Instant Polymorphic Type Systems for Mobile Process Calculi: Just - PowerPoint PPT Presentation

Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close Henning Makholm and Joe Wells Heriot-Watt University ESOP’05 – April 8, 2005 U seful L ogics, T ypes, Work supported by EU/IST/FET grant 2001-33477 (DART) R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.1/29

Mobile process calculi Mobility and process calculi are used to model and reason about systems with mobile devices , mobile code , dynamically changing networks , . . . and to model biological systems and business processes . Many such calculi exist: The π -calculus – and variants Mobile Ambients – and variants Safe Ambients, Boxed Ambients, Seal – and variants D π , Higher-order π -calculus – and variants Join calculus – and variants There is no obvious best calculus. For different purposes one may need different calculi, and needs are likely to change. U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.2/29

Types for process calculi Any process and mobility calculus can benefit from having a type system . For pinpointing programming errors To prove that programs or systems are safe To provide flow information for automatic analyses Traditionally each new calculus has a type system designed specifically for it. We present the re-targetable type system Poly ✶ which automatically adapts to new calculi or variants. Allows easy experimentation with calculus variants Just write down your reduction rules. Poly ✶ does the rest. Experimenting with type system features : Which features do I need to handle this kind of code? U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.3/29

Plan Poly ✶ example Case study: Evolution of calculi Spatial polymorphism Theoretical properties Conclusion U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.4/29

An example Poly ✶ type for an ambient term a a a Term: ( x ) . b b b b b | b in x . 0 | � a � . 0 | → in a . 0 | → ֒ ֒ 0 0 0 0 Type: (x) b[] in x x:={a} x:={a} x:={a} ROOT <{a}> b[] in a a[] b[] Black edges with labels define the possible term structure. Red edges encode flow , which is the same as subtyping . U seful L ogics, Green edges encode flow/subtyping with substitutions . T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.5/29

Drawing the type graph more compactly (x) b[] in x x:={a} x:={a} x:={a} ROOT <{a}> b[] in a a[] b[] When all black edges leading to a node have the same label, we write the label inside the target node: (x) b[] in x ROOT <{a}> x:={a} x:={a} x:={a} b[] in a U seful a[] b[] L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.6/29

Plan Poly ✶ example Case study: Evolution of calculi Spatial polymorphism Theoretical properties Conclusion U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.7/29

The siege of Troy horse Ulysses Troy Term: in Troy in horse . out horse 0 Input to Poly ✶ type inference tool: ✝ ✠ ✡ ☛ ✄ ✞ ✞ ✂ � ✁ ☎ ✆ � ✟ ✝ ✠ ✓ ✡ ✓ ✠ ✡ ✠ ✠ ✓ ✡ ✓ ✡ ☛ ✎ ✌ ✄ ✎ ✑ ✞ ✔ ✑ ✕ ✑ ✞ ✔ ✕ ✗ ☞ ✆ ✍ ✁ ✆ � ✏ � ✖ ✖ ✒ ✝ ✠ ✠ ✓ ✡ ✓ ✡ ✠ ✡ ✓ ✠ ✓ ✡ ☛ ✌ ✑ ✎ ✎ ✞ ✔ ✕ ✕ ✑ ✞ ✔ ✗ ✂ ☞ ✆ ✍ ✁ ✆ � ✘ ✍ � � ✖ ✖ ✒ ✝ ✓ ✠ ✡ ✓ ☛ ✌ ✎ ✎ ✞ ✚ ✞ ✚ ✗ � � ☞ ✆ ✍ ✁ ✆ ✘ ✙ ✆ ✏ ✖ ✖ ✒ ✝ ✠ ✡ ✜ ✄ ✣ ✂ ✆ ☞ ✛ ✘ ☞ ✢ ✆ ✏ ☞ ✘ ✤ ✓ ✠ ✡ ✥ ✦ ✄ ✜ ✜ ✂ ✤ ✢ ✢ ✆ ✢ ✏ ✘ ☞ ✢ ✆ ✘ ✍ ✘ ☞ ✢ ✆ ✒ ✓ ✠ ✡ ✣ ✧ ☞ ✘ ✤ ☛ U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.8/29

The siege of Troy horse Ulysses Troy Term: in Troy in horse . out horse 0 Inferred type: Ulysses[] Troy[] horse[] in Troy ROOT horse[] in horse out horse Ulysses[] U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.9/29

Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.10/29 What if one needed permission to enter and exit ambients? ☛ ☛ ☛ ✚ ✚ ✡ ✚ Safe Ambients, first try ✡ � ✒ ✓ ✚ ✓ ✎ ✡ ✑ ✒ ✕ ✂ ✔ ✡ ✍ ✕ ✓ ✘ ✞ ✎ ✓ � ✏ ✠ ✒ ✖ ✕ ✘ ✑ ✄ ✡ ✔ ✁ ✖ ✎ ✓ ✏ ✘ ✎ ✓ ✓ ✚ ✞ ✁ ✆ ✡ ✠ ✙ ✆ ✎ ✓ ✡ ✡ � ✕ ✘ ✢ ✆ ✓ ✠ ✠ ☞ ✖ ✢ ✓ ✡ � ✑ ✕ ✘ ✞ ✘ ☞ ✜ ✕ ✁ ✘ ✠ ✜ ✗ ✗ ✗ ✎ ✓ ✏ ✑ ✖ ✖ ✖ ✄ ✖ ✖ ✖ ✂ ✡ ✓ ✍ ✔ ✖ ✓ ✚ ✘ ✘ ✡ ✠ ✁ ✧ ✡ ✓ � ✒ ✔ ✆ ✞ ✤ ✒ ✢ ✓ ☞ ✘ ✓ ✓ � ✒ ✤ ☞ ✘ ✞ ✞ ✜ ✣ ☛ ✘ ☞ ✎ � ✣ ✏ ✏ ✑ ✒ ✂ ✒ ✡ ✞ ✍ ✄ ✄ ✠ ✘ ✏ ✠ ✖ ✎ ✎ � ✏ ✏ ✄ ✢ ✘ ✎ ✠ ✠ ✄ ✆ ✆ ✁ ✑ ✙ ✢ ✠ ✆ ✎ ✟ ✤ ✠ ✠ ✘ ✢ ✢ � � ☞ ✤ ✞ ✘ ✎ ☞ ✘ ✦ ✜ ✥ ✣ ✝ ✝ ✝ ✝ R ewriting, and their ✆ ✆ ✆ ✆ A utomation ☎ ✁ ✁ ✁ ✝ ☛ ✓ ✓ ✍ ✍ ✍ ✛ ✄ ✌ ✌ ✌ ☞ ✂ ✁ ✆ ✆ ✆ ✆ T ypes, � ☞ ☞ ☞ L ogics, ✂ U seful

Safe Ambients, first try What if one needed permission to enter and exit ambients? Inferred type: co-in Troy Ulysses[] in horse out horse Troy[] horse[] co-in horse ROOT horse[] in Troy Ulysses[] Hmm. This seems to work. Or does it? U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.11/29

Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.12/29 ☛ ☛ ☛ The first try did not work If the horse can use the “ co - in Troy ”, then so can he. ✚ ✚ ✡ ✚ ✡ � ✒ ✓ ✡ ✚ ✓ ✤ ✎ ✡ ✑ ✒ ✕ ✂ ✔ ✘ ✡ ✍ ✕ ☞ ✓ ✣ ✘ ✞ ✎ ✓ � ✏ ✠ ✒ ✖ ✕ ✘ ✑ ✏ ✄ ✡ ✔ ✁ ✄ ✖ ✎ ✓ ✏ ✘ ✎ ✓ ✓ Unfortunately, Ulysses is rather clever. ✚ ✞ ✁ ✆ ✡ ✓ ✠ ✙ ✆ ✎ ✓ ✡ � ✕ ✘ ✢ ✆ ✓ ✠ ✠ ☞ ✖ ✢ ✓ ✡ � ✑ ✕ ✘ ✞ ✘ ☞ ✜ ✕ ✁ ✘ ✠ ✜ ✗ ✗ ✗ ✎ ✓ ✏ ✑ ✖ ✖ ✖ ✄ ✂ ✖ ✖ ✖ ✡ ✓ ✍ ✔ ✖ ✓ ✚ ✘ ✘ ✡ ✠ ✁ ✧ ✡ ✓ � ✒ ✔ ✆ ✞ ✢ ✤ ✒ ✓ ☞ ✘ ✓ ✓ � ✒ ✤ ☞ ✘ ✞ ✞ ✜ ✣ ☛ ✘ ☞ ✎ � ✒ ✣ ✏ ✏ ✑ ✒ ✂ ✡ ✞ ✍ ✄ ✄ ✏ ✠ ✘ ✠ ✎ ✖ ✎ � ✏ ✏ ✄ ✢ ✘ ✎ ✠ ✠ ✄ ✆ ✆ ✁ ✑ ✙ ✢ ✠ ✆ ✎ ✟ ✤ ✠ ✠ ✘ ✢ ✢ � � ☞ ✤ ✞ ✘ ✎ ✘ ✦ ☞ ✜ ✥ ✣ ✝ ✝ ✝ ✝ R ewriting, and their ✆ ✆ ✆ ✆ A utomation ☎ ✁ ✁ ✝ ✁ ☛ ✓ ✓ ✍ ✍ ✍ ✛ ✄ ✌ ✌ ✌ ☞ ✂ ✁ ✆ ✆ ✆ ✆ T ypes, � ☞ ☞ ☞ L ogics, ✂ U seful

The first try did not work Unfortunately, Ulysses is rather clever. If the horse can use the “ co - in Troy ”, then so can he. Inferred type: Ulysses[] Ulysses[] co-in Troy Troy[] co-in horse horse[] in Troy in horse out horse ROOT horse[] Ulysses[] in Troy Ulysses[] U seful L ogics, T ypes, R ewriting, and their A utomation Instant Polymorphic Type Systems for Mobile Process Calculi: Just Add Reduction Rules and Close – p.13/29