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

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approximating optimal bounds in prompt ltl realizability

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

Dec 08, 2023 276 likes •1.22k views

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

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SLIDE 1

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

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SLIDE 2

Realizability: a Toy Example

Setting: an arbiter with 4 clients Requests ri from client i (controlled by the environment) Grants gi for client i (controlled by the system) Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Goal: Formal specification of arbiter’s behavior

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

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SLIDE 3

Realizability: a Toy Example

Setting: an arbiter with 4 clients Requests ri from client i (controlled by the environment) Grants gi for client i (controlled by the system) Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Goal: Formal specification of arbiter’s behavior

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

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SLIDE 4

Realizability: a Toy Example

Setting: an arbiter with 4 clients Requests ri from client i (controlled by the environment) Grants gi for client i (controlled by the system) Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Goal: Formal specification of arbiter’s behavior

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

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SLIDE 5

Realizability: a Toy Example

Setting: an arbiter with 4 clients Requests ri from client i (controlled by the environment) Grants gi for client i (controlled by the system) Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Goal: Formal specification of arbiter’s behavior

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 2/18

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SLIDE 6

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

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SLIDE 7

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

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SLIDE 8

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 9

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 10

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: Sys: . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 11

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 Sys: . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 12

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 Sys: g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 13

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 Sys: g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 14

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 Sys: g1 − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 15

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − Sys: g1 − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 16

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − Sys: g1 − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 17

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 Sys: g1 − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 18

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 Sys: g1 − g1 − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 19

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − Sys: g1 − g1 − − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 20

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − Sys: g1 − g1 − − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 21

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 Sys: g1 − g1 − − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 22

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 Sys: g1 − g1 − − g1 − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 23

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 − − Sys: g1 − g1 − − g1 − − − . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 24

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 − − − Sys: g1 − g1 − − g1 − − − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 25

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 − − − r1 Sys: g1 − g1 − − g1 − − − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 26

Continuing the Example: Specification

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Specification: 4

i=1 G (ri → F gi)

Admissible execution: Env: r1 r1 − r1 − − r1 − − − r1 Sys: g1 − g1 − − g1 − − − g1 . . .

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 4/18

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SLIDE 27

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 28

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 29

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 30

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 31

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 32

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 33

Prompt-LTL

Problem: F ϕ does not guarantee when ϕ holds true. Solution: Add prompt-eventually operator FP : ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | F ϕ | FP ϕ Semantics: Given some word α, k ∈ N (α, k) | = FP ϕ if, and only if, ϕ holds true within at most k steps

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 5/18

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SLIDE 34

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

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SLIDE 35

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

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SLIDE 36

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

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SLIDE 37

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

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SLIDE 38

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

slide-39

SLIDE 39

Prompt-LTL Example

Before: 4

i=1 G (ri → F gi)

Now: 4

i=1 G (ri → FP gi)

Execution α: Env: Sys: r1 r1 − r1 − − r1 − − − r1 g1 − g1 − − g1 − − − g1 . . . 1 2 3 α | = 4

i=1 G (ri → F gi)

There exists no k such that (α, k) | = 4

i=1 G (ri → FP gi)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 6/18

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SLIDE 40

Prompt-LTL Realizability

Theorem (Kupferman, Piterman, Vardi ’07)

The following problem is 2Exptime-complete: Input: Prompt-LTL formula ϕ over I ∪ O Question: Does there exist a strategy σ: (2I)+ → 2O and a bound k, such that every word consistent with σ models ϕ w.r.t. k? Now: Prompt-LTL realizability as optimization problem

Theorem (Z. ’11)

The minimal k such that there exists a strategy σ: (2I)+ → 2O such that every word consistent with σ models ϕ w.r.t. k can be determined in triply-exponential time.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 7/18

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SLIDE 41

Prompt-LTL Realizability

Theorem (Kupferman, Piterman, Vardi ’07)

