On Boundedness Problems for Pushdown Vector Addition Systems J er - PowerPoint PPT Presentation

On Boundedness Problems for Pushdown Vector Addition Systems J´ erˆ ome Leroux Gr´ egoire Sutre Patrick Totzke September 21, 2015 1 / 12

Vector Addition Systems – Recap Definition A VAS is a finite set of vectors a ∈ Z d . For v , v ′ : N d it has a step a v ′ = v + a . → v ′ v − − if ◮ Equivalent to Petri Nets (concurrency, weak counters, event systems) ◮ Reachability: decidable Mayr’81 , Kosaraju’82 , . . . Leroux and Schmitz’15 ◮ Coverability, Boundedness: ExpSpace -complete Lipton’76 , Rackoff’78 ◮ Most Games/Equivalences undecidable (e.g. Bisimulation) Janˇ car’95 2 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � − 1 � � 2 � push( A ), pop( A ), � 0 0 0 � nop, − 1 q s 3 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � − 1 � � 2 � push( A ), pop( A ), � 0 0 0 � nop, − 1 q s � � 2 s , ⊥ , 1 3 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � − 1 � � 2 � push( A ), pop( A ), � 0 0 0 � nop, − 1 q s � � � 0 � 2 s , ⊥ , − − →− − → s , AA ⊥ , 1 1 3 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � − 1 � � 2 � push( A ), pop( A ), � 0 0 0 � nop, − 1 q s � � � 0 � � 0 � 2 s , ⊥ , − − →− − → s , AA ⊥ , − − → q , AA ⊥ , 1 1 0 3 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � − 1 � � 2 � push( A ), pop( A ), � 0 0 0 � nop, − 1 q s � � � 0 � � 0 � � 4 � 2 s , ⊥ , − − →− − → s , AA ⊥ , − − → q , AA ⊥ , − − →− − → q , ⊥ , 1 1 0 0 3 / 12

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. They can for example model recursive prorams with variables over N . 1: x ← n start 2 6 2: procedure DoubleX if ( ⋆ ∧ x > 0) then 3: x ← ( x − 1) 4: push( A ) pop( A ) 3 7 DoubleX 5: end if 6: − 1 +2 x ← ( x + 2) 7: 5 8 8: end procedure 3 / 12

Pushdown Vector Addition Systems ◮ Reachability = Coverability (= State-Reachability) Tower -hard Lazic’13 4 / 12

Pushdown Vector Addition Systems ◮ Reachability d dim. = Coverability d + 1 dim. Tower -hard Lazic’13 4 / 12

Pushdown Vector Addition Systems ◮ Reachability d dim. = Coverability d + 1 dim. Tower -hard Lazic’13 ◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 4 / 12

Pushdown Vector Addition Systems ◮ Reachability d dim. = Coverability d + 1 dim. Tower -hard Lazic’13 ◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14 Theorem [LSP’14] If a PVAS configuration ( p , ⊥ , n ) is bounded then the cardinality of the reachability set is at most F ω d ·| Q | ( d · n ). 4 / 12

Pushdown Vector Addition Systems ◮ Reachability d dim. = Coverability d + 1 dim. Tower -hard Lazic’13 ◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14 ◮ Counter-, Stack-, and Combined Boundedness Problems. Combined Stack Counter 4 / 12

Pushdown Vector Addition Systems ◮ Reachability d dim. = Coverability d + 1 dim. Tower -hard Lazic’13 ◮ Coverability in 1 dim. is decidable Leroux, Sutre, and T.’15 ◮ Boundedness: decidable with Hyper-Ackermannian bounds Leroux, Praveen, and Sutre’14 ◮ Counter-, Stack-, and Combined Boundedness Problems. Combined Stack Counter The following is in ExpTime . 1-PVAS Counter-Boundedness Given : 1-dim. PVAS, initial configuration ( p , w , a ). ∗ → ( p ′ , w ′ , b ) } infinite? Question : is { b | ( p , w , a ) − − 4 / 12

Another Perspective Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k a 1 a 2 . . . a k ∈ L and − − → − − → · · · − − → t . s 5 / 12

Another Perspective Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k a 1 a 2 . . . a k ∈ L and − − → − − → · · · − − → t . s Theorem For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness. 5 / 12

Another Perspective Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k a 1 a 2 . . . a k ∈ L and − − → − − → · · · − − → t . s Theorem For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness. Observation Relevant for the PVAS boundedness problem is the trace language { w ∈ A ∗ | ( p 0 , ⊥ ) w − − →} defined by the PDA. 5 / 12

Another Perspective Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k a 1 a 2 . . . a k ∈ L and − − → − − → · · · − − → t . s Theorem For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness. Observation Relevant for the PVAS boundedness problem is the trace language { w ∈ A ∗ | ( p 0 , ⊥ ) w − − →} defined by the PDA. prefix-closed! 5 / 12

Another Perspective Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k a 1 a 2 . . . a k ∈ L and − − → − − → · · · − − → t . s Theorem For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness. Observation Relevant for the PVAS boundedness problem is the trace language { w ∈ A ∗ | ( p 0 , ⊥ ) w − − →} defined by the PDA. Main Theorem Boundedness of 1-dim VAS controlled by a prefix-closed language is in ExpTime . 5 / 12

Another Perspective given as GfG Definition (Context-free Controlled VAS) a VAS A ⊆ Z d together with a context-free language L ⊆ A ∗ . X → t between s , t ∈ N d if There is a step s − − a 1 a 2 a k ∗ X = = ⇒ a 1 a 2 . . . a k and s − − → − − → · · · − − → t . Theorem For Cf-Controlled VAS, Coverability (and Reachability) logspace reduces to Boundedness. Observation Relevant for the PVAS boundedness problem is the trace language { w ∈ A ∗ | ( p 0 , ⊥ ) w − − →} defined by the PDA. Main Theorem Boundedness of 1-dim VAS controlled by a prefix-closed language is in ExpTime . 5 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X − 1 Y Z 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X − 1 Y Z 1 Y 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X − 1 Y Z 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X − 1 Y Z − 1 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z − 1 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 − 1 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 − 1 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 4 − 1 1 Y 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 4 − 1 1 Y 4 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 4 − 1 1 Y 4 5 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 4 − 1 1 Y 4 5 5 1 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 − 1 Y Z 5 4 4 − 1 1 Y 4 5 5 1 5 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 5 − 1 Y Z 5 4 4 6 6 5 − 1 1 Y 4 5 5 6 6 5 1 5 6 6 / 12

Flow Trees A derivation tree with consistent in/out labels in Z ∪ {−∞} . X 5 5 − 1 Y Z 5 4 4 6 6 5 − 1 1 Y 4 5 5 6 6 5 1 5 6 → b ′ ≥ b ; → b ′ ≥ b . X X b means a − − b means ∃ a ∈ N . a − − X X −∞ a 6 / 12