Towards a Regular Theory of Parameterized Concurrent Systems - PowerPoint PPT Presentation

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l r l r l l r s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l s 6 r s 4 l r l l r s 5 s 6 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l s 6 r s 4 l r l l r s 5 s 6 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 Acceptance condition: � r !1 l ?0 r !1 l ?0 MSO formula over rings whose nodes are labeled with states. � l ?1 l ?1 r l y s ( x ) s 1 s 2 s 3 s 1 s 2 s 3 Signature: � x � r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 Thus, there are no constant processes (e.g., no «first» or «last» process). r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) s 6 r s 4 l r l l r s 5 s 6 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4

Parameterized Communicating Automata (PCA) over Rings r l | 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) s 6 r = s 4 l r l l r s 5 s 6 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4 4

Parameterized Communicating Automata (PCA) over Rings r l | 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) s 6 r = s 4 l r l l r s 5 s 6 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4 Token-Ring Protocol 4

Parameterized Communicating Automata (PCA) over Rings r l | 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) s 6 s 6 r = l r l l r s 5 s 4 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 4 4

Parameterized Communicating Automata (PCA) over Rings r l 6 | = 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) s 6 r s 4 l r l l r s 6 s 4 s 0 s 0 r !1 l ?0 r !1 l ?0 s 0 s 0 l ?1 l ?1 r !1 l ?0 r !1 l ?0 l ?1 l ?1 s 1 s 2 s 3 s 1 s 2 s 3 r !0 r !0 s 1 s 2 s 3 s 1 s 2 s 3 l ?0 r !1 r !0 l ?0 r !1 r !0 r !0 r !0 l ?0 r !1 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 s 4 s 5 s 6 47 47 4

Parameterized Communicating Automata (PCA) over Rings s 0 l ?0 r !1 l ?1 L s 1 s 3 s 2 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) 5

Parameterized Communicating Automata (PCA) over Rings     l r                                 s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 5

Parameterized Communicating Automata (PCA) over Rings     l r                                 s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 5

Parameterized Communicating Automata (PCA) over Rings     l r                                 s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 5

Parameterized Communicating Automata (PCA) over Rings     l r                                 s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                           …       5

Complementation s 0 l ?0 r !1 l ?1 L s 1 s 3 s 2 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) 6

Complementation                                     s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 6

Complementation                                     s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 6

Complementation                                     s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 6

Complementation                                     s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                                 6

Complementation                                     s 0           l ?0 r !1   l ?1       = L       s 1 s 3 s 2   r !0   l ?0 r !1 r !0               s 4 s 5 s 6         9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y )))                           …       6

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. 7

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. Proof: … … … … … … … … … 7

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. Proof: … … … … … … … … … … … … … … … Behaviors encode grids. 7

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. Proof: … … … … … … … … … … … … … … … Behaviors encode grids. Grid automata are not closed under complementation [Matz-Schweikardt-Thomas ’02]. 7

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. Proof: … … … … … … … … … … … … … … … Behaviors encode grids. Grid automata are not closed under complementation [Matz-Schweikardt-Thomas ’02]. Theorem [Emerson-Namjoshi 2003]: � Emptiness is undecidable for PCAs over rings � (even token-passing systems, unless ). | Msg | = 1 7

Negative Results Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � PCAs over rings are not complementable. Proof: … … … … … … … … … … … … … … … Behaviors encode grids. Grid automata are not closed under complementation [Matz-Schweikardt-Thomas ’02]. Theorem [Emerson-Namjoshi 2003]: � Emptiness is undecidable for PCAs over rings � (even token-passing systems, unless ). | Msg | = 1 7

Context-Bounded PCAs 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 8

Context-Bounded PCAs Idea: Every process is contrained to a bounded number of contexts. There are several possible definitions of a context that lead to positive results. Here: Process only sends XOR only receives from one fixed neighbor. 3-bounded 8

Context-Bounded PCAs s 0 r !1 l ?0 l ?1 s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) Definition: A PCA is k -bounded if the finite automaton restricts to k contexts. 9

Context-Bounded PCAs s 0 r !1 l ?0 l ?1 s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) 2-bounded PCA Definition: A PCA is k -bounded if the finite automaton restricts to k contexts. 9

Context-Bounded PCAs s 0 r !1 l ?0 l ?1 s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) 2-bounded PCA Definition: A PCA is k -bounded if the finite automaton restricts to k contexts. Theorem [B.-Gastin-Kumar; FSTTCS 2014]: � A L ( B ) = L ( A ) For every bounded PCA , there is a PCA such that . B 9

Proof Outline disambiguation � nondeterminism complementation every behavior has a unique run s 0 r !1 l ?0 l ?1 s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) k -bounded 10

Proof Outline disambiguation � nondeterminism complementation every behavior has a unique run s 0 r !1 l ?0 l ?1 A s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 ϕ 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) k -bounded 10

Proof Outline disambiguation � nondeterminism complementation every behavior has a unique run s 0 r !1 l ?0 l ?1 ! A A s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 ϕ ¬ ϕ 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) k -bounded 10

Proof Outline disambiguation � nondeterminism complementation every behavior has a unique run s 0 r !1 l ?0 l ?1 ? ! A A s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 ϕ ¬ ϕ 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) k -bounded Powerset construction not applicable due to message contents. 10

Proof Outline disambiguation � nondeterminism complementation every behavior has a unique run s 0 r !1 l ?0 l ?1 ? ! A A s 1 s 2 s 3 r !0 l ?0 r !1 r !0 s 4 s 5 s 6 ϕ ¬ ϕ 9 x ( s 4 ( x ) ^ 8 y ( y 6 = x ! s 5 ( y ) _ s 6 ( y ))) k -bounded Powerset construction not applicable due to message contents. Disambiguation through summaries: � Alur-Madhusudan: Visibly pushdown languages. STOC 2004. La Torre-Madhusudan-Parlato: The language theory of bounded context switching. LATIN 2010. La Torre-Napoli-Parlato: Scope-bounded pushdown languages. DLT 2014. 10

Disambiguation of context-bounded PCAs 11

Disambiguation of context-bounded PCAs 11

Disambiguation of context-bounded PCAs 11

Disambiguation of context-bounded PCAs 11

Disambiguation of context-bounded PCAs 11

Disambiguation of context-bounded PCAs Every process traverses a bounded number of zones. 11

Disambiguation of context-bounded PCAs Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 6 = Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 1, 2 ,3 2, 3 ,1 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 1, 2 ,3 2, 3 ,1 2, 2 ,3 0, 2 ,2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs 0, 0 ,0 0, 0 ,0 0, 0 ,0 0, 0 ,0 r l 0, 1 ,1 1, 1 ,1 2, 1 ,0 1, 2 ,1 1, 2 ,3 2, 3 ,1 2, 2 ,3 0, 2 ,2 2, 2 ,3 2, 3 ,1 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously by a PCA. 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously. Sending processes deterministically compute summaries for zones. 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously. Sending processes deterministically compute summaries for zones. 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously. Sending processes deterministically compute summaries for zones. Acceptance condition checks if summaries correspond to accepting run. 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously. Sending processes deterministically compute summaries for zones. Acceptance condition checks if summaries correspond to accepting run. 11

Disambiguation of context-bounded PCAs R 1 R 3 R i ⊆ S 3 × S 3 R 2 Every process traverses a bounded number of zones. Zone numbers can be computed unambiguously. Sending processes deterministically compute summaries for zones. Acceptance condition checks if summaries correspond to accepting run. 11