The following problem is 2Exptime-complete: Input: Prompt-LTL formula ϕ over I ∪ O Question: Does there exist a strategy σ: (2I)+ → 2O and a bound k, such that every word consistent with σ models ϕ w.r.t. k? Now: Prompt-LTL realizability as optimization problem

Theorem (Z. ’11)

The minimal k such that there exists a strategy σ: (2I)+ → 2O such that every word consistent with σ models ϕ w.r.t. k can be determined in triply-exponential time.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 7/18

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SLIDE 42

Prompt-LTL Realizability

Theorem (Kupferman, Piterman, Vardi ’07)

The following problem is 2Exptime-complete: Input: Prompt-LTL formula ϕ over I ∪ O Question: Does there exist a strategy σ: (2I)+ → 2O and a bound k, such that every word consistent with σ models ϕ w.r.t. k? Now: Prompt-LTL realizability as optimization problem

Theorem (Z. ’11)

The minimal k such that there exists a strategy σ: (2I)+ → 2O such that every word consistent with σ models ϕ w.r.t. k can be determined in triply-exponential time.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 7/18

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SLIDE 43

Prompt-LTL Realizability

Theorem

The minimal k as defined previously can be approximated within a factor of 2 in doubly-exponential time.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 8/18

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SLIDE 44

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . Approximation Algorithm ϕ k k = 1, . . .

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

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SLIDE 45

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . Approximation Algorithm ϕ k k = 1, . . .

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

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SLIDE 46

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . .

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

slide-47

SLIDE 47

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . .

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

slide-48

SLIDE 48

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . .

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

slide-49

SLIDE 49

Prompt-LTL Approximation

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax

Theorem (Kupferman, Piterman, Vardi ’07)

For every Prompt-LTL formula ϕ and each bound k ∈ N, there exists an LTL formula ϕk, such that if ϕk is realizable, then (ϕ, 2k) is realizable, and if (ϕ, k) is realizable, then ϕk is realizable

Theorem (Kupferman, Piterman, Vardi ’07)

If (ϕ, k) is realizable for some k ∈ N, then (ϕ, k′) is realizable for some k′ doubly exponential in |ϕ|.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 9/18

slide-50

SLIDE 50

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

slide-51

SLIDE 51

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

slide-52

SLIDE 52

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

slide-53

SLIDE 53

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

slide-54

SLIDE 54

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

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SLIDE 55

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

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SLIDE 56

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

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SLIDE 57

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

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SLIDE 58

Construction of ϕk: Alternating Color

Given: Prompt-LTL formula ϕ, bound k ∈ N. Wanted: LTL formula ϕk. Idea: Use fresh proposition r / ∈ P, “color” α. α = α0 α1 α2 α3 α4 α5 α6 α7 α8 α9 r r r r r r ¬r ¬r ¬r ¬r

1. Replace each FP ψ by LTL formula rel(FP ψ) stating

“ϕ holds within one color change”

2. Add ψk stating

“The coloring changes after at most k steps”

ϕ

rel(ϕ) ∧ ψk

(Prompt-LTL) (LTL)

Correctness due to (Kupferman, Piterman, Vardi ’07)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 10/18

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SLIDE 59

The Algorithm

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax

1: if ϕ unrealizable then 2:

return “ϕ unrealizable”

3: for k = 0, 1, 2, . . . , 22|ϕ| do 4:

if rel(ϕ) ∧ ψk realizable then

5:

return 2k Run-time: doubly-exponential in |ϕ|: Lines 1 and 4: doubly-exponential time. At most doubly-exponentially many iterations.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 11/18

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SLIDE 60

The Algorithm

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax

1: if ϕ unrealizable then 2:

return “ϕ unrealizable”

3: for k = 0, 1, 2, . . . , 22|ϕ| do 4:

if rel(ϕ) ∧ ψk realizable then

5:

return 2k Run-time: doubly-exponential in |ϕ|: Lines 1 and 4: doubly-exponential time. At most doubly-exponentially many iterations.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 11/18

