communicating recursive programs: control and …acts/2015/slides/aiswarya.pdfc. aiswarya, and p....
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![Page 1: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/1.jpg)
C. Aiswarya Uppsala University, Sweden
Paul Gastin LSV, ENS Cachan, France
K. Narayan Kumar Chennai Mathematical Institute, India
Joint work with
ACTS!09/02/2015, Chennai
COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT-WIDTH
INFORMEL Indo-French Formal Methods Lab
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VERIFICATIONModel Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
![Page 3: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/3.jpg)
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
Undecidable in many cases
VERIFICATION
![Page 4: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/4.jpg)
UNDER-APPROXIMATE VERIFICATION
![Page 5: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/5.jpg)
UNDER-APPROXIMATE VERIFICATION
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UNDER-APPROXIMATE VERIFICATIONParametrised
![Page 7: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/7.jpg)
UNDER-APPROXIMATE VERIFICATIONParametrised
![Page 8: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/8.jpg)
UNDER-APPROXIMATE VERIFICATIONParametrised
![Page 9: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/9.jpg)
UNDER-APPROXIMATE VERIFICATION
…
Parametrised
![Page 10: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/10.jpg)
UNDER-APPROXIMATE VERIFICATION
…
Parametrised
Exhaustive
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UNDER-APPROXIMATE VERIFICATION
…
Parametrised
Exhaustive
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UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 13: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/13.jpg)
UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 14: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/14.jpg)
UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 15: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/15.jpg)
UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 16: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/16.jpg)
UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 17: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/17.jpg)
COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT-WIDTH
![Page 18: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/18.jpg)
COMMUNICATING DISTRIBUTED SYSTEMS
Process 1 Process 2
Process 3Process 4
Network
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BEHAVIOURS :! MESSAGE SEQUENCE CHARTS
time
Proc 1
Proc 2
Proc 3
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Emptiness or Reachability!
Inclusion or Universality!
Satisfiability φ!
Model Checking: S ⊨ φ!
Temporal logics!
Propositional dynamic logics!
Monadic second order logic
VERIFICATION PROBLEMS
![Page 21: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/21.jpg)
COMMUNICATING RECURSIVE PROGRAMS:
• Turing powerful: verification undecidable!• Under-upproximations!
• Decidable!• Controllable
![Page 22: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/22.jpg)
• Turing powerful: verification undecidable!• Under-upproximations!
• Decidable!• Controllable
COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT-WIDTH
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CONTROLLERS FOR VERIFICATION
OF COMMUNICATING SYSTEMS
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Process 2 Process 3
Network
Process 1
From ToCOMMUNICATING
DISTRIBUTED SYSTEMS
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CONTROLLERS FOR DISTRIBUTED SYSTEMS
Process 2 Process 3
Network
Process 1
From To
Controller 1 Controller 2 Controller 3
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Process 2 Process 3
Network
Process 1
From To
Controller 1 Controller 2 Controller 3From To
Heavy
Fragile
CONTROLLERS FOR DISTRIBUTED SYSTEMS
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Process 2 Process 3Process 1
Controller 1 Controller 2 Controller 3
Network
![Page 28: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/28.jpg)
Process 2 Process 3Process 1
Controller 1 Controller 2 Controller 3
Network
![Page 29: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/29.jpg)
CONTROLLERS FOR DISTRIBUTED SYSTEMS
Collection of local controllers!Communication via piggy-backing!Privacy: Do NOT read states/messages
![Page 30: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/30.jpg)
UNDER-APPROXIMATION: BOUNDED (K) PHASE
LET’S DESIGN A CONTROLLER
![Page 31: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/31.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 32: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/32.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 33: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/33.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 34: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/34.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 35: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/35.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 36: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/36.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 37: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/37.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
![Page 38: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/38.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
1. At most k phases on each process!2. No cycles
k-BOUNDED PHASE
![Page 39: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/39.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
1. At most k phases on each process!2. No cycles
k-BOUNDED PHASE
![Page 40: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/40.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
1. At most k phases on each process!2. No cycles
k-BOUNDED PHASE
![Page 41: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/41.jpg)
time
Proc 1
Proc 2
Proc 3
Receive from one process, send to all processesPHASE
1. At most k phases on each process!2. No cycles
k-BOUNDED PHASE
![Page 42: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/42.jpg)
DISTRIBUTED CONTROLLER FOR K-BOUNDED PHASE U-A
A local controller for each process
Has a Phase Counter
Remembers current sender
Different sender?
Detect Cycle?Increment counter,
Update sender
State
Transitions
![Page 43: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/43.jpg)
Detect Cycle?
