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Exploiting the Temporal Logic Hierarchy and the Non-Confluence - - PowerPoint PPT Presentation

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Problem Det. via Automata Hierarchy Det. Non-Confluent Automata Conclusion Exploiting the Temporal Logic Hierarchy and the Non-Confluence Property for Efficient LTL Synthesis Andreas Morgenstern GandALF 2010 Andreas Morgenstern Symbolic

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SLIDE 1

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Exploiting the Temporal Logic Hierarchy and the Non-Confluence Property for Efficient LTL Synthesis

Andreas Morgenstern GandALF 2010

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 1

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SLIDE 2

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Overview

1

Motivation: What is LTL Synthesis

2

Symbolic Determinisation via the Automata Hierarchy

3

Symbolic Determinisation via Non-Confluent Automata

4

Experiments and Conclusion

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 2

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SLIDE 3

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Model Checking

System S I O

| =

? LTL formula Φ Specification: Formula Φ in Temporal-Logic LTL Question: S | = Φ ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 3

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SLIDE 4

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

LTL Synthesis ?

I O

| =

LTL formula Φ Specification: Formula Φ in Temporal-Logic LTL Question: ∃ System S. S | = Φ ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 3

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SLIDE 5

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Model Checking

System I O

| =

? LTL formula Φ Question: S | = Φ ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

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SLIDE 6

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Model Checking

System I O

| =

? LTL formula ¬Φ (S | = Φ) ↔ (S | = ¬Φ) ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

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SLIDE 7

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Automata based Model Checking

ξ1 ξ3 ξ2 a b c d a b c

×

1 2 3 a a b b b c Non-terminating systems ! B¨ uchi-Automata Automata read infinite words Automata accept, whenever a F state is visited ∞ often ! Graphsearch for one Non-Accepting run ! (S | = Φ) ↔ (S | = ¬Φ) ↔ L(S × A¬Φ) = ∅) ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

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SLIDE 8

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Symbolic Model Checking

R = p0 ∧ a ∨ p1 ∧ ¬b . . .

∧

R = r0 ↔ (a ∨ ¬b) ∧ r1 ↔ (c ∨ d) ∧ . . . Using propositional logic to represent

System and B¨ uchi Automata

Advantages:

Represent large state spaces Efficient methods like BDD / SAT

Industry-sized problems managable:

Verification at Intel

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 4

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SLIDE 9

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Automata based LTL Synthesis

?

I O |

=

1 2 3 a ¬a b b b ¬c (∃S. S | = Φ) ↔ ∃S. L(S) ⊆ L(AΦ) Idea: Search for valid sub-automaton for each input on B¨ uchi automaton! Infinite Game between Environment and System !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 5

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SLIDE 10

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Automata based LTL Synthesis

p1 p2 p3 a a b c Idea: Search for satisfying automaton for each input on specification automaton! Problem: nondeterminism intuitively: a priori not known whether b or c comes deterministic system from nondeterministic B¨ uchi automaton

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 5

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SLIDE 11

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Determinisation of B¨ uchi Automata: Facts

Rabin-Scott Subset construction not sufficient ! Safra (1988): Determinisation of B¨ uchi automata First Implementation : 2006 State space: Trees of sets of states No fully symbolic implementation known Only small examples managable

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 6

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SLIDE 12

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Core of this Work

LTL NDet Det symbolic: exists Determinisation : symbolic Minimizing Minimizing symbolic translation LTL → NDet √ symbolic Algorithms for infinite games √ minimizing automata symbolically √ symbolic determinization (shown in [MoSc08,MoSc08a])

How well does it work in practice ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 7

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SLIDE 13

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Overview

1

Motivation: What is LTL Synthesis

2

Symbolic Determinisation via the Automata Hierarchy

3

Symbolic Determinisation via Non-Confluent Automata

4

Experiments and Conclusion

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 8

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SLIDE 14

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

ω-Automata

1 2 3 a a b b b c ω-Automata ω-Automata read infinite Worte. Different acceptance conditions:

B¨ uchi : visit F states infinitely often

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

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SLIDE 15

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

ω-Automata

1 2 3 a a b b b c ω-Automata ω-Automata read infinite Worte. Different acceptance conditions:

B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

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SLIDE 16

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

ω-Automata

1 2 3 a a b b b c ω-Automata ω-Automata read infinite Worte. Different acceptance conditions:

B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often Streett: boolean combination of (co)-B¨ uchi (in Normalform)

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

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SLIDE 17

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

ω-Automata

1 2 3 a a b b b c ω-Automata ω-Automata read infinite Worte. Different acceptance conditions:

B¨ uchi : visit F states infinitely often Co-B¨ uchi: visit ¬F states finitely often Streett: boolean combination of (co)-B¨ uchi (in Normalform) Safety : visit only F states Liveness : visit F states at least once Prefix : boolean combination of Safety und Liveness (in Normalform)

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 9

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SLIDE 18

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

The Automata Hierarchy (Wagner, 1979)

(N)DetSafety NDettotal

Liveness

DetLiveness NDetPrefix DetPrefix DetB¨

uchi

(N)DetCo-B¨

uchi

NDetB¨

uchi

(N)DetStreett

  • bool. comb.
  • bool. comb.
  • bool. comb
  • bool. comb

C1 C2 := automaton from C1 can be translated to one from C2

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 10

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SLIDE 19

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

The Temporallogic Hierarchy (Manna&Pnueli, 1987)

(N)DetSafety NDettotal

Liveness

DetLiveness NDetPrefix DetPrefix DetB¨

uchi

(N)DetCo-B¨

uchi

NDetB¨

uchi

(N)DetStreett

  • TLSafety

TLLiveness TLPrefix TLStreett TLB¨

uchi

TLCo-B¨

uchi

  • bool. comb.
  • bool. comb.
  • bool. comb
  • bool. comb

C1 C2 := automaton from C1 can be translated to one from C2

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 10

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SLIDE 20

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Symbolic Determinisation via Automata Hierarchy

(N)DetSafety NDettotal

Liveness

DetLiveness NDetPrefix DetPrefix DetB¨

uchi

(N)DetCo-B¨

uchi

NDetB¨

uchi

(N)DetStreett TLSafety TLLiveness TLPrefix TLStreett TLB¨

uchi

TLCo-B¨

uchi

  • Subset

Safra Breakpoint BDD-represented Automata for TLSafety and TLCo-B¨

uchi

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 11

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SLIDE 21

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Symbolic Determinisation via Automata Hierarchy

(N)DetSafety NDettotal

Liveness

DetLiveness NDetPrefix DetPrefix DetB¨

uchi

(N)DetCo-B¨

uchi

NDetB¨

uchi

(N)DetStreett TLSafety TLLiveness TLPrefix TLStreett TLB¨

uchi

TLCo-B¨

uchi

Dual Dual

  • bool. comb.
  • bool. comb.
  • bool. comb
  • bool. comb

BDD-represented Automata for TLLiveness, TLPrefix TLB¨

uchi und

TLStreett

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 11

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SLIDE 22

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Determinisation via Automata Hierarchy: Conclusion

Main Idea Locate formula syntactically in Hierarchy Subset (Breakpoint) construction symbolically boolean combination of Formulas / Automata Advantages Deterministic automata never explicitely represented Efficient: due to boolean combination subautomata very small (less than < 20 ndet states) Nearly all formula belong to TLStreett Disadvantages Not every formula is in TLStreett!