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SLIDE 61

The Algorithm

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax

1: if ϕ unrealizable then 2:

return “ϕ unrealizable”

3: for k = 0, 1, 2, . . . , 22|ϕ| do 4:

if rel(ϕ) ∧ ψk realizable then

5:

return 2k Run-time: doubly-exponential in |ϕ|: Lines 1 and 4: doubly-exponential time. At most doubly-exponentially many iterations.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 11/18

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SLIDE 62

The Algorithm

LTL synthesis Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax

1: if ϕ unrealizable then 2:

return “ϕ unrealizable”

3: for k = 0, 1, 2, . . . , 22|ϕ| do 4:

if rel(ϕ) ∧ ψk realizable then

5:

return 2k Run-time: doubly-exponential in |ϕ|: Lines 1 and 4: doubly-exponential time. At most doubly-exponentially many iterations.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 11/18

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SLIDE 63

Back to the Example

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Parameters: Number of clients: r Number of prioritized clients: rp

1. Answer every request of clients 1 through rp promptly:
1≤i≤rp G (ri → FP gi)
2. Answer every other request eventually:

rp<i G (ri → F gi)

3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 12/18

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SLIDE 64

Back to the Example

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Parameters: Number of clients: r Number of prioritized clients: rp

1. Answer every request of clients 1 through rp promptly:
1≤i≤rp G (ri → FP gi)
2. Answer every other request eventually:

rp<i G (ri → F gi)

3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 12/18

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SLIDE 65

Back to the Example

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Parameters: Number of clients: r Number of prioritized clients: rp

1. Answer every request of clients 1 through rp promptly:
1≤i≤rp G (ri → FP gi)
2. Answer every other request eventually:

rp<i G (ri → F gi)

3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 12/18

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SLIDE 66

Back to the Example

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Parameters: Number of clients: r Number of prioritized clients: rp

1. Answer every request of clients 1 through rp promptly:
1≤i≤rp G (ri → FP gi)
2. Answer every other request eventually:

rp<i G (ri → F gi)

3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 12/18

slide-67

SLIDE 67

Back to the Example

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Parameters: Number of clients: r Number of prioritized clients: rp

1. Answer every request of clients 1 through rp promptly:
1≤i≤rp G (ri → FP gi)
2. Answer every other request eventually:

rp<i G (ri → F gi)

3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 12/18

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SLIDE 68

LTL synthesis vs. Prompt-LTL synthesis

Resources Prioritized Resources LTL [s] Prompt-LTL [s] 3 1 2 3 4 1 2 3 4

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 13/18

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SLIDE 69

LTL synthesis vs. Prompt-LTL synthesis

Resources Prioritized Resources LTL [s] Prompt-LTL [s] 3 0.26 1 2 3 4 0.32 1 2 3 4

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 13/18

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SLIDE 70

LTL synthesis vs. Prompt-LTL synthesis

Resources Prioritized Resources LTL [s] Prompt-LTL [s] 3 0.26 0.37 1 0.47 2 0.64 3 0.72 4 0.32 0.47 1 1.32 2 1.52 3 1.72 4 1.72

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 13/18

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SLIDE 71

Bounded Prompt-LTL Approximation

LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

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SLIDE 72

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

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SLIDE 73

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

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SLIDE 74

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

slide-75

SLIDE 75

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

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SLIDE 76

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

slide-77

SLIDE 77

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

slide-78

SLIDE 78

Bounded Prompt-LTL Approximation

Bounded LTL synthesis

(Finkbeiner, Schewe ’13)

Prompt-LTL → LTL

(Kupferman, Piterman, Vardi ’07)

ϕ k ϕk k = 1, . . . , kmax n = 1, . . . , nmax k n 1 3 4 5 6 7 8 1 2 3 Pareto Positions