Phase Vectors best info about phase number of other processes
Sends: tag with phase vector
Receives: update phase vector by taking MAX
DISTRIBUTED CONTROLLER FOR K-BOUNDED PHASE U-A
![Page 44: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/44.jpg)
CONTROLLERS FOR BOUNDED PHASE
DISTRIBUTED SYSTEMS
Collection of local controllers!Communication via piggy-backing!Privacy: Do NOT read states/messages
System independent!Generic!Deterministic!Finite state
![Page 45: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/45.jpg)
DECIDABILITY OF K BOUNDED PHASE
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Polynomial SPLIT-WIDTH
PSPACE
PDLTemporal Logics
Reachability
Decidable MSO
![Page 47: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/47.jpg)
Polynomial SPLIT-WIDTH
Refine phases to tree-like
bound split-width
![Page 48: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/48.jpg)
ACYCLIC PHASE DECOMPOSITION
time
Proc2
Proc1
Proc3
![Page 49: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/49.jpg)
INDUCED GRAPH ON PHASE
![Page 50: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/50.jpg)
INDUCED GRAPH ON PHASE
![Page 51: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/51.jpg)
PHASE DECOMPOSITION
time
Proc2
Proc1
Proc3
![Page 52: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/52.jpg)
PHASE DECOMPOSITION
time
Proc2
Proc1
Proc3
Tree-like
![Page 53: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/53.jpg)
Polynomial SPLIT-WIDTH
![Page 54: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/54.jpg)
Split-width
a a b c d
b a c d c
![Page 55: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/55.jpg)
Split-width
a a b c d
b a c d c
BUDGET
![Page 56: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/56.jpg)
b c d
a
a a
b c d c
a a b c d
b a c d c
![Page 57: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/57.jpg)
b c d
a
a a
b c d c
![Page 58: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/58.jpg)
b c d
a
a a
b c d c
![Page 59: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/59.jpg)
b c d
a
a a
b c d c
![Page 60: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/60.jpg)
b c d
a
![Page 61: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/61.jpg)
b c d
a
b dc
a
![Page 62: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/62.jpg)
b dc
a
![Page 63: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/63.jpg)
b dc
a
![Page 64: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/64.jpg)
b c d
a
a a
b c d c
![Page 65: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/65.jpg)
a a
b c d c
![Page 66: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/66.jpg)
a
b c d
a
c
a a
b c d c
![Page 67: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/67.jpg)
a
b c d
a
c
![Page 68: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/68.jpg)
a
b c d
a
c
![Page 69: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/69.jpg)
a
b c d
a
c
![Page 70: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/70.jpg)
a
b c d
![Page 71: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/71.jpg)
b d
a
c
a
b c d
![Page 72: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/72.jpg)
b d
a
c
![Page 73: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/73.jpg)
b d
a
c
![Page 74: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/74.jpg)
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
SPLIT TREE !OF THE FULL DECOMPOSITION
Split-width
![Page 75: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/75.jpg)
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 76: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/76.jpg)
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 77: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/77.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 78: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/78.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 79: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/79.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Vertices TREE INTERPRETATION
![Page 80: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/80.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Vertices TREE INTERPRETATION
![Page 81: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/81.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Vertices TREE INTERPRETATION
![Page 82: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/82.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Vertices TREE INTERPRETATION
![Page 83: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/83.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Vertices TREE INTERPRETATION
![Page 84: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/84.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 85: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/85.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Data edges TREE INTERPRETATION
![Page 86: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/86.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Data edges TREE INTERPRETATION
![Page 87: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/87.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
1
Data edges TREE INTERPRETATION
![Page 88: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/88.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
TREE INTERPRETATION
![Page 89: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/89.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 90: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/90.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 91: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/91.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 92: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/92.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 93: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/93.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 94: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/94.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 95: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/95.jpg)
b d
b d
a ca c
a c
C. Aiswarya, and P. Gastin 13
M
p
q
a a b c d
b a c d c
M′
p
q
a a b c d
b a c d c
M1
p
q
a a
b c d c
M′
1
p
q
a a
b c d c
M3
a
b c d
M′
3
a
b c d
a
c b d
a
c
M2
b c d
a
M′
2
b c d
a
c
a
b d
Figure 4 A split decomposition of width 3.
split-!
div-ÛÛ
split-!
div-ÛÛ
split-!
div-ÛÛ
Bd2
a c
Bd3
b d
Bd2
a c
split-!
div-ÛÛ
Bd4
a c
Bd1
b d
(n) split-!(m, mm)
(nÕ) div-ÛÛ(¸r, ¸r¸)
(n1) split-!(m, im)
(nÕ1) div-ÛÛ
(¸r, ¸¸r)
(n3) split-!(Á, im)
(nÕ3) div-ÛÛ
(¸, r¸r)
Bd2
(¸, r)
(n4) a, p c, q
Bd3
(Á, ¸r)
b, q d, q
Bd2
(¸, r)
(n5) a, p c, q
(n2) split-!(mm, Á)
(nÕ2) div-ÛÛ
(r¸r, ¸)
Bd4
(r, ¸)
a, q c, p
Bd1
(¸r, Á)
b, p d, p
Figure 5 A split term s (left) and a labelled term t (right) corresponding to Figure 4.