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 12

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SLIDE 23

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Overview

1

Motivation: What is LTL Synthesis

2

Symbolic Determinisation via the Automata Hierarchy

3

Symbolic Determinisation via Non-Confluent Automata

4

Experiments and Conclusion

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 13

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SLIDE 24

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Backwards-Determinism

Determinism q0 q1 q2 a ¬a Backwards Determinism q3 q1 q2 a ¬a every B¨ uchi automaton from LTL is backwards deterministic !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 14

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SLIDE 25

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Non-Confluent Automata

q q q q q q q q q q q q a a a a ba ba ba ba ba ba ba Backwards determinism implies non-confluence Non-Confluence: two different runs never merge Therefore: Every run uniquely determined by last visited state No Need for Tree Structure in Deterministic Automaton !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 15

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SLIDE 26

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Determinization: Example

NDet 1 2 3 a a b b b c Deterministic Automaton {1}, ∅, ∅ {1, 2}, ∅, {2} a a {1, 2, 3}, ∅, ∅ b a {1, 2, 3}, ∅, ∅ ∅, ∅, ∅ {3}, ∅, ∅ b b c c a, b, c a c c a, b Accept, whenever some set ( ¬∞ often red) and ( ∞ blue) !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 16

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SLIDE 27

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Overview

1

Motivation: What is LTL Synthesis

2

Symbolic Determinisation via the Automata Hierarchy

3

Symbolic Determinisation via Non-Confluent Automata

4

Experiments and Conclusion

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 17

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SLIDE 28

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Other Tools

Opal [Morgenstern] Symbolic determinisation for full LTL Symbolic minimization for different ω-automata Symbolic synthesis algorithm for generalized Parity Games[Chatterjee et al, 2007]

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 18

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SLIDE 29

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Other Tools

Opal [Morgenstern] Symbolic determinisation for full LTL Symbolic minimization for different ω-automata Symbolic synthesis algorithm for generalized Parity Games[Chatterjee et al, 2007] Lily[Jobstmann et al.] Based on Safraless approach of Vardi and Kupferman replace determinisim by universal tree automata explicit implementation first Tool for full LTL

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 18

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SLIDE 30

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Lily Examples

5 10 15 2 4 6 8 10 Running times in seconds Lily Opal Nonconfluent Opal Hierarchy

Example Nr.

23 Medium-Sized Spezifications: Arbiter, Mutex Temporal operators: 4 − 40, boolean operators: 3 − 38

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 19

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SLIDE 31

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Other Tools

Opal [Morgenstern] Symbolic determinisation for full LTL Symbolic minimization for different ω-automata Symbolic synthesis algorithm for generalized Parity Games[Chatterjee et al, 2007] Anzu[Jobstmann et al.] Symbolic implementation of Generalised Streett (1) Approach (Piterman et al.) Limited set of specifications (implication of B¨ uchi-conditions) Possible: determinise B¨ uchi automata manually nevertheless: much more restricted than TLStreett

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 20

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SLIDE 32

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Industry Specification: AMBA Arbiter

2 4 6 8 10 12 14 1 2 3 4 5 ·104 # Clients Running time in 10000 Seconds Anzu Opal NonConfluent

Arbiter for AMBA Bus: connects Caches, CPUs, . . . Each Client 10 Safety / 1 Streett Condition Total: 40 Temporal operators / Client

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 21

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SLIDE 33

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Conclusion

Summary Presented a tool for LTL Synthesis starting with plain formula Running time competitive to anzu (where deterministic automata are generated manually)

LTL Synthesis for full LTL is possible !

Future ?

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 22

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SLIDE 34

Problem

  • Det. via Automata Hierarchy
  • Det. Non-Confluent Automata

Conclusion

Thank you!

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 23

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SLIDE 35

Temporallogic-Hierarchy

PG ::= VΣ | ¬PF | PG ∧ PG | PG ∨ PG | XPG | GPG | [PG U PG] + Past PF ::= VΣ | ¬PG | PF ∧ PF | PF ∨ PF | XPF | FPF | [PF U PF] + Past PPrefix ::= PG | PF | ¬PPrefix | PPrefix ∧ PPrefix | PPrefix ∨ PPrefix+ Past PGF ::= PPrefix | ¬PFG | PGF ∧ PGF | PGF ∨ PGF | XPGF | GPGF | [PGF U PGF] | [PGF U PF] + Past PFG ::= PPrefix | ¬PGF | PFG ∧ PFG | PFG ∨ PFG | XPFG | FPFG | [PFG U PFG] | [PG U PFG] + Past PStreett ::= PGF | PFG | ¬PStreett | PStreett ∧ PStreett | PStreett ∨ PStreett+ Past Example for LTL, not TLStreett : GF [ϕ U ψ] ≡ FG [ϕ U ψ] ∧ GFψ

