L, Xu, Xc, L′) =

ℓ∈L

ℓ′ ⇔ fℓ(L, Xu, Xc) and then set UPRE(S) = ∃Xu, ∀Xc, ∃L′. T(L, Xu, Xc, L′) ∧ S(L′).

(solved approximately 150 out of 530 benchmarks from last year’s competition)

2 Keep a partitioned transition relation, and substitute fℓ for each ℓ in S

UPRE(S) = ∃Xu, ∀Xc : S(L′)[ℓ′ ← fℓ(Xu, Xc, L)]ℓ∈L.

(solved approximately 500 benchmarks in 500 seconds)

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 5 / 17

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Idea of the decomposition

Often: specifications are big conjunctions of smaller specifications

Example from amba2b9

assign sys safe err = sys safe err0 | sys safe err1 | sys safe err2 | ...| sys safe err19; assign o err = ∼env safe err & ∼env safe err happened & (sys safe err | fair err);

err can be rewritten:

(∼env safe err & ∼env safe err happened & fair err) | φ0 | ...| φ19 where φi = ∼env safe err & ∼env safe err happened & sys safe erri

we define a game Gi for each formula φi to win the “big” game, we must win each “small” game Gi

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 6 / 17

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Decomposition of AIGs

We must recover the structure of the specifications from the AIG Explore the graph until encountering a negation This corresponds to a disjunction, and it can be distributed over a1 a2 ℓ1 err ∧ ∧ ∨ a1 a2 ℓ1 err ∨ ∧ ∧ a2 ∧ ℓ1 a1 a2 err

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 7 / 17

SLIDE 11

Decomposition of AIGs

We must recover the structure of the specifications from the AIG Explore the graph until encountering a negation This corresponds to a disjunction, and it can be distributed over a1 a2 ℓ1 err ∧ ∧ ∨ a1 a2 ℓ1 err ∨ ∧ ∧ a2 ∧ ℓ1 a1 a2 err

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 7 / 17

SLIDE 12

Decomposition of AIGs

We must recover the structure of the specifications from the AIG Explore the graph until encountering a negation This corresponds to a disjunction, and it can be distributed over a1 a2 ℓ1 err ∧ ∧ ∨ a1 a2 ℓ1 err ∨ ∧ ∧ a2 ∧ ℓ1 a1 a2 err

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 7 / 17

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Subgames

We obtain a decomposition err = e1 ∨ e2 ∨ · · · ∨ en If formula ei does not depend on all latches, solving the game for ei can be more efficient

Cone of influence

cone(ei): set of variables on which ei depends (directly or indirectly) → can be over-approximated efficiently by exploring the AIG We consider the game Gi where the error function is given by ei and we

nly consider variables in cone(ei)

Brenguier, P´ erez, Raskin, Sankur (ULB) Compositional Algo. for Succ. Safety Games SYNT’15 8 / 17

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