Robust, Expressive, and Quantitative Linear Temporal Logics: Pick - - PowerPoint PPT Presentation

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Robust, Expressive, and Quantitative Linear Temporal Logics: Pick any Two for Free Joint work with Daniel Neider and Alexander Weinert Martin Zimmermann University of Liverpool September 3rd, 2019 GandALF 2019, Bordeaux, France Martin

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SLIDE 1

Robust, Expressive, and Quantitative Linear Temporal Logics: Pick any Two for Free

Joint work with Daniel Neider and Alexander Weinert

Martin Zimmermann

University of Liverpool

September 3rd, 2019

GandALF 2019, Bordeaux, France

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 1/12

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SLIDE 2

Linear Temporal Logic (LTL)

The most prominent and most important specification language for reactive systems.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 2/12

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SLIDE 3

Linear Temporal Logic (LTL)

The most prominent and most important specification language for reactive systems. Examples (q → p): every request is responded to eventually. a → g: if assumption holds always, then guarantee holds always.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 2/12

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SLIDE 4

Linear Temporal Logic (LTL)

The most prominent and most important specification language for reactive systems. Exponential Compilation Property (ECP): every LTL formula can be translated into a Büchi automaton of exponential size. ECP yields model checking in PSpace and synthesis in 2ExpTime.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 2/12

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SLIDE 5

Linear Temporal Logic (LTL)

The most prominent and most important specification language for reactive systems. Exponential Compilation Property (ECP): every LTL formula can be translated into a Büchi automaton of exponential size. ECP yields model checking in PSpace and synthesis in 2ExpTime. Shortcomings Inability to express timing constraints Limited expressiveness (weaker than Büchi automata) Inability to capture robustness

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 2/12

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SLIDE 6

Linear Temporal Logic (LTL)

The most prominent and most important specification language for reactive systems. Exponential Compilation Property (ECP): every LTL formula can be translated into a Büchi automaton of exponential size. ECP yields model checking in PSpace and synthesis in 2ExpTime. Shortcomings Inability to express timing constraints Limited expressiveness (weaker than Büchi automata) Inability to capture robustness All three shortcomings have been addressed before..

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 2/12

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SLIDE 7

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 3/12

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SLIDE 8

Prompt-LTL

Kupferman, Piterman, Vardi (’09): Add timing constraints to LTL Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ U ϕ | ϕ R ϕ |

p ϕ

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 4/12

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SLIDE 9

Prompt-LTL

Kupferman, Piterman, Vardi (’09): Add timing constraints to LTL Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ U ϕ | ϕ R ϕ |

p ϕ

Semantics via evaluation function V p mapping a trace w, a bound k, and a formula ϕ to a truth value in {0,1}. V p(w, k,

p ϕ) = 1 iff

w :

1 2 3 k − 1 · · · k

ϕ

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 4/12

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SLIDE 10

Prompt-LTL

Kupferman, Piterman, Vardi (’09): Add timing constraints to LTL Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ U ϕ | ϕ R ϕ |

p ϕ

Example (q →

p p): every request is responded to within k steps.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 4/12

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SLIDE 11

Linear Dynamic Logic

Vardi (’11): Add guards to and to restrict their scope Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | r ϕ | [r ] ϕ r ::= φ | ϕ? | r + r | r ; r | r ∗ where φ ranges over boolean formulas over the atomic propositions.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 5/12

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SLIDE 12

Linear Dynamic Logic

Vardi (’11): Add guards to and to restrict their scope Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | r ϕ | [r ] ϕ r ::= φ | ϕ? | r + r | r ; r | r ∗ where φ ranges over boolean formulas over the atomic propositions. Semantics V d(w, r ϕ) = 1 iff

w : ϕ

∈ L(r) ∈ L(r)

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 5/12

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SLIDE 13

Linear Dynamic Logic

Vardi (’11): Add guards to and to restrict their scope Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | r ϕ | [r ] ϕ r ::= φ | ϕ? | r + r | r ; r | r ∗ where φ ranges over boolean formulas over the atomic propositions. Semantics V d(w, [r ] ϕ) = 1 iff

w : ϕ ϕ

∈ L(r) ∈ L(r)

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 5/12

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SLIDE 14

Linear Dynamic Logic

Vardi (’11): Add guards to and to restrict their scope Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | r ϕ | [r ] ϕ r ::= φ | ϕ? | r + r | r ; r | r ∗ where φ ranges over boolean formulas over the atomic propositions. Example [r ] p with r = (tt ; tt)∗: p holds at every even position.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 5/12

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SLIDE 15

Robust LTL

Tabuada and Neider (’16): Capture robustness in LTL semantics Consider the five (canonical) ways a can be satisfied/violated:

  • 1. a holds always (

a)

  • 2. a holds almost always (

a)

  • 3. a holds infinitely often (

a)

  • 4. a holds at least once (

a)

  • 5. a holds never (

¬a)

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 6/12

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SLIDE 16

Robust LTL

Tabuada and Neider (’16): Capture robustness in LTL semantics Consider the five (canonical) ways a can be satisfied/violated:

  • 1. a holds always (

a)

  • 2. a holds almost always (

a)

  • 3. a holds infinitely often (

a)

  • 4. a holds at least once (

a)

  • 5. a holds never (

¬a) Note that 1. ⇒ 2. ⇒ 3. ⇒ 4.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 6/12

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SLIDE 17

Robust LTL

Tabuada and Neider (’16): Capture robustness in LTL semantics Consider the five (canonical) ways a can be satisfied/violated:

  • 1. a holds always (

a) 1111

  • 2. a holds almost always (

a) 0111

  • 3. a holds infinitely often (

a) 0011

  • 4. a holds at least once (

a) 0001

  • 5. a holds never (

¬a) 0000 Note that 1. ⇒ 2. ⇒ 3. ⇒ 4.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 6/12

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SLIDE 18

Robust LTL

Tabuada and Neider (’16): Capture robustness in LTL semantics Consider the five (canonical) ways a can be satisfied/violated:

  • 1. a holds always (

a) 1111

  • 2. a holds almost always (

a) 0111

  • 3. a holds infinitely often (

a) 0011

  • 4. a holds at least once (

a) 0001

  • 5. a holds never (

¬a) 0000 Note that 1. ⇒ 2. ⇒ 3. ⇒ 4. Basis of five-valued robust semantics for LTL.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 6/12

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SLIDE 19

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000}

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 20

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 21

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4 Negation based on 1111 representing satisfaction and all other truth values representing shades of violation

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 22

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4 Negation based on 1111 representing satisfaction and all other truth values representing shades of violation Implication “ψ → ϕ” satisfied (1111) if truth value of consequence ϕ not smaller than truth value of antecedent ψ (otherwise truth value of consequence)

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 23

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4 Negation based on 1111 representing satisfaction and all other truth values representing shades of violation Implication “ψ → ϕ” satisfied (1111) if truth value of consequence ϕ not smaller than truth value of antecedent ψ (otherwise truth value of consequence) Eventually classical

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 24

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4 Negation based on 1111 representing satisfaction and all other truth values representing shades of violation Implication “ψ → ϕ” satisfied (1111) if truth value of consequence ϕ not smaller than truth value of antecedent ψ (otherwise truth value of consequence) Eventually classical Always based on intuition from last slide

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 25

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Truth value for atomic propositions always in {1111, 0000} Conjunction and disjunction via minimization and maximization over B4 Negation based on 1111 representing satisfaction and all other truth values representing shades of violation Implication “ψ → ϕ” satisfied (1111) if truth value of consequence ϕ not smaller than truth value of antecedent ψ (otherwise truth value of consequence) Eventually classical Always based on intuition from last slide Until and release ignored for simplicity

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 26

Robust Semantics

Truth values B4 = {1111 > 0111 > 0011 > 0001 > 0000} Example a → g: the level of satisfaction of the guarantee is at least as large as the level of satisfaction of the assumption.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 7/12

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SLIDE 27

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL All three extensions have the ECP..

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 8/12

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SLIDE 28

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL All three extensions have the ECP.. hence model checking is still in PSpace and synthesis is still in 2ExpTime!

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 8/12

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SLIDE 29

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL What about combinations of the extensions?

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 8/12

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SLIDE 30

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL Prompt-LDL Faymonville and Z. (’14): the combination of Prompt-LTL and LDL has the ECP, i.e., model checking is still in PSpace and synthesis is still in 2ExpTime!

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 8/12

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SLIDE 31

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL Prompt-LDL Here: investigate the remaining combinations

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 8/12

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SLIDE 32

rPrompt-LTL

Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ |

p ϕ

Semantics Via evaluation function V rp (defined as expected).

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 9/12

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SLIDE 33

rPrompt-LTL

Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ |

p ϕ

Example V rp(w, k,

p s) = b1b2b3b4

b1 = 1: distance between synchronizations is bounded by k, b2 = 1: from some point onwards, the distance between synchronizations is bounded by k, b3 = 1: there are infinitely many synchronizations, and b4 = 1: there is at least one synchronization.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 9/12

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SLIDE 34

rPrompt-LTL

Syntax ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ | ϕ |

p ϕ

Theorem

For every rPrompt-LTL formula ϕ and every truth value β ∈ B4, there is a Prompt-LTL formula ϕβ of size O(|ϕ|) such that V rp(w, k, ϕ) ≥ β if and only if V p(w, k, ϕβ) = 1. Hence, rPrompt-LTL has the ECP, i.e., model checking is still in PSpace and synthesis is still in 2ExpTime!

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 9/12

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SLIDE 35

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL rPrompt-LTL Prompt-LDL

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 10/12

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SLIDE 36

rLDL

Syntax Add dots to LDL operators. Semantics V rd(w, [·r·] a) in case r has infinitely many matches in w:

  • 1. a holds at every match

1111

  • 2. a holds at almost all matches

0111

  • 3. a holds at infinitely many matches

0011

  • 4. a holds at some match

0001

  • 5. a holds at no match

0000 Additionally: rules for case of finitely many matches.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 11/12

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SLIDE 37

rLDL

Syntax Add dots to LDL operators. Example [·r·] q → [·tt ; r·] p) with r = (tt ; tt)∗: the level of satisfaction of p at odd positions is at least as large as the level of satisfaction of q at even positions.

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 11/12

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SLIDE 38

rLDL

Syntax Add dots to LDL operators.

Theorem

Let ϕ be an rLDL formula, n = |ϕ|, and β ∈ B4. There is a non-deterministic Büchi automaton with 2O(n log n) states recognizing the language {w ∈ (2P)ω | V rd(w, ϕ) ≥ β}. Hence, rLDL has the ECP, i.e., model checking is still in PSpace and synthesis is still in 2ExpTime!

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 11/12

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SLIDE 39

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL rPrompt-LTL rLDL Prompt-LDL All these logics have the ECP, i.e., model checking is still in PSpace and synthesis is still in 2ExpTime!

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 12/12

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SLIDE 40

The Big Picture

LTL rLTL( , ) Prompt-LTL LDL rPrompt-LTL rLDL Prompt-LDL rPrompt-LDL Open problem what about the combination of all three extensions?

Martin Zimmermann University of Liverpool Robust, Expressive, and Quantitative Temporal Logics 12/12