Antichain algorithms. Using Antichains to solve reachabillity - PowerPoint PPT Presentation

Antichain algorithms. Using Antichains to solve reachabillity problems on non-deterministic finite automata. Albert-Ludwigs-Universität Freiburg Samuel Roth Proseminar on Automata Theory at the chair of Software Engineering. Supervised by Alexander Nutz.

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 2 / 28

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 3 / 28

Partial orders V be a finite set � a binary relation � ⊆ V × V � reflexive, transitive and anti-symmetric then it is called a partial order ( � , V ) is called a partially ordered set. February 2017 Samuel Roth – Antichain algorithms. 4 / 28

Partial orders V be a finite set � a binary relation � ⊆ V × V � reflexive, transitive and anti-symmetric then it is called a partial order ( � , V ) is called a partially ordered set. Example ( ≤ , { 1 , 2 , 3 , 42 } ), the ≤ order of the natural numbers. February 2017 Samuel Roth – Antichain algorithms. 4 / 28

Partial orders V be a finite set � a binary relation � ⊆ V × V � reflexive, transitive and anti-symmetric then it is called a partial order ( � , V ) is called a partially ordered set. Example ( ≤ , { 1 , 2 , 3 , 42 } ), the ≤ order of the natural numbers. ( ⊆ , 2 V )), subset-inclusion in a powerset. February 2017 Samuel Roth – Antichain algorithms. 4 / 28

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 5 / 28

Antichain subsets of V pairwise incompatible with regard to � . February 2017 Samuel Roth – Antichain algorithms. 6 / 28

Antichain subsets of V pairwise incompatible with regard to � . Example Powerset of { x , y , z } with ⊆ as partial order. February 2017 Samuel Roth – Antichain algorithms. 6 / 28

Antichain subsets of V pairwise incompatible with regard to � . Example Powerset of { x , y , z } with ⊆ as partial order. {{ x , y } , { x , z }} is a antichain. February 2017 Samuel Roth – Antichain algorithms. 6 / 28

Antichain subsets of V pairwise incompatible with regard to � . Example Powerset of { x , y , z } with ⊆ as partial order. {{ x , y } , { x , z }} is a antichain. {{ x } , { x , z }} is not a antichain. February 2017 Samuel Roth – Antichain algorithms. 6 / 28

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 7 / 28

Downward closure, Maximum of S ⊆ V Downward closure Down ( � , S ) := { v ′ ∈ V | ∃ v ∈ Sv ′ � v } February 2017 Samuel Roth – Antichain algorithms. 8 / 28

Downward closure, Maximum of S ⊆ V Downward closure Down ( � , S ) := { v ′ ∈ V | ∃ v ∈ Sv ′ � v } Examples Down ( ⊆ , { x , y } ) = { / 0 , { x } , { y } , { x , y }} February 2017 Samuel Roth – Antichain algorithms. 8 / 28

Downward closure, Maximum of S ⊆ V Downward closure Down ( � , S ) := { v ′ ∈ V | ∃ v ∈ Sv ′ � v } Maximum Max ( � , S ) := { v ∈ S | ∀ v ′ ∈ S : v � v ′ ⇒ v ′ � v } Examples February 2017 Samuel Roth – Antichain algorithms. 8 / 28

Downward closure, Maximum of S ⊆ V Downward closure Down ( � , S ) := { v ′ ∈ V | ∃ v ∈ Sv ′ � v } Maximum Max ( � , S ) := { v ∈ S | ∀ v ′ ∈ S : v � v ′ ⇒ v ′ � v } Examples Max ( ⊆ , {{ x } , { x , y } , { x , z }} ) = {{ x , y } , { x , z }} February 2017 Samuel Roth – Antichain algorithms. 8 / 28

Downward closure, Maximum of S ⊆ V Downward closure Down ( � , S ) := { v ′ ∈ V | ∃ v ∈ Sv ′ � v } Maximum Max ( � , S ) := { v ∈ S | ∀ v ′ ∈ S : v � v ′ ⇒ v ′ � v } Examples Max ( ⊆ , {{ x } , { x , y } , { x , z }} ) = {{ x , y } , { x , z }} Max ( ⊆ , {{ x } , { x , y } , { x , z } , { x , y , z }} ) = {{ x , y , z }} February 2017 Samuel Roth – Antichain algorithms. 8 / 28

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 9 / 28

Antichain as a representation for a downward closed set S ⊆ V . Use S ′ := Max ( � , S ) to represent S . February 2017 Samuel Roth – Antichain algorithms. 10 / 28