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 14/18

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SLIDE 79

Strategies: Slow, but Small

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Always assume the worst: All requests in each step g1 g2 g3 g4 4 states, maximal delay 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 15/18

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SLIDE 80

Strategies: Slow, but Small

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g3 g4 4 states, maximal delay 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 15/18

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SLIDE 81

Strategies: Slow, but Small

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g3 g4 4 states, maximal delay 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 15/18

slide-82

SLIDE 82

Strategies: Slow, but Small

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g3 g4 4 states, maximal delay 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 15/18

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SLIDE 83

Strategies: Slow, but Small

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g3 g4 4 states, maximal delay 3

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 15/18

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SLIDE 84

Strategies: Fast, but Large

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g1 g3 g1 g4 6 states, maximal delay 2

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 16/18

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SLIDE 85

Strategies: Fast, but Large

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g1 g3 g1 g4 6 states, maximal delay 2

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 16/18

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SLIDE 86

Strategies: Fast, but Large

Arbiter Client2 Client1 Client4 Client3 r1 r2 r3 r4 g1 g2 g3 g4 Client1 Always assume the worst: All requests in each step g1 g2 g1 g3 g1 g4 6 states, maximal delay 2

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 16/18

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SLIDE 87

Pareto Positions

Theorem

There exists a family of Prompt-LTL formulas ϕb of size linear in b such that the output player has: a positional strategy realizing ϕb w.r.t. k = 2b, and a strategy of size n = 22b realizing ϕb w.r.t. k = 0. k n 2b − 1 2b 1 2 3 22b − 1 22b

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 17/18

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SLIDE 88

Pareto Positions

Theorem

There exists a family of Prompt-LTL formulas ϕb of size linear in b such that the output player has: a positional strategy realizing ϕb w.r.t. k = 2b, and a strategy of size n = 22b realizing ϕb w.r.t. k = 0. k n 2b − 1 2b 1 2 3 22b − 1 22b

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 17/18

slide-89

SLIDE 89

Pareto Positions

Theorem

There exists a family of Prompt-LTL formulas ϕb of size linear in b such that the output player has: a positional strategy realizing ϕb w.r.t. k = 2b, and a strategy of size n = 22b realizing ϕb w.r.t. k = 0. k n 2b − 1 2b 1 2 3 22b − 1 22b

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 17/18

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SLIDE 90

Pareto Positions

Theorem

There exists a family of Prompt-LTL formulas ϕb of size linear in b such that the output player has: a positional strategy realizing ϕb w.r.t. k = 2b, and a strategy of size n = 22b realizing ϕb w.r.t. k = 0. k n 2b − 1 2b 1 2 3 22b − 1 22b

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 17/18

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SLIDE 91

Pareto Positions

Theorem

There exists a family of Prompt-LTL formulas ϕb of size linear in b such that the output player has: a positional strategy realizing ϕb w.r.t. k = 2b, and a strategy of size n = 22b realizing ϕb w.r.t. k = 0. k n 2b − 1 2b 1 2 3 22b − 1 22b

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 17/18

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SLIDE 92

Conclusion

Our contribution: The first approximation algorithm for Prompt-LTL realizability with doubly-exponential running time Computes a realizing strategy Applicable to stronger logics as well Prototypical implementation Upper and lower bounds on tradeoff time vs. memory Take-away: Relaxing the optimality requirement for Prompt-LTL yields exponentially better runtime In general, memory can be traded for response time and vice versa.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 18/18

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SLIDE 93

Conclusion

Our contribution: The first approximation algorithm for Prompt-LTL realizability with doubly-exponential running time Computes a realizing strategy Applicable to stronger logics as well Prototypical implementation Upper and lower bounds on tradeoff time vs. memory Take-away: Relaxing the optimality requirement for Prompt-LTL yields exponentially better runtime In general, memory can be traded for response time and vice versa.

Alexander Weinert Saarland University Approximating Prompt-LTL Realizability 18/18