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Overview Foundations Definitions & Constructions Main proof Complexity of B uchi automata minimization Dejan Kostyszyn Chair of Software Engineering February 3, 2018 Dejan Kostyszyn Complexity of B uchi automata minimization 1

  1. Overview Foundations Definitions & Constructions Main proof Complexity of B¨ uchi automata minimization Dejan Kostyszyn Chair of Software Engineering February 3, 2018 Dejan Kostyszyn Complexity of B¨ uchi automata minimization 1

  2. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Short overview ◮ Proof by Sven Schewe in 2010 Dejan Kostyszyn Complexity of B¨ uchi automata minimization 2

  3. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Short overview ◮ Proof by Sven Schewe in 2010 ◮ Minimization of deterministic B¨ uchi automata (MIN) is NP-complete Dejan Kostyszyn Complexity of B¨ uchi automata minimization 2

  4. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Short overview ◮ Proof by Sven Schewe in 2010 ◮ Minimization of deterministic B¨ uchi automata (MIN) is NP-complete ◮ Reduction from vertex cover problem to MIN Dejan Kostyszyn Complexity of B¨ uchi automata minimization 2

  5. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Short overview ◮ Proof by Sven Schewe in 2010 ◮ Minimization of deterministic B¨ uchi automata (MIN) is NP-complete ◮ Reduction from vertex cover problem to MIN Dejan Kostyszyn Complexity of B¨ uchi automata minimization 2

  6. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  7. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations ◮ Deterministic B¨ uchi automata Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  8. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations ◮ Deterministic B¨ uchi automata ◮ NP-completeness Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  9. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations ◮ Deterministic B¨ uchi automata ◮ NP-completeness ◮ The vertex cover problem Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  10. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations ◮ Deterministic B¨ uchi automata ◮ NP-completeness ◮ The vertex cover problem ◮ Definitions & Constructions ◮ ’Nice graph G v 0 ’ ◮ Language of the nice graph L ( G v 0 ) ◮ DBA that recognises L ( G v 0 ) Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  11. Overview Overview & Theorem Foundations Roadmap Definitions & Constructions Main proof Roadmap ◮ Foundations ◮ Deterministic B¨ uchi automata ◮ NP-completeness ◮ The vertex cover problem ◮ Definitions & Constructions ◮ ’Nice graph G v 0 ’ ◮ Language of the nice graph L ( G v 0 ) ◮ DBA that recognises L ( G v 0 ) ◮ The proof Dejan Kostyszyn Complexity of B¨ uchi automata minimization 3

  12. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) Deterministic B¨ uchi automaton B := (Σ , Q , q 0 , δ, F ), where Σ = finite set of symbols Q = finite set of states Q + = Q ∪ {⊥ , ⊤} q 0 ∈ Q + is initial state δ : Q + × Σ → Q + , δ ( ⊥ , a ) = ⊥ ∧ δ ( ⊤ , a ) = ⊤ , a ∈ Σ F ⊆ Q + , finite set of final states. Dejan Kostyszyn Complexity of B¨ uchi automata minimization 4

  13. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) Deterministic B¨ uchi automaton B := (Σ , Q , q 0 , δ, F ), where Σ = finite set of symbols Q = finite set of states Q + = Q ∪ {⊥ , ⊤} q 0 ∈ Q + is initial state δ : Q + × Σ → Q + , δ ( ⊥ , a ) = ⊥ ∧ δ ( ⊤ , a ) = ⊤ , a ∈ Σ F ⊆ Q + , finite set of final states. ρ = q 0 q 1 q 2 . . . , where i ∈ N 0 ∧ q i ∈ Q + , a run. B accepts exactly those runs in which at least one of the infinitely often occurring states is in F Dejan Kostyszyn Complexity of B¨ uchi automata minimization 4

  14. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) Σ ∗ is infinite set of finite words. Contains all possible finite combinations of symbols in Σ Dejan Kostyszyn Complexity of B¨ uchi automata minimization 5

