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

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INFORMEL Indo-French Formal Methods Lab COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT -WIDTH C. Aiswarya Uppsala University, Sweden Joint work with Paul Gastin K. Narayan Kumar LSV, ENS Cachan, France Chennai Mathematical Institute,

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SLIDE 1
  • 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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VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

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Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs) Undecidable in many cases

VERIFICATION

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UNDER-APPROXIMATE VERIFICATION

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UNDER-APPROXIMATE VERIFICATION

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UNDER-APPROXIMATE VERIFICATION

Parametrised

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UNDER-APPROXIMATE VERIFICATION

Parametrised

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SLIDE 8

UNDER-APPROXIMATE VERIFICATION

Parametrised

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SLIDE 9

UNDER-APPROXIMATE VERIFICATION

… Parametrised

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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

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UNDER-APPROXIMATE VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

k

Decidable

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UNDER-APPROXIMATE VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

k

Decidable

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SLIDE 15

UNDER-APPROXIMATE VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

k

Decidable

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UNDER-APPROXIMATE VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

k

Decidable

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COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT

  • WIDTH
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COMMUNICATING DISTRIBUTED SYSTEMS

Process 1 Process 2 Process 3 Process 4 Network

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SLIDE 19

BEHAVIOURS :! MESSAGE SEQUENCE CHARTS

time Proc 1 Proc 2 Proc 3

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SLIDE 20

Emptiness or Reachability! Inclusion or Universality! Satisfiability φ! Model Checking: S ⊨ φ! Temporal logics! Propositional dynamic logics! Monadic second order logic

VERIFICATION PROBLEMS

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SLIDE 21

COMMUNICATING RECURSIVE PROGRAMS:

  • Turing powerful: verification undecidable!
  • Under-upproximations!
  • Decidable!
  • Controllable
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SLIDE 22
  • Turing powerful: verification undecidable!
  • Under-upproximations!
  • Decidable!
  • Controllable

COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT

  • WIDTH
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SLIDE 23

CONTROLLERS FOR VERIFICATION OF COMMUNICATING SYSTEMS

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Process 2 Process 3 Network Process 1

From To

COMMUNICATING 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 3

From To

Heavy Fragile

CONTROLLERS FOR DISTRIBUTED SYSTEMS

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SLIDE 27

Process 2 Process 3 Process 1 Controller 1 Controller 2 Controller 3 Network

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SLIDE 28

Process 2 Process 3 Process 1 Controller 1 Controller 2 Controller 3 Network

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CONTROLLERS FOR DISTRIBUTED SYSTEMS

Collection of local controllers! Communication via piggy-backing! Privacy: Do NOT read states/messages

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UNDER-APPROXIMATION: BOUNDED (K) PHASE

LET’S DESIGN A CONTROLLER

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

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time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

  • 1. At most k phases on each process!
  • 2. No cycles

k-BOUNDED PHASE

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SLIDE 39

time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

  • 1. At most k phases on each process!
  • 2. No cycles

k-BOUNDED PHASE

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SLIDE 40

time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

  • 1. At most k phases on each process!
  • 2. No cycles

k-BOUNDED PHASE

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SLIDE 41

time Proc 1 Proc 2 Proc 3 Receive from one process, send to all processes PHASE

  • 1. At most k phases on each process!
  • 2. No cycles

k-BOUNDED PHASE

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SLIDE 42

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

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SLIDE 43

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

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SLIDE 44

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

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DECIDABILITY OF K BOUNDED PHASE

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Polynomial SPLIT-WIDTH PSPACE PDL Temporal Logics Reachability Decidable MSO

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Polynomial SPLIT-WIDTH Refine phases to tree-like bound split-width

