Reachability Problems on (Partially Lossy) Queue Automata 13 th - PowerPoint PPT Presentation

Reachability Problems on (Partially Lossy) Queue Automata 13 th International Conference on Reachability Problems, Brussels Chris K¨ ocher Automata and Logics Group Technische Universit¨ at Ilmenau September 11, 2019 1

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶= { a ∣ a ∈ A } , A ∶= { a ∣ a ∈ A } q 2 q 3 Σ ∶= A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Queue Automata Example b b a q 0 q 1 Let A be an alphabet. Two actions for each a ∈ A : a write letter a ↝ a a a a b read letter a ↝ a a A ∶ = { a ∣ a ∈ A } , A ∶ = { a ∣ a ∈ A } q 2 q 3 Σ ∶ = A ⊎ A a a b a a b b b 2

Reachability Problem Example b b a Inputs: q 0 q 1 T ⊆ Σ ∗ regular language of transformation sequences a L ⊆ A ∗ regular language of queue a a a b contents a Compute: q 2 q 3 R each ( L , T ) ∶ = the set of all queue contents afer application of T on L a a b a a b b 3

Reachability Problem Example b b a Inputs: q 0 q 1 T ⊆ Σ ∗ regular language of transformation sequences a L ⊆ A ∗ regular language of queue a a a b contents a Compute: q 2 q 3 R each ( L , T ) ∶ = the set of all queue contents afer application of T on L a a b a a b b 3

Turing-Completeness Teorem (Brand, Zafiropulo 1983) Queue Automata can simulate Turing-machines. R each ( L , T ) can be any recursively enumerable language holds already for some fixed T = { t 1 , . . . , t n } ∗ with t 1 , . . . , t n ∈ Σ ∗ 4

Approximations of the Reachability Problem Iterative approach: for i = 0, 1, 2, . . . do compute the prefixes T i of length i from T apply T i on L Faster approach: Teorem (Boigelot, Godefroid, Willems, Wolper 1997) Let L ⊆ A ∗ be regular and t ∈ Σ ∗ . Ten R each ( L , t ∗ ) is effectively regular. ⇒ Combine multiple iterations of a loop to a meta-transformation Aim Generalize this result. 5

Te Main Teorem Teorem Let L , W , R ⊆ A ∗ be regular. Ten R each ( L , ( WR ) ∗ ) is effectively regular (in polynomial time). We slightly modify W and R : Let $ ∉ A be some new letter. Set W ′ ∶ = $ W and R ′ ∶ = shuffle ( R , $ ∗ ) . Easy: R each ( L , ( WR ) ∗ ) = proj A ( R each ( L , ( W ′ R ′ ) ∗ )) . We prove that R each ( L , ( W ′ R ′ ) ∗ ) is regular. From now on, we write W and R instead of W ′ and R ′ , resp. 6

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ a NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (1) Consider the following example: NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a, b a a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a , b a b $ 7

Proof Idea (2) NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a a, b a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a a , b b $ A configuration of the queue automaton can be abstracted as follows: the current state in ( WR ) ∗ 1 control state of C 2 the starting state of the path in LW ∗ 3 the ending state of the path in LW ∗ 4 the number of $ s on the path counter of C ⇒ Te queue automaton can be simulated by a one-counter automaton C 8

Proof Idea (2) NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a a, b a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a a , b b $ A configuration of the queue automaton can be abstracted as follows: the current state in ( WR ) ∗ 1 control state of C 2 the starting state of the path in LW ∗ 3 the ending state of the path in LW ∗ 4 the number of $ s on the path counter of C ⇒ Te queue automaton can be simulated by a one-counter automaton C 8

Proof Idea (2) NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a a, b a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a a , b b $ A configuration of the queue automaton can be abstracted as follows: the current state in ( WR ) ∗ 1 control state of C 2 the starting state of the path in LW ∗ 3 the ending state of the path in LW ∗ 4 the number of $ s on the path counter of C ⇒ Te queue automaton can be simulated by a one-counter automaton C 8

Proof Idea (2) NFA ( WR ) ∗ accepting ( WR ) ∗ : b $ a a, b a $ $ $ b $ a b NFA LW ∗ accepting LW ∗ : a a b $ a a , b b $ A configuration of the queue automaton can be abstracted as follows: the current state in ( WR ) ∗ 1 control state of C 2 the starting state of the path in LW ∗ 3 the ending state of the path in LW ∗ 4 the number of $ s on the path counter of C ⇒ Te queue automaton can be simulated by a one-counter automaton C 8

Semantics of C C ’s configurations consist of: the current state in ( WR ) ∗ 1 control state of C 2 the starting state of the path in LW ∗ 3 the ending state of the path in LW ∗ 4 the number of $ s on the path counter of C Let ( p , q , r , n ) ∈ Conf C be a configuration of C . q → r ) ∩ shuffle ( $ n , A ∗ ) � p , q , r , n � ∶ = L ( LW ∗ Proposition ⋃ R each ( L , ( WR ) ∗ ) = � σ � , σ ∈ Conf C , reac h. + acc. i.e., R each ( L , ( WR ) ∗ ) is a rational image of the set of reachable and accepting configurations of C . 9

Finishing the Proof Consider the set of reachable and accepting configurations of C . By [Bouajjani, Esparza, Maler 1997] this set is semilinear. Using a rational transduction implies effective regularity of ◻ R each ( L , ( WR ) ∗ ) . ⇒ We have seen: Teorem (Main Teorem) Let L , W , R ⊆ A ∗ be regular . T en R each ( L , ( WR ) ∗ ) is effectively regular (in polynomial time). 10

Consequences Corollary Let L ⊆ A ∗ and T ⊆ Σ ∗ be regular . T en R each ( L , T ∗ ) is regular if 1 T = R 1 WR 2 for regular W , R 1 , R 2 ⊆ A ∗ , 2 T = W ∪ R for regular W , R ⊆ A ∗ , 3 T = { t } for t ∈ Σ ∗ (cf. [Boigelot et al. 1997]), or 4 T = shuffle ( W , R ) for regular W , R ⊆ A ∗ . Remark: Proofs of 3 and 4 use some result from [K. 2018, cf. STACS’18] Tank you! 11