Process edges TREE INTERPRETATION
![Page 96: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/96.jpg)
Split-widthC. Aiswarya, and P. Gastin 17
ProblemComplexity
bound on split-widthpart of the input (inunary)
bound on split-widthfixed
CPDS emptiness ExpTime-Complete PTime-CompleteCPDS inclusion or universality 2ExpTime ExpTime-CompleteLTL / CPDL satisfiability or model checking ExpTime-CompleteICPDL satisfiability or model checking 2ExpTime -CompleteMSO satisfiability or model checking Non-elementaryTable 2 Summary of the complexities for bounded split-width verification.
Now, let Ï be a sentence in MSO(A, �). Using the MSO interpretation (�valid
, �vertex
,
(�a)aœ�
, (�p)pœProcs
, (�d)dœDS
, �æ) for k-bounded split-width, we can construct a formula Ï
k
from Ï such that for all trees t œ STkvalid
, we have t |= Ï
k if and only if cbm(t) |= Ï. By [31],from the MSO formula Ï
k we can construct an equivalent tree automaton AkÏ. Therefore, the
satisfiability problem for the MSO formula Ï restricted to CBMksplit
reduces to the emptinessproblem of the tree automaton Ak
valid
fl AkÏ.
Finally, we deduce easily that Lcbm
(S) fl CBMksplit
™ Lcbm
(Ï) if and only if for all trees t
accepted by AkS we have t |= Ï
k. Therefore, the model checking problem S |= Ï restricted toCBMk
split
reduces to the emptiness problem for the tree automaton Akvalid
fl AkS fl Ak
¬Ï.We have described above uniform decision procedures for an array of verification problems.
We refer to [2, 15,16] for more details and we summarise the computational complexities ofthese procedures in Table 2.
Verification procedures for other under-approximation classes. Our approach is generic inyet another sense. Under-approximation classes which admit a bound on split-width alsomay benefit from the uniform decision procedures described above, provided these classescorrespond to regular sets of split-terms.
More precisely, let Cm be an under-approximation class with Cm ™ CBMksplit
. For instance,we have seen that existentially m-bounded CBMs have split-width at most k = m + 1 (Ex. 6)and m-bounded phase MNWs have split-width at most k = 2m (Ex. 7). Assume that we canconstruct3 a tree automaton Ak
Cmwhich accepts a tree t œ STk
valid
if and only if cbm(t) œ Cm.Then, the decision procedures can be restricted to the class Cm with a further intersectionwith the tree automaton Ak
Cm. For instance, the emptiness problem for S restricted to Cm
reduces to the emptiness problem of Akvalid
fl AkCm
fl AkS . The model checking problem S |= Ï
restricted to Cm reduces to the emptiness problem of Akvalid
fl AkCm
fl AkS fl Ak
¬Ï.Clearly, the bound k on split-width in terms of m as well as the size of Ak
Cmwill impact
on the complexity of the decision procedures. We give below several examples.First, nested words have split-width bounded by a constant 2, and the set of nested words
can be recognised by a trivial 1-state CPDS. Hence the complexities of various problemsfollow the right-most column of Table 2. Notice that already for this simple case, thecomplexities match the corresponding lower bounds for all problems.
3 One way to obtain AkCm
is to provide a CPDS Sm which accepts the class Cm, then the automaton AkSm
serves as AkCm
. Similarly, if there is a formula Ïm in MSO(A, �) characterising the under-approximationthen the automaton Ak
Ïmserves as Ak
Cm.
![Page 97: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/97.jpg)
SPLIT-WIDTH
Proc2
Proc1
Proc3
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SPLIT-WIDTH
Proc2
Proc1
Proc3
![Page 99: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/99.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 100: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/100.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
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SPLIT-WIDTH 3
Proc2
Proc1
Proc3
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SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 103: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/103.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 104: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/104.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 105: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/105.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 106: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/106.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 107: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/107.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 108: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/108.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 109: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/109.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 110: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/110.jpg)
SPLIT-WIDTH 3
Proc2
Proc1
Proc3
![Page 111: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/111.jpg)
Polynomial SPLIT-WIDTH
PSPACE
PDLTemporal Logics
Reachability
Decidable MSO
![Page 112: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/112.jpg)
UNDER-APPROXIMATE VERIFICATION
Model Checking
S ⊨ φ?✗
✓System S!
Specification φ
Refine S !(Fix bugs)
k
Decidable
![Page 113: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/113.jpg)
OTHER UNDER-APPROXIMATIONS
Bounded channel size!
Existentially bounded [Genest et al.]!
Acyclic Architectures [La Torre et al., Heußner et al. Clemente et al.]!
