Symbolic Encodings of Bounded Synthesis Saarland University Peter - - PowerPoint PPT Presentation

Symbolic Encodings of Bounded Synthesis

Peter Faymonville1, Bernd Finkbeiner1, Markus N. Rabe2, Leander Tentrup1

1Reactive Systems Group

Saarland University

2UC Berkeley

Reactive synthesis φ Synthesis S

realizable unrealizable bound [Schewe/F. 2007]

Bounded synthesis φ Synthesis S

realizable unrealizable bound n [Schewe/F. 2007]

Bounded synthesis

▸ synthesis of systems with minimal # states ▸ basis for other output-sensitive synthesis algorithms,

e.g. bounded cycle synthesis [F./Klein 2016]

▸ undecidable synthesis problems, e.g., distributed synthesis, become decidable ▸ complexity split between input and output

TBURST4 (synthesized with Acacia+ v2.3)

TBURST4 (synthesized with bounded synthesis)

TBURST4 (synthesized with bounded cycle synthesis)

Complexity of standard synthesis (in specification)

1-process architectures — 2EXPTIME

Env

1 a b Pipeline architectures — NONELEMENTARY

Env

1 2 3 a b c d 2-process arbiter architecture — UNDECIDABLE

Env

1 2 r1 r2 g1 g2

Complexity of bounded synthesis (in bound)

1-process architectures — NP

Env

1 a b Pipeline architectures — NP

Env

1 2 3 a b c d 2-process arbiter architecture — NP

Env

1 2 r1 r2 g1 g2

Constraint-based bounded synthesis

specification φ Co-Büchi automaton universal Büchi automaton nondeterministic deterministic automaton emptiness game constraint system

bounded standard

Constraint-based bounded synthesis

specification φ Co-Büchi automaton universal Büchi automaton nondeterministic deterministic automaton emptiness game constraint system doubly exponential in specification exponential in specification NP in bound

bounded standard

Constraint-based bounded synthesis

specification φ Co-Büchi automaton universal Büchi automaton nondeterministic deterministic automaton emptiness game constraint system symbolic encoding BDDs, antichains, etc standard encoding (2007): explicit

bounded standard

Constraint-based bounded synthesis

specification φ Co-Büchi automaton universal Büchi automaton nondeterministic deterministic automaton emptiness game constraint system symbolic encoding BDDs, antichains, etc standard encoding (2007): explicit new encoding (2017): symbolic

bounded standard

Bounded synthesis

universal co-Büchi automaton Aφ q0 q1 q2 qe ⊺ r1 r2 g1g2 g1 g2 ⊺ φ = (r1 → g1) ∧ (r2 → g2) ∧ ¬(g1 ∧ g2) Transition System T t0 t1 ⊺/g1 ⊺/g2

Run graph of automaton and transition system

Run graph with annotation λ q0 q1 q2 qe ⊺ r1 r2 g1g2 g1 g2 ⊺

t0 t1 ⊺/g1 ⊺/g2

⟨t0, q0⟩ λ ∶ 0 ⟨t1, q0⟩ λ ∶ 0 ⟨t1, q1⟩ λ ∶ 1 ⟨t1, q2⟩ λ ∶ 2 ⟨t0, q1⟩ λ ∶ 2 ⟨t0, q2⟩ λ ∶ 1 ⟨t0, qe⟩ λ ∶ ⟨t1, qe⟩ λ ∶

Run graph of automaton and transition system

Run graph with annotation λ q0 q1 q2 qe ⊺ r1 r2 g1g2 g1 g2 ⊺

t0 t1 ⊺/g1 ⊺/g2

⟨t0, q0⟩ λ ∶ 0 ⟨t1, q0⟩ λ ∶ 0 ⟨t1, q1⟩ λ ∶ 1 ⟨t1, q2⟩ λ ∶ 2 ⟨t0, q1⟩ λ ∶ 2 ⟨t0, q2⟩ λ ∶ 1 ⟨t0, qe⟩ λ ∶ ⟨t1, qe⟩ λ ∶