Antichain as a representation for a downward closed set S ⊆ V . Use S ′ := Max ( � , S ) to represent S . The question v ∈ S becomes ∃ v ′ ∈ S ′ : v � v ′ February 2017 Samuel Roth – Antichain algorithms. 10 / 28

Antichain as a representation for a downward closed set S ⊆ V . Use S ′ := Max ( � , S ) to represent S . The question v ∈ S becomes ∃ v ′ ∈ S ′ : v � v ′ Example 0 , { x } , { y }} so S ′ S 1 := { / 1 = {{ x } , { y }} 0 , { x } , { y } , { x , y }} so S ′ S 2 := { / 2 = {{ x , y }} February 2017 Samuel Roth – Antichain algorithms. 10 / 28

Content Preliminaries Partial orders Antichains Downward closure, Maximum Antichains as representations of closed sets Powerset determinization of Non-deterministic finite automata Reachability problem Backward reachability fixpoint algorithm Antichain Backward reachability algorithm Conclusion February 2017 Samuel Roth – Antichain algorithms. 11 / 28

Powerset determinization of Non-deterministic finite automata Let A := ( Loc , Init , Fin , δ , Σ) be a finite automaton. G ( A ) := ( V , E , In , Fin ) is the corresponding powerset automaton. February 2017 Samuel Roth – Antichain algorithms. 12 / 28

Powerset determinization of Non-deterministic finite automata Let A := ( Loc , Init , Fin , δ , Σ) be a finite automaton. G ( A ) := ( V , E , In , Fin ) is the corresponding powerset automaton. V := 2 Loc February 2017 Samuel Roth – Antichain algorithms. 12 / 28

Powerset determinization of Non-deterministic finite automata Let A := ( Loc , Init , Fin , δ , Σ) be a finite automaton. G ( A ) := ( V , E , In , Fin ) is the corresponding powerset automaton. V := 2 Loc In := { v ∈ 2 Loc | Init ∈ v } February 2017 Samuel Roth – Antichain algorithms. 12 / 28

Powerset determinization of Non-deterministic finite automata Let A := ( Loc , Init , Fin , δ , Σ) be a finite automaton. G ( A ) := ( V , E , In , Fin ) is the corresponding powerset automaton. V := 2 Loc In := { v ∈ 2 Loc | Init ∈ v } Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } February 2017 Samuel Roth – Antichain algorithms. 12 / 28

Powerset determinization of Non-deterministic finite automata Let A := ( Loc , Init , Fin , δ , Σ) be a finite automaton. G ( A ) := ( V , E , In , Fin ) is the corresponding powerset automaton. V := 2 Loc In := { v ∈ 2 Loc | Init ∈ v } Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } ( v 1 , v 2 ) ∈ E iff there exists a σ ∈ Σ such that � q ∈ v 1 δ ( q , σ ) = v 2 . February 2017 Samuel Roth – Antichain algorithms. 12 / 28

Example A := ( Loc , Init , Fin , δ , Σ) to G ( A ) := ( V , E , In , Fin ) V := 2 Loc Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } In := { v ∈ 2 Loc | Init ∈ v } ( v 1 , v 2 ) ∈ E iff there exists a σ ∈ Σ such that � q ∈ v 1 δ ( q , σ ) = v 2 . February 2017 Samuel Roth – Antichain algorithms. 13 / 28

Example A := ( Loc , Init , Fin , δ , Σ) to G ( A ) := ( V , E , In , Fin ) V := 2 Loc Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } In := { v ∈ 2 Loc | Init ∈ v } ( v 1 , v 2 ) ∈ E iff there exists a σ ∈ Σ such that � q ∈ v 1 δ ( q , σ ) = v 2 . February 2017 Samuel Roth – Antichain algorithms. 13 / 28

Example A := ( Loc , Init , Fin , δ , Σ) to G ( A ) := ( V , E , In , Fin ) V := 2 Loc Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } In := { v ∈ 2 Loc | Init ∈ v } ( v 1 , v 2 ) ∈ E iff there exists a σ ∈ Σ such that � q ∈ v 1 δ ( q , σ ) = v 2 . February 2017 Samuel Roth – Antichain algorithms. 13 / 28

Example A := ( Loc , Init , Fin , δ , Σ) to G ( A ) := ( V , E , In , Fin ) V := 2 Loc Fin := { v ∈ 2 Loc | v ⊆ Loc \ Fin } In := { v ∈ 2 Loc | Init ∈ v } ( v 1 , v 2 ) ∈ E iff there exists a σ ∈ Σ such that � q ∈ v 1 δ ( q , σ ) = v 2 . February 2017 Samuel Roth – Antichain algorithms. 13 / 28