Boundedness and Coverability for Pushdown Vector Addition Systems - PowerPoint PPT Presentation

Boundedness and Coverability for Pushdown Vector Addition Systems Grégoire Sutre LaBRI, CNRS & University of Bordeaux, France ACTS, CMI, Chennai — February 2017 Based on joint works with J. Leroux, M. Praveen and P. Totzke.

Table of Contents Pushdown Vector Addition Systems 1 Boundedness for Pushdown VAS 2 Coverability for 1-dim Pushdown VAS 3 Conclusion 4 2 / 31

Table of Contents Pushdown Vector Addition Systems 1 Boundedness for Pushdown VAS 2 Coverability for 1-dim Pushdown VAS 3 Conclusion 4 3 / 31

Vector Addition Systems Definition A VAS is a finite set of vectors a ∈ Z d . For u , v ∈ N d it has a step a − − → v if v = u + a . u ( 1 , 1 ) ( 0 , 3 ) ( 3 , 0 ) a = ( − 1 , 2 ) a b ( 2 , 2 ) ( 1 , 1 ) ( 1 , 4 ) ( 4 , 1 ) b = ( 2 , − 1 ) ( 0 , 6 ) ( 3 , 3 ) ( 6 , 0 ) 4 / 31

Vector Addition Systems Definition A VAS is a finite set of vectors a ∈ Z d . For u , v ∈ N d it has a step a − − → v if v = u + a . u Equivalent to Petri nets Many decidable verification questions ∗ Reachability: does u − − → v ? Coverability: does there exist v ′ ≥ v such that u ∗ → v ′ ? − − ∗ Boundedness: is { v | u − − → v } finite ? . . . 4 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � � � � − 1 2 push ( A ) , pop ( A ) , 0 0 � � 0 nop , − 1 p q 5 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � � � � − 1 2 push ( A ) , pop ( A ) , 0 0 � � 0 nop , − 1 p q � � 2 p , ⊥ , 1 5 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � � � � − 1 2 push ( A ) , pop ( A ) , 0 0 � � 0 nop , − 1 p q � � � � 2 0 p , ⊥ , − − →− − → p , AA ⊥ , 1 1 5 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � � � � − 1 2 push ( A ) , pop ( A ) , 0 0 � � 0 nop , − 1 p q � � � � � � 2 0 0 p , ⊥ , − − →− − → p , AA ⊥ , − − → q , AA ⊥ , 1 1 0 5 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. � � � � − 1 2 push ( A ) , pop ( A ) , 0 0 � � 0 nop , − 1 p q � � � � � � � � 2 0 0 4 p , ⊥ , − − →− − → p , AA ⊥ , − − → q , AA ⊥ , − − →− − → q , ⊥ , 1 1 0 0 5 / 31

Pushdown Vector Addition Systems . . . are products of VAS with pushdown automata. They can for example model recursive programs 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 5 / 31

Pushdown Vector Addition Systems — Definition Definition A pushdown VAS is a triple � Q , Γ , ∆ � where Q : finite set of states Γ : finite stack alphabet ∆ ⊆ Q × ( Op × Z d ) × Q : finite set of transitions, with Op = { nop } ∪ { push ( γ ) , pop ( γ ) | γ ∈ Γ } Configurations: ( q , σ, v ) with q ∈ Q , σ ∈ Γ ∗ and v ∈ N d Steps: as expected ∗ Reachability: does ( p , ε, u ) − − → ( q , ε, v ) ? Coverability: does there exist v ′ ≥ v with ( p , ε, u ) ∗ → ( q , ε, v ′ ) ? − − ∗ Boundedness: is { ( q , σ, v ) | ( p , ε, u ) − − → ( q , σ, v ) } finite ? 6 / 31

Pushdown Vector Addition Systems — Motivations VAS + ≃ Petri net ➪ Richer model for the verification of concurrent systems Multi-threaded recursive programs One recursive server + unboundedly many finite-state clients 7 / 31

Pushdown Vector Addition Systems — Motivations VAS + ≃ Petri net ➪ Richer model for the verification of concurrent systems Multi-threaded recursive programs One recursive server + unboundedly many finite-state clients ➪ Is the model too powerful? VAS + zero-tests Multi-PDA VAS PDA 7 / 31

