COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT -WIDTH C. - PowerPoint PPT Presentation

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d TREE INTERPRETATION Vertices

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d TREE INTERPRETATION Vertices

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Data edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Data edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a 1 a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Data edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

a a c p b d M a c c q b d a a c p b d M ′ a c c q b d p a a c b d M 1 M 2 q c c b d a a a p c b d M ′ 1 M ′ c c q 2 b d a a b d c a M 3 c b d a c a a c b d a c M ′ 3 c b d a split- ! ( n ) ( m, mm ) b d c div- Û Û a c b d Process edges TREE INTERPRETATION

Split-width Complexity Problem bound on split-width bound on split-width part of the input (in fixed unary) CPDS emptiness ExpTime -Complete PTime -Complete CPDS inclusion or universality ExpTime -Complete 2ExpTime LTL / CPDL satisfiability or model checking ExpTime -Complete ICPDL satisfiability or model checking 2ExpTime -Complete MSO satisfiability or model checking Non-elementary Table 2 Summary of the complexities for bounded split-width verification.

SPLIT-WIDTH Proc 1 Proc 2 Proc 3

SPLIT-WIDTH Proc 1 Proc 2 Proc 3

SPLIT-WIDTH 3 Proc 1 Proc 2 Proc 3

SPLIT-WIDTH 3 Proc 1 Proc 2 Proc 3