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 24

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SLIDE 36

Symbolic Subset-Construktion: 1. Onehot-Kodierung

Nondeterministic automaton

q0q1 q1q0 q0q1 q0q1 ¬b ¬b b b ¬(a ∨ b) ¬(a ∨ b) a ∨ b

R = (q0 ↔ b ∨ a ∧ q0′)∧ (q1 ↔ q0′) 4 Onehot encoded automaton

p0 p1 p2 p3 ¬b ¬b b b ¬(a ∨ b) ¬(a ∨ b) a ∨ b

R = (p0 ∧ ¬b ∨ p2 ∧ b) → p′

1∨

(p1 ∧ (¬(a ∨ b))) → p′

3∨

. . .

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 25

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SLIDE 37

Symbolic Subset-Construktion: 2. Determinisation

Onehot-kodierter Automat

p0 p1 p2 p3 ¬b ¬b b b ¬(a ∨ b) ¬(a ∨ b) a ∨ b

R = (p0 ∧ ¬b ∨ p2 ∧ b) → p′

1∨

(p1 ∧ (¬(a ∨ b))) → p′

3∨

. . . Deterministic automaton

p0p1p2p3 p0p1p2p3} p0p1p2p3 p0p1p2p3} 1 ¬(a ∨ b) b 1 a ∧ ¬b 1

R = (p0 ∧ ¬b ∨ p2 ∧ b)↔p′

1∨

(p1 ∧ (¬(a ∨ b)))↔p′

3∨

. . .

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 26

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SLIDE 38

Symbolic Subset-Construktion: BDD-Implementation

Definition (Symbolic Subset Construction) H :≡

n

  • j=1

pj ∧ mtQdet(ϑj) Idet :≡

n

  • i=1

pi ↔ ∃q1 . . . qm. mtQdet(ϑi) ∧ I Fdet :≡ ∃q1 . . . qm. H ∧ F Rdet :≡

n

  • i=1

pi ′ ↔ ηi, mit ηi :≡ ∃q1 . . . qmq1′ . . . qm′. H ∧ R ∧ [mtQdet(ϑi)]q1,...,qm

q′

1,...,q′ m

nicht-symbolischer Schritt: Aufz¨ ahlen der n nichtdeterministischen Zust¨ ande

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 27

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SLIDE 39

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ Parallel Subset-Construktionen in sets S0 . . . Sn−1

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 40

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ a Dead-Ends → red marking

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 41

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a F states → new Subset-Construktion rightt

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 42

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a {1, 2}, ∅, ∅ a Dead-Ends → red marking

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 43

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a split off F states → same state !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 44

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a a same state!

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 45

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a a {1, 2, 3}, ∅, {1, 2, 3} b parallel Subset-Construktion!

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 46

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, ∅ {1, 2}, ∅, {2} a a a {1, 2, 3}, ∅, {1, 2, 3} b {1, 2, 3}, ∅, ∅ b runs unique → blue marking and removal right

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28

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SLIDE 47

Determinizing Non-Confluent Automata

NDet 1 2 3 a a b b b c Deterministic automaton {1}, ∅, ∅ {1, 2}, ∅, {2} a a {1, 2, 3}, ∅, ∅ b a {1, 2, 3}, ∅, ∅ ∅, ∅, ∅ {3}, ∅, ∅ b b c c a, b, c a c c a, b Accept, whenever some set ( ¬∞ often red) and ( ∞ blue) !

Andreas Morgenstern Symbolic LTL Synthesis via Hierarchy and Non-Confluence 28