Encoding bounded synthesis with constraints

Encode the existence of a transition system and a valid annotation

▸ Representation of transition system

▸ states t ∈ T ▸ transitions τt,i,t′ ▸ output labeling ot,i

▸ Representation of annotation on run graph T × Q

▸ state occurrence λB ▸ rejecting bound λ#

λB

t0,q0 ∧ ⋀ q∈Q

⋀

t∈T

⎛ ⎝λB

t,q → ⋀ q′∈Q

⋀

i∈2I

(δt,q,i,q′ → ⋀

t′∈T

(τt,i,t′ → λB

t′,q′ ∧ λ# t′,q′ ⊳q′ λ# t,q))⎞

⎠

Target logics

SAT

▸ Only existentially quantified boolean variables permitted. ▸ No symbolic encoding of functions.

QBF

▸ Universally and existentially quantified boolean variables in total order. ▸ Symbolic encoding of functions with single applications.

DQBF

▸ Universally and existentially quantified boolean variables in partial order. ▸ Symbolic encoding of functions with multiple applications.

Results of difgerent encodings

LTL realizability, SYNTCOMP 2016 experiment set

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 10−2 10−1 100 101 102 103 # instances time (sec.) fully-symbolic state-symbolic SMT basic input-symbolic

Basic encoding (SAT)

I T Q Symbolic ✗ ✗ ✗ ∃{λB

t,q, λ# t,q ∣ t ∈ T, q ∈ Q}

∃{τt,i,t′ ∣ (t, t′) ∈ T × T, i ∈ 2I} ∃{ot,i ∣ o ∈ O, t ∈ T, i ∈ 2I}

Basic encoding (SAT)

I T Q Symbolic ✗ ✗ ✗ ∃{λB

t,q, λ# t,q ∣ t ∈ T, q ∈ Q}

∃{τt,i,t′ ∣ (t, t′) ∈ T × T, i ∈ 2I} ∃{ot,i ∣ o ∈ O, t ∈ T, i ∈ 2I} λB

t0,q0 ∧ ⋀ t∈T

⋀

i∈2I ⋁ t′∈T

τt,i,t′ ⋀

q∈Q

⋀

t∈T

⎛ ⎝λB

t,q → ⋀ q′∈Q

⋀

i∈2I

(δt,q,i,q′ → ⋀

t′∈T

(τt,i,t′ → λB

t′,q′ ∧ λ# t′,q′ ⊳q′ λ# t,q))⎞

⎠

Input-symbolic encoding (QBF)

I T Q Symbolic ✓ ✗ ✗ ∃{λB

t,q, λ# t,q ∣ t ∈ T, q ∈ Q}

∀I ∃{τt,t′ ∣ (t, t′) ∈ T × T} ∃{ot ∣ o ∈ O, t ∈ T}

Input-symbolic encoding (QBF)

I T Q Symbolic ✓ ✗ ✗ ∃{λB

t,q, λ# t,q ∣ t ∈ T, q ∈ Q}

∀I ∃{τt,t′ ∣ (t, t′) ∈ T × T} ∃{ot ∣ o ∈ O, t ∈ T} λB

t0,q0 ∧ ⋀ t∈T

⋁

t′∈T

τt,t′ ⋀

q∈Q

⋀

t∈T

⎛ ⎝λB

t,q → ⋀ q′∈Q

(δt,q,q′ → ⋀

t′∈T

(τt,t′ → λB

t′,q′ ∧ λ# t′,q′ ⊳q′ λ# t,q))⎞

⎠

State-symbolic encoding (DQBF)

I T Q Symbolic ✓ ✓ ✗ ∃{λB

q ∶2T → B, λ# q∶2T → Bb ∣ q ∈ Q}

∃τ ∶ 2T × 2I → 2T ∃{o∶2T × 2I → B ∣ o ∈ O} ∀I. ∀T, T′.

