[PPT] - Complementing Bchi automata via alternating automata Guillaume PowerPoint Presentation

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Complementing Büchi automata via alternating automata

Guillaume Sadegh

LRDE – EPITA Research and Development Laboratory

January 06, 2010

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Context

Automata-theoretic approach to model checking

1. We have an automaton for the system AS,
2. We have an automaton for the properties AP,
3. Synchronized-product between AS and ¬AP.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Context

Automata-theoretic approach to model checking

1. We have an automaton for the system AS,
2. We have an automaton for the properties AP,
3. Synchronized-product between AS and ¬AP.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Context

Automata-theoretic approach to model checking

1. We have an automaton for the system AS,
2. We have an automaton for the properties AP,
3. Synchronized-product between AS and ¬AP.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Context

Automata-theoretic approach to model checking

1. We have an automaton for the system AS,
2. We have an automaton for the properties AP,
3. Synchronized-product between AS and ¬AP.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Complementing Büchi Automata

Why not

◮ Complementation is unrealistic in practice,

Best algorithms produce 2O(n log n) states for an automaton with n states.

◮ Model checking avoids the complementation. (¬Aϕ ≡ A¬ϕ).

Why

◮ When properties are not formulæ, ◮ To enrich our library, ◮ To complete Safra’s complementation, not adapted to

TGBA.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

The complementation

Nondeterministic Büchi Automata Universal co- Büchi Automata Weak Alternating Automata Nondeterministic Büchi Automata Changing interpretation (Obvious) Transformation Kupferman and Vardi (1997) Direct transformation Kupferman and Vardi (1997) Transformation Miyano and Hayashi (1984) A ¬A

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Outline

1

Overview on ω-automata Mode of transitions Ranks

2

The complementation

3

Implementation

4

Conclusion and Perspectives

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Outline

1

Overview on ω-automata Mode of transitions Ranks

2

The complementation

3

Implementation

4

Conclusion and Perspectives

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Büchi (1962) acceptance condition

◮ The Büchi acceptance condition is a set of states. A run

must visit infinitely often some states of this acceptance condition to be accepting.

◮ Formally, a run π is accepting iff inf(π) ∩ F= ∅ with F ⊆ Q.

1 2 3

States in the accepting set are marked with .

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

co-Büchi (1962) acceptance condition

◮ The co-Büchi acceptance condition is a set of states. A run

must visit finitely often all states of this acceptance condition to be accepting.

◮ Formally, a run π is accepting iff inf(π) ∩ F= ∅ with F ⊆ Q.

1 2 3

States in the accepting set are marked with .

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Generalized Büchi (1962) acceptance condition

◮ The Generalized Büchi acceptance condition has more

than one set of states. A run must visit infinitely often some states of each set of acceptance conditions to be accepting.

◮ Formally, a run π is accepting iff ∀i, inf(π) ∩ Fi = ∅ with

F = {F1, F2, · · · , Fn} and Fi ⊆ Q.

1 2 3

States in accepting sets are marked with and .

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Generalized co-Büchi (1962) acceptance condition

◮ The Generalized co-Büchi acceptance condition has more

than one set of states. A run must visit finitely often every states of some set of acceptance conditions to be accepting.

◮ Formally, a run π is accepting iff ∃i, inf(π) ∩ Fi = ∅ with

F = {F1, F2, · · · , Fn} and Fi ⊆ Q.

1 2 3

States in accepting sets are marked with and .

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Mode of transitions

Mode of transitions

deterministic

1 2 b a a b

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Mode of transitions

Mode of transitions

non-deterministic (existential) deterministic

1 2 a, b a a a, b

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Mode of transitions

Mode of transitions

non-deterministic (existential) universal deterministic

1 2 a, b a a a, b

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Mode of transitions

Mode of transitions

alternating non-deterministic (existential) universal deterministic

1 2 3 a a

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

DAG

The run of a word over an automaton can be represented by a Direct Acyclic Graph.

1 2 a a

Universal co-Büchi automaton

1, 0

DAG

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

DAG

The run of a word over an automaton can be represented by a Direct Acyclic Graph.

1 2 a a

Universal co-Büchi automaton

1, 0 2, 1 2, 2 2, 3 a a a a

DAG

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

DAG

The run of a word over an automaton can be represented by a Direct Acyclic Graph.

1 2 a a

Universal co-Büchi automaton

1, 0 2, 1 2, 2 2, 3 a a a a F-free vertices F vertex

DAG

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

DAG

The run of a word over an automaton can be represented by a Direct Acyclic Graph.

1 2 a a

Universal co-Büchi automaton

1, 0 Finite

DAG

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

DAG

The run of a word over an automaton can be represented by a Direct Acyclic Graph.

1 2 a a

Universal co-Büchi automaton

1, 0 2, 1 2, 2 2, 3 a a a a

DAG

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

Is a run accepting?