Brief State of the Art Boundedness Coverability Reachability ExpSpace -c 1 , 2 ExpSpace -c 1 , 2 Decidable 3 , 4 , 5 VAS Decidable 7 Decidable 6 + full counter Decidable 9 Tower -h 8 + stack ExpTime -e 11 Decidable 10 1-VAS + stack ? [ 1 ] Lipton 1976 [ 7 ] Finkel, Sangnier 2010 [ 2 ] Rackoff 1978 [ 8 ] Lazić 2012 [ 3 ] Mayr 1981 [ 9 ] Leroux, Praveen, S. 2014 [ 4 ] Kosaraju 1982 [ 10 ] Leroux, S., Totzke 2015 [ 5 ] Leroux, Schmitz 2015 [ 11 ] Leroux, S., Totzke 2015 [ 6 ] Reinhardt 2008 8 / 31

Brief State of the Art Boundedness Coverability Reachability ExpSpace -c 1 , 2 ExpSpace -c 1 , 2 Decidable 3 , 4 , 5 VAS Decidable 7 Decidable 6 + full counter Decidable 9 Tower -h 8 + stack ExpTime -e 11 Decidable 10 1-VAS + stack ? Subclasses of pushdown VAS with decidable reachability Multiset pushdown systems [Sen, Viswanathan 2006] VAS ∩ CFL of finite index [Atig, Ganty 2011] Related decidable models with counters and recursion BPA( Z ) [Bouajjani, Habermehl, Mayr 2003] 8 / 31

Table of Contents Pushdown Vector Addition Systems 1 Boundedness for Pushdown VAS 2 Coverability for 1-dim Pushdown VAS 3 Conclusion 4 9 / 31

Reachability Tree of a Pushdown VAS q init , ε, v init q , σ, v q 1 , σ 1 , v 1 q n , σ n , v n ➪ Exhaustive and enumerative forward exploration from ( q init , ε, v init ) ➪ Potentially infinite, need to truncate 10 / 31

Reduced Reachability Tree for VAS [Karp, Miller 1969] a = ( − 1 , 2 ) Truncation Rule ( 1 , 1 ) v init b = ( 2 , − 1 ) v ( 1 , 1 ) ( 0 , 3 ) ( 3 , 0 ) v ′ if v ≤ v ′ ( 2 , 2 ) ( 2 , 2 ) ➪ The reduced reachability tree is finite ➪ It contains enough information to decide boundedness ➪ Crucial ingredient: the strict order < is a simulation relation 11 / 31

Tentative Simulation-Based Truncation for Pushdown VAS Truncation Rule q init , ε, v init q , σ, v q ′ , σ ′ , v ′ if q = q ′ , v ≤ v ′ and σ ≤ prefix σ ′ ➪ No loss of information to decide boundedness But... 12 / 31

Tentative Simulation-Based Truncation for Pushdown VAS Truncation Rule push ( A ) p q push ( B ) q init , ε, v init ( p , ⊥ ) ( q , A ⊥ ) q , σ, v ( q , BA ⊥ ) ( q , BBA ⊥ ) q ′ , σ ′ , v ′ if q = q ′ , v ≤ v ′ and σ ≤ prefix σ ′ ➪ No loss of information to decide boundedness But... The reduced reachability tree may be infinite! 12 / 31

Reduced Reachability Tree for Pushdown VAS Truncation Rule push ( A ) , − 1 1 q init , ε, v init p q push ( B ) , 1 q , σ, v ( p , ⊥ , 1 ) ( _ , _ , ρ ) ( q , A ⊥ , 0 ) q ′ , σ ′ , v ′ q = q ′ and v ≤ v ′ � if ( q , BA ⊥ , 1 ) σ ≤ suffix ρ for all ρ ➪ The reduced reachability tree is finite ➪ It contains enough information to decide boundedness 13 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · q q · · · v ′ ≥ v v 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · q q · · · v ′ ≥ v v 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · 14 / 31

Finiteness of the Reduced Reachability Tree Proposition The reduced reachability tree of a pushdown VAS is finite. Proof . By contradiction, assume that it is infinite. The tree is finitely branching. So, by König’s Lemma, there is an infinite branch ( q init , ε, v init ) → ( q 1 , σ 1 , v 1 ) → ( q 2 , σ 2 , v 2 ) · · · q v ′ ≥ v q v 14 / 31