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ACYCLIC PHASE DECOMPOSITION

time Proc2 Proc1 Proc3

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SLIDE 49

INDUCED GRAPH ON PHASE

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INDUCED GRAPH ON PHASE

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SLIDE 51

PHASE DECOMPOSITION

time Proc2 Proc1 Proc3

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PHASE DECOMPOSITION

time Proc2 Proc1 Proc3 Tree-like

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Polynomial SPLIT-WIDTH

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Split-width

a a b c d b a c d c

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Split-width

a a b c d b a c d c

BUDGET

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SLIDE 56

b c d a a a b c d c a a b c d b a c d c

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b c d a a a b c d c

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b c d a a a b c d c

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b c d a a a b c d c

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b c d a

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b c d a b d c a

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b d c a

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b d c a

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b c d a a a b c d c

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a a b c d c

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a b c d a c a a b c d c

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a b c d a c

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a b c d a c

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a b c d a c

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a b c d

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b d a c a b c d

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b d a c

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b d a c

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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

(n) split-! (m, mm) div-Û Û

SPLIT TREE ! OF THE FULL DECOMPOSITION

Split-width

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SLIDE 75

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 76

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 77

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 78

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 79

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Vertices TREE INTERPRETATION

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SLIDE 80

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Vertices TREE INTERPRETATION

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SLIDE 81

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Vertices TREE INTERPRETATION

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SLIDE 82

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Vertices TREE INTERPRETATION

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SLIDE 83

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Vertices TREE INTERPRETATION

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SLIDE 84

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 85

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Data edges TREE INTERPRETATION

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SLIDE 86

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Data edges TREE INTERPRETATION

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SLIDE 87

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

1

Data edges TREE INTERPRETATION

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SLIDE 88

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

TREE INTERPRETATION

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SLIDE 89

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 90

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 91

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 92

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 93

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 94

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 95

b d b d a c a c a c

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

(n) split-! (m, mm) div-Û Û

Process edges TREE INTERPRETATION

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SLIDE 96

Split-width

Problem Complexity bound on split-width part of the input (in unary) bound on split-width fixed CPDS emptiness ExpTime-Complete PTime-Complete CPDS inclusion or universality 2ExpTime ExpTime-Complete 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.

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SLIDE 97

SPLIT-WIDTH

Proc2 Proc1 Proc3

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SPLIT-WIDTH

Proc2 Proc1 Proc3

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SLIDE 99

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 101

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 103

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 104

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 105

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 106

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 107

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 108

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 109

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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SLIDE 110

SPLIT-WIDTH 3

Proc2 Proc1 Proc3

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Polynomial SPLIT-WIDTH PSPACE PDL Temporal Logics Reachability Decidable MSO

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UNDER-APPROXIMATE VERIFICATION

Model Checking S ⊨ φ? ✗ ✓ System S! Specification φ Refine S ! (Fix bugs)

k

Decidable

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SLIDE 113

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.]

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SLIDE 114

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

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SLIDE 115

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

Many of the above classes have bounded tree-width [Parlato, Madhusudhan]

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SLIDE 116

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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SLIDE 117

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 T ree-Width t Clique-Width c

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SLIDE 118

Width: split vs tree vs clique

Split-Width k T ree-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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SLIDE 119

Split-Width k T ree-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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SLIDE 120

COMMUNICATING RECURSIVE PROGRAMS: CONTROL AND SPLIT

  • WIDTH
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SLIDE 121

AUTONOMOUS COMPUTATIONS

  • Recursive computations which does not read from
  • ther stacks/queues.!
  • A stretch of computation in which all incoming

edges are on a single stack

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SLIDE 122

AUTONOMOUS COMPUTATIONS

  • Recursive computations which does not read from
  • ther stacks/queues.!
  • A stretch of computation in which all incoming

edges are on a single stack

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SLIDE 123

PHASE

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SLIDE 124

PHASE

  • A stretch of computation which reads from at

most one stack/queue

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SLIDE 125

PHASE

  • A stretch of computation which reads from at

most one stack/queue

  • free (unlimited) autonomous computations
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SLIDE 126

PHASE

  • A stretch of computation which reads from at

most one stack/queue

  • free (unlimited) autonomous computations
  • no loops
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SLIDE 127

K-BOUNDED PHASE

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SLIDE 128

K-BOUNDED PHASE

Phase 1 Phase 2

Phase 3

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SLIDE 129

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.

1 s?0 s?1 ¯ s? else s?0 s!0 else s!1 s?1 ¯ s?

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SLIDE 130
  • C. A., Paul Gastin, and K. Narayan Kumar. Controllers for the verification
  • f 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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SLIDE 131