State-symbolic encoding (DQBF)

I T Q Symbolic ✓ ✓ ✗ ∃{λB

q ∶2T → B, λ# q∶2T → Bb ∣ q ∈ Q}

∃τ ∶ 2T × 2I → 2T ∃{o∶2T × 2I → B ∣ o ∈ O} ∀I. ∀T, T′. (T = 0 → λB

q0(T))

⋀

q∈Q

⎛ ⎝λB

q (T) → ⋀ q′∈Q

(δq,q′ ∧ (τ(T, I) ⇒ T′) → λB

q′(T′) ∧ λ# q′(T′) ⊳q′ λ# q(T))⎞

⎠

Fully-symbolic encoding (DQBF)

I T Q Symbolic ✓ ✓ ✓ ∃λB∶2T × 2Q → B, λ#∶2T × 2Q → Bb ∃τ ∶ 2T × 2I → 2T ∃{o∶2T × 2I → B ∣ o ∈ O} ∀I. ∀T, T′. ∀Q, Q′.

Fully-symbolic encoding (DQBF)

I T Q Symbolic ✓ ✓ ✓ ∃λB∶2T × 2Q → B, λ#∶2T × 2Q → Bb ∃τ ∶ 2T × 2I → 2T ∃{o∶2T × 2I → B ∣ o ∈ O} ∀I. ∀T, T′. ∀Q, Q′. (T = 0 ∧ Q = 0 → λB(T, Q)) ∧ (λB(T, Q) → (δ ∧ (τ(T, I) ⇒ T′) → λB(T′, Q′) ∧ λ#(T′, Q′) ⊳ λ#(T, Q)))

Realizability results on selected instances

Maximal parameter value, cumulative solving time (timeout 1h) basic input-sym state-sym Acacia Party instance max k sum t max k sum t max k sum t max k sum t max k sum t simple-arbiter 7 1008.7 8 2.7 3 100.5 8 59.2 6 902.7 full-arbiter 4 2994.5 3 0.6 2 13.3 5 2683.4 3 111.7 roundrob-arbiter 4 143.1 4 227.0 2 11.0 4 345.6 4 19.2 loadfull 5 268.7 8 44.2 2 25.1 4 83.7 4 213.5 prio-arbiter 4 176.5 4 1.6 2 0.4 6 701.2 3 69.0 loadcomp 5 36.9 6 639.4 3 432.1 5 387.8 5 212.7 genbuf 2 1840.3 2 2711.8 – 5 159.3 – generalized-bufger 2 2093.8 2 3542.8 – 6 3194.8 2 792.5 load-balancer 5 1148.8 8 83.2 2 75.3 5 270.8 – detector 6 1769.0 8 1010.7 3 239.4 8 261.6 5 370.3

Realizability results in comparison

LTL realizability, SYNTCOMP 2016 experiment set

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 10−2 10−1 100 101 102 103 # instances time (sec.) fully-symbolic state-symbolic Party elli rally SMT basic input-symbolic Acacia

Implementation extraction

• 1. Use certification feature of solvers to compute witness for o and τ

▸ Assignments from SAT solver ▸ Skolem functions from QBF solver

• 2. Create AIGER circuit and minimize using abc

Implementation size

Input-symbolic vs. basic encoding, circuit size (# AND-Gates)

100 101 102 103 100 101 102 103 basic input-symbolic

Implementation size

Input-symbolic encoding vs. state-of-the-art tools, circuit size (# AND-Gates) 100 101 102 103 104 105 100 101 102 103 104 105

Acacia BoSy (input-symbolic)

100 101 102 103 100 101 102 103

Party elli rally BoSy (input-symbolic)

Conclusions

▸ Symbolic encoding for bounded synthesis pays ofg ▸ Today, QBF = input symbolic encoding is the sweetspot:

Runtime competitive with non-bounded symbolic synthesis Implementation size > order of magnitude better

▸ Significant potential for DQBF solvers