We can prune the DAG

◮ G0 = Gr ◮ G2i+1 = G2i \ {q, l|q, l is finite in G2i} ◮ G2i+2 = G2i+1 \ {q, l|q, l is F-free in G2i+1}

If G2n is empty, the run was accepting rank(q, l) =

2i

If q, l is finite in G2i 2i + 1 If q, l is F-free in G2i+1

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

1, 0 2, 1 2, 2 2, 3 a a a a G0 = Gr

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

1, 0 2, 1 2, 2 2, 3 a a a a G1 = G0 \ {q, l|q, l is finite in G0}

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

1, 0 Vertices were F-free G2 = G1 \ {q, l|q, l is F-free in G1}

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

Vertex was finite G3 = G2 \ {q, l|q, l is finite in G2}

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

No vertices in the graph G4 = ∅. So the DAG is accepting.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References Ranks

Ranks

1, 0 2, 1 2, 2 2, 3 rank: 1 rank: 1 rank: 1 rank: 2 a a a a Define the rank of each vertex.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Outline

1

Overview on ω-automata

2

The complementation

3

Implementation

4

Conclusion and Perspectives

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

The complementation

Nondeterministic Büchi Automata Universal co- Büchi Automata Weak Alternating Automata Nondeterministic Büchi Automata Changing interpretation (Obvious) Transformation Kupferman and Vardi (1997) Direct transformation Kupferman and Vardi (1997) Transformation Miyano and Hayashi (1984) A ¬A

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

The algorithm

Ranks

The knowledge of ranks makes the complementation easy: all runs that visit infinitely often an odd rank are accepting. But we cannot find the rank: we need to guess it.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

The algorithm

◮ A state: {Q × N} ◮ Transition: δ′({Q × N}) = δ(Q) × n, ∀n ≤ N ◮ if Q ∈ F, Q was not F-free then N cannot be odd. ◮ Odd ranks are accepting.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Example

A universal co-Büchi with { } as acceptance condition

A B a a

Construction

A4 B4 A3 B3 A2 B2 A1 B1 A0 B0

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Example

A universal co-Büchi with { } as acceptance condition

A B a a

Construction

A4 B4 A3 A2 B2 A1 A0 B0

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Dealing with generalized acceptance conditions

Ranks

F-free vertices are Fj-free vertices. For example -free vertices and

free vertices.

1 2 3 a a a a a

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Outline

1

Overview on ω-automata

2

The complementation

3

Implementation

4

Conclusion and Perspectives

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Implementation

A new interface for alternating automata

◮ A new interface s❛❜❛ for alternating automata. ◮ Inspired by t❣❜❛ interface.

Some algorithms implemented

◮ Universal co-Büchi to Weak Alternating Büchi. ◮ Universal Generalized co-Büchi to Non-deterministic

Büchi.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Testing

ϕ

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Testing

ϕ ¬ϕ

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Testing

ϕ ¬ϕ Aϕ A¬ϕ

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Testing

ϕ ¬ϕ Aϕ A¬ϕ ¬Aϕ ¬A¬ϕ

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Testing

ϕ ¬ϕ Aϕ A¬ϕ ¬Aϕ ¬A¬ϕ L (Aϕ) = L (¬A¬ϕ) ⇒ Aϕ ∩ ¬A¬ϕ = ∅ L (A¬ϕ) = L (¬Aϕ) ⇒ A¬ϕ ∩ ¬Aϕ = ∅

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Benchmarks

Original Ranks Safra ¬ϕ st st tr acc st tr acc st tr acc 1 4.3 10.7 1.0 6.0 13.7 1.0 3.0 5.7 1.3 2 19.6 236.9 1.0 11.3 44.2 1.2 2.6 4.9 0.9 3 1595.8 387510.6 1.0 19.4 99.7 1.4 2.7 4.8 1.2 4 6486.3 3757235.0 1.0 33.2 273.3 2.2 3.3 6.5 1.4 5 9060.5 4689062.5 1.0 141.0 730.0 3.0 2.5 6.0 1.0 6 12107.7 81361931.3 1.0 27.2 195.8 2.0 3.0 5.7 1.7 7 x x x 157.0 1325.5 5.0 3.0 7.0 1.0

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Outline

1

Overview on ω-automata

2

The complementation

3

Implementation

4

Conclusion and Perspectives

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Conclusion and Perspectives

Nondeterministic Büchi Automata Universal Co- Büchi Automata Weak Alternating Automata Nondeterministic Büchi Automata Changing interpretation (Obvious) Transformation Kupferman and Vardi (1997) D i r e c t t r a n s f

r

m a t i

n

K u p f e r m a n a n d V a r d i ( 1 9 9 7 ) Transformation Miyano and Hayashi (1984) Simplification Gurumurthy et al. (2003) A ¬A

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Questions

1, 0 2, 1 2, 2 2, 3 a a a a 3, 1 3, 2 3, 3 a a a a a a a

A Dag

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Bibliography I

Büchi, J. R. (1962). On a decision method in restricted second

rder arithmetic. In Proceedings of the International

Congress on Logic, Methodology, and Philosophy of Science, Berkley, 1960, pages 1–11. Standford University Press. Republished in Lane and Siefkes (1990). Gurumurthy, S., Kupferman, O., Somenzi, F ., and Vardi, M. Y. (2003). On complementing nondeterminisitic Büchi

automata. In Proceedings of the 12th Advanced Research

Working Conference on Correct Hardware Design and Verification Methods (CHARME’03), volume 2860 of Lecture Notes in Computer Science, pages 96–110. Springer-Verlag.

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Introduction Overview on ω-automata The complementation Implementation Conclusion and Perspectives References

Bibliography II

Kupferman, O. and Vardi, M. Y. (1997). Weak alternating automata are not that weak. In Proceedings of the 5st Israeli Symposium on Theory of Computing and Systems (ISTC’97), pages 147–158. IEEE Computer Society Press. Lane, S. M. and Siefkes, D., editors (1990). The Collected Works of J. Richard Büchi. Springer-Verlag. Miyano, S. and Hayashi, T. (1984). Alternating finite automata

n ω-words. Theoretical Computer Science, 32:321–330.

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