Bounded context switching [Qadeer, Rehof], [LaTorre et al.], …!
Bounded phase [LaTorre et al.]!
Bounded scope [LaTorre et al.]!
Priority ordering [Atig et al., Saivasan et al.]
![Page 114: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/114.jpg)
OTHER UNDER-APPROXIMATIONS
Bounded channel size!
Existentially bounded [Genest et al.]!
Acyclic Architectures [La Torre et al., Heußner et al. Clemente et al.]!
Bounded context switching [Qadeer, Rehof], [LaTorre et al.], …!
Bounded phase [LaTorre et al.]!
Bounded scope [LaTorre et al.]!
Priority ordering [Atig et al., Saivasan et al.]
Tree-width
![Page 115: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/115.jpg)
OTHER UNDER-APPROXIMATIONS
Bounded channel size!
Existentially bounded [Genest et al.]!
Acyclic Architectures [La Torre et al., Heußner et al. Clemente et al.]!
Bounded context switching [Qadeer, Rehof], [LaTorre et al.], …!
Bounded phase [LaTorre et al.]!
Bounded scope [LaTorre et al.]!
Priority ordering [Atig et al., Saivasan et al.]
Tree-widthMany of the above classes have bounded tree-width [Parlato, Madhusudhan]
![Page 116: COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND …acts/2015/Slides/aiswarya.pdfC. Aiswarya, and P. Gastin 13 M p q aab c d b acd c M′ p q aab c d b acd c M 1 p q aa b c d c M′ 1](https://reader034.vdocuments.net/reader034/viewer/2022042921/5f68e4edac592d417c19b61e/html5/thumbnails/116.jpg)
Split-width!
Acyclic Architectures
Bounded channel size
Existentially bounded
Bounded context switching
Bounded scope
Bounded phase
Priority ordering
Bounded Tree-width
Constant
Bound + 2
2Bound
Linear
OTHER UNDER-APPROXIMATIONS
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Width: split vs tree vs clique
Let C be a class of bounded degree MSO definable graphs.!TFAE!1. C has a decidable MSO theory!2. C can be interpreted in binary trees!3. C has bounded tree-width!4. C has bounded clique-width!5. C has bounded split-width (for concurrent recursive behaviors)
Split-Width k
Tree-Width t Clique-Width c
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Width: split vs tree vs cliqueSplit-Width k
Tree-Width t Clique-Width c
t ≤ 2(k + |Procs|) - 1 c ≤ 2(k + |Procs|) + 1
Let C be a class of bounded degree MSO definable graphs.!TFAE!1. C has a decidable MSO theory!2. C can be interpreted in binary trees!3. C has bounded tree-width!4. C has bounded clique-width!5. C has bounded split-width (for concurrent recursive behaviors)
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Split-Width k
Tree-Width t Clique-Width c
k ≤ 120(t + 1) k ≤ 2c - 3
Width: split vs tree vs clique
Let C be a class of bounded degree MSO definable graphs.!TFAE!1. C has a decidable MSO theory!2. C can be interpreted in binary trees!3. C has bounded tree-width!4. C has bounded clique-width!5. C has bounded split-width (for concurrent recursive behaviors)
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COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT-WIDTH
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AUTONOMOUS COMPUTATIONS
• Recursive computations which does not read from other stacks/queues.!
• A stretch of computation in which all incoming edges are on a single stack
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AUTONOMOUS COMPUTATIONS
• Recursive computations which does not read from other stacks/queues.!
• A stretch of computation in which all incoming edges are on a single stack
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PHASE
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PHASE
• A stretch of computation which reads from at most one stack/queue
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PHASE
• A stretch of computation which reads from at most one stack/queue
• free (unlimited) autonomous computations
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PHASE
• A stretch of computation which reads from at most one stack/queue
• free (unlimited) autonomous computations• no loops
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K-BOUNDED PHASE
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K-BOUNDED PHASE
Phase 1 Phase 2 Phase 3
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IDENTIFYING AUTONOMOUS POPS
• Possible by tagging the values on stacks!
• Deterministic controller for each stack !
• The phase controller simulates one such automaton for each stack.
0 1
s?0
s?1
s̄?
else
s?0
s!0
else
s!1
s?1
s̄?
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C. A., Paul Gastin, and K. Narayan Kumar. Controllers for the verification of communicating multi-pushdown systems. In CONCUR 2014.
A. C. Verification of Communicating Recursive Programs via Split-width. PhD thesis, ENS Cachan, 2014.
C. A., Paul Gastin, and K. Narayan Kumar. Verifying communicating multi pushdown systems via Split-width. In ATVA 2014.
A. C., Paul Gastin, and K. Narayan Kumar. MSO decidability of multi-pushdown systems via Split-width. In CONCUR 2012.
COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT-WIDTH
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