  15. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) Σ ∗ is infinite set of finite words. Contains all possible finite combinations of symbols in Σ Σ ω is infinite set of infinite words. Contains all possible infinite combinations of symbols in Σ Dejan Kostyszyn Complexity of B¨ uchi automata minimization 5

  16. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) Dejan Kostyszyn Complexity of B¨ uchi automata minimization 6

  17. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) L = { w ∈ Σ ω | w contains infinitely many a ′ s } Dejan Kostyszyn Complexity of B¨ uchi automata minimization 6

  18. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Deterministic B¨ uchi automata (DBA) L = { w ∈ Σ ω | w contains infinitely many a ′ s } ⇒ Minimal, equivalent, but non-isomorphic Dejan Kostyszyn Complexity of B¨ uchi automata minimization 6

  19. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof NP-completeness By Behnam Esfahbod, CC BY-SA 3.0, https:commons.wikimedia.org/w/index.php?curid=3532181 Dejan Kostyszyn Complexity of B¨ uchi automata minimization 7

  20. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof NP-completeness By Behnam Esfahbod, CC BY-SA 3.0, https:commons.wikimedia.org/w/index.php?curid=3532181 NP is the set of problems that can be solved in non-deterministic polynomial time. Dejan Kostyszyn Complexity of B¨ uchi automata minimization 7

  21. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof NP-completeness By Behnam Esfahbod, CC BY-SA 3.0, https:commons.wikimedia.org/w/index.php?curid=3532181 NP is the set of problems that can be solved in non-deterministic polynomial time. A problem H is NP-hard if every problem L ∈ NP can be reduced in polynomial time to H . Dejan Kostyszyn Complexity of B¨ uchi automata minimization 7

  22. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof NP-completeness By Behnam Esfahbod, CC BY-SA 3.0, https:commons.wikimedia.org/w/index.php?curid=3532181 NP is the set of problems that can be solved in non-deterministic polynomial time. A problem H is NP-hard if every problem L ∈ NP can be reduced in polynomial time to H . A problem is NP-complete if it belongs to NP and NP-hard. Dejan Kostyszyn Complexity of B¨ uchi automata minimization 7

  23. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Vertex cover problem Let G = ( E , V ) be an undirected graph. S ⊆ V is called a vertex cover if ( u , v ) ∈ E ⇒ u ∈ S ∨ v ∈ S . Dejan Kostyszyn Complexity of B¨ uchi automata minimization 8

  24. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Vertex cover problem Let G = ( E , V ) be an undirected graph. S ⊆ V is called a vertex cover if ( u , v ) ∈ E ⇒ u ∈ S ∨ v ∈ S . A minimal vertex cover (MCOVER) is a vertex cover of minimal size. Dejan Kostyszyn Complexity of B¨ uchi automata minimization 8

  25. Overview DBA Foundations NP-completeness Definitions & Constructions Vertex cover problem Main proof Vertex cover problem Let G = ( E , V ) be an undirected graph. S ⊆ V is called a vertex cover if ( u , v ) ∈ E ⇒ u ∈ S ∨ v ∈ S . A minimal vertex cover (MCOVER) is a vertex cover of minimal size. Dejan Kostyszyn Complexity of B¨ uchi automata minimization 8

  26. Overview Nice graph Foundations Characteristic language of nice graph L ( G v 0 ) Definitions & Constructions DBA that recognises L ( G v 0 ) Main proof Next steps ◮ Definition ’nice graph’ Dejan Kostyszyn Complexity of B¨ uchi automata minimization 9

  27. Overview Nice graph Foundations Characteristic language of nice graph L ( G v 0 ) Definitions & Constructions DBA that recognises L ( G v 0 ) Main proof Next steps ◮ Definition ’nice graph’ ◮ Definition characteristic language of nice graph L ( G v 0 ) Dejan Kostyszyn Complexity of B¨ uchi automata minimization 9

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