Approximating Optimal Bounds in Prompt-LTL Realizability in - PowerPoint PPT Presentation

Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time Joint work with Leander Tentrup and Martin Zimmermann Alexander Weinert Saarland University September, 16th 2016 GandALF ’16 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 1/18

Realizability: a Toy Example Setting: an arbiter with 4 clients Requests r i from client i (controlled by the environment) Grants g i for client i (controlled by the system) r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Goal: Formal specification of arbiter’s behavior Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

Realizability: a Toy Example Setting: an arbiter with 4 clients Requests r i from client i (controlled by the environment) Grants g i for client i (controlled by the system) r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Goal: Formal specification of arbiter’s behavior Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

Realizability: a Toy Example Setting: an arbiter with 4 clients Requests r i from client i (controlled by the environment) Grants g i for client i (controlled by the system) r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Goal: Formal specification of arbiter’s behavior Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

Realizability: a Toy Example Setting: an arbiter with 4 clients Requests r i from client i (controlled by the environment) Grants g i for client i (controlled by the system) r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Goal: Formal specification of arbiter’s behavior Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

Linear Temporal Logic ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ + typical shorthands where p ranges over a finite set P of atomic propositions. Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 3/18

Linear Temporal Logic ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ + typical shorthands where p ranges over a finite set P of atomic propositions. Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 3/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: . . . Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: . . . Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: Sys: . . . Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: r 1 Sys: . . . Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: r 1 Sys: . . . g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: r 1 r 1 Sys: . . . g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: r 1 r 1 Sys: − . . . g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − r 1 r 1 Sys: − . . . g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − r 1 r 1 Sys: − . . . g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − r 1 r 1 r 1 Sys: − . . . g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − r 1 r 1 r 1 Sys: − − . . . g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − r 1 r 1 r 1 Sys: − − − . . . g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − r 1 r 1 r 1 Sys: − − − . . . g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − r 1 r 1 r 1 r 1 Sys: − − − . . . g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − r 1 r 1 r 1 r 1 Sys: − − − − . . . g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − − − r 1 r 1 r 1 r 1 Sys: − − − − − − . . . g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − − − − r 1 r 1 r 1 r 1 Sys: − − − − − − . . . g 1 g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − − − − r 1 r 1 r 1 r 1 r 1 Sys: − − − − − − . . . g 1 g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Continuing the Example: Specification r 1 r 3 Client 1 Client 3 g 1 g 3 Arbiter g 2 g 4 Client 2 Client 4 r 2 r 4 Specification: � 4 i =1 G ( r i → F g i ) Admissible execution: Env: − − − − − − r 1 r 1 r 1 r 1 r 1 Sys: − − − − − − . . . g 1 g 1 g 1 g 1 Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

Prompt-LTL Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator F P : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | F P ϕ Semantics: Given some word α , k ∈ N ( α, k ) | = F P ϕ if, and only if, ϕ holds true within at most k steps Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

Prompt-LTL Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator F P : ϕ ::= p | ¬ p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | F P ϕ Semantics: Given some word α , k ∈ N ( α, k ) | = F P ϕ if, and only if, ϕ holds true within at most k steps Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18