Limit Your Consumption! Finding Bounds in Average-energy Games - - PowerPoint PPT Presentation

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Limit Your Consumption! Finding Bounds in Average-energy Games - - PowerPoint PPT Presentation

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Limit Your Consumption! Finding Bounds in Average-energy Games Joint work with Kim G. Larsen and Simon Laursen (Aalborg University) Martin Zimmermann Saarland University April, 3nd 2016 QAPL 16 Martin Zimmermann Saarland University Finding

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SLIDE 1

Limit Your Consumption! Finding Bounds in Average-energy Games

Joint work with Kim G. Larsen and Simon Laursen (Aalborg University)

Martin Zimmermann

Saarland University

April, 3nd 2016

QAPL 16

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 1/14

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SLIDE 2

Motivation

Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

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SLIDE 3

Motivation

Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive Successful approach to verification and synthesis: an infinite game between the system and its environment: two players infinite duration perfect information system player wins if specification is satisfied

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

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SLIDE 4

Motivation

Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive Successful approach to verification and synthesis: an infinite game between the system and its environment: two players infinite duration perfect information system player wins if specification is satisfied Here: graph-based games with quantitative winning conditions modeling consumption of a ressource

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

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SLIDE 5

An Example

v2 v1 v0

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 6

An Example

v2 v1 v0 A play: v0

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 7

An Example

v2 v1 v0 A play: v0 v2

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 8

An Example

v2 v1 v0 A play: v0 v2 v1

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 9

An Example

v2 v1 v0 A play: v0 v2 v1 v0

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 10

An Example

v2 v1 v0 A play: v0 v2 v1 v0 v2

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 11

An Example

v2 v1 v0 A play: v0 v2 v1 v0 v2 v0

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 12

An Example

v2 v1 v0 A play: v0 v2 v1 v0 v2 v0 v1

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 13

An Example

v2 v1 v0 A play: v0 v2 v1 v0 v2 v0 v1 · · ·

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 14

An Example

v2 v1 v0

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2

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3

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 15

An Example

v2 v1 v0

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2

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3 A play (with energy levels): (v0, 0)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 16

An Example

v2 v1 v0

  • 3

2

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  • 1

3 A play (with energy levels): (v0, 0) (v2, 3)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 17

An Example

v2 v1 v0

  • 3

2

  • 1
  • 1

3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 18

An Example

v2 v1 v0

  • 3

2

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3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0) (v0, 0)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 19

An Example

v2 v1 v0

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2

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3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0) (v0, 0) (v2, 3)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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An Example

v2 v1 v0

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2

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  • 1

3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0) (v0, 0) (v2, 3) (v0, 5)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 21

An Example

v2 v1 v0

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2

  • 1
  • 1

3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0) (v0, 0) (v2, 3) (v0, 5) (v1, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 22

An Example

v2 v1 v0

  • 3

2

  • 1
  • 1

3 A play (with energy levels): (v0, 0) (v2, 3) (v1, 0) (v0, 0) (v2, 3) (v0, 5) (v1, 4) · · ·

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 23

An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 24

An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0) (v2, 3)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 25

An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 26

An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0) (v1, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 27

An Example

v2 v1 v0

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2

  • 1
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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0) (v1, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 28

An Example

v2 v1 v0

  • 3

2

  • 1
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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0) (v1, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 29

An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0) (v1, 4) Energy level always between 0 and 5

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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An Example

v2 v1 v0

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2

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3 A strategy: (v0, 0) (v2, 3) (v0, 5) (v1, 0) (v1, 4) Energy level always between 0 and 5 Average energy level at most 4

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

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SLIDE 31

Previous Work

  • bjective

Complexity Memory Requirements EGL NP ∩ co-NP memoryless EGLU ExpTime-complete pseudopolynomial

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

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Previous Work

  • bjective

Complexity Memory Requirements EGL NP ∩ co-NP memoryless EGLU ExpTime-complete pseudopolynomial AE NP ∩ co-NP memoryless AELU ExpTime-complete pseudopolynomial

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

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SLIDE 33

Previous Work

  • bjective

Complexity Memory Requirements EGL NP ∩ co-NP memoryless EGLU ExpTime-complete pseudopolynomial AE NP ∩ co-NP memoryless AELU ExpTime-complete pseudopolynomial AELU (U − L poly) NP ∩ co-NP polynomial

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

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SLIDE 34

Previous Work

  • bjective

Complexity Memory Requirements EGL NP ∩ co-NP memoryless EGLU ExpTime-complete pseudopolynomial AE NP ∩ co-NP memoryless AELU ExpTime-complete pseudopolynomial AELU (U − L poly) NP ∩ co-NP polynomial AEL ExpTime-hard ≥ pseudopolynomial

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

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SLIDE 35

Previous Work

  • bjective

Complexity Memory Requirements EGL NP ∩ co-NP memoryless EGLU ExpTime-complete pseudopolynomial AE NP ∩ co-NP memoryless AELU ExpTime-complete pseudopolynomial AELU (U − L poly) NP ∩ co-NP polynomial AEL ExpTime-hard ≥ pseudopolynomial W.l.o.g.: fix lower bound 0 In all problems, lower and upper bounds part of the input. Here: upper bound existentially quantified.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

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Objectives

Capacity cap ∈ N, threshold t ∈ N EGL = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn)} EGLU(cap) = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn) ≤ cap}

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

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Objectives

Capacity cap ∈ N, threshold t ∈ N EGL = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn)} EGLU(cap) = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn) ≤ cap} AE(t) = {v0v1 · · · | lim sup

n→∞ 1 n

n−1

i=0 EL(v0 · · · vi) ≤ t}

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

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SLIDE 38

Objectives

Capacity cap ∈ N, threshold t ∈ N EGL = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn)} EGLU(cap) = {v0v1 · · · | ∀n. 0 ≤ EL(v0 · · · vn) ≤ cap} AE(t) = {v0v1 · · · | lim sup

n→∞ 1 n

n−1

i=0 EL(v0 · · · vi) ≤ t}

AEL(t) = EGL ∩ AE(t) AELU(cap, t) = EGLU(cap) ∩ AE(t)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

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SLIDE 39

Finding Bounds in Average-energy Games

Input: Weighted arena A Question: Exists a threshold t ∈ N s.t. Player 0 wins (A, AEL(t))?

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 6/14

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SLIDE 40

Finding Bounds in Average-energy Games

Input: Weighted arena A Question: Exists a threshold t ∈ N s.t. Player 0 wins (A, AEL(t))? We show this to be equivalent to.. Input: Weighted arena A Question: Exists a capacity cap ∈ N s.t. Player 0 wins (A, EGLU(cap))? .. which is in 2ExpTime [Juhl, Larsen, Raskin ’13].

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 6/14

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Finding Bounds in Average-energy Games

Input: Weighted arena A Question: Exists a threshold t ∈ N s.t. Player 0 wins (A, AEL(t))? We show this to be equivalent to.. Input: Weighted arena A Question: Exists a capacity cap ∈ N s.t. Player 0 wins (A, EGLU(cap))? .. which is in 2ExpTime [Juhl, Larsen, Raskin ’13]. Note: The direction ∃cap ⇒ ∃t is trivial.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 6/14

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SLIDE 42

∃t ⇒ ∃cap

Obstacle: average can be bounded while energy level is unbounded EL

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 7/14

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∃t ⇒ ∃cap

But: every time energy level increases above threshold t on average, it drops below t later Crossings are characterized by vertex v and energy level in range t + 1, . . . , t + W For every such combination play like in situation with smallest maximal energy level before next drop below t t EL

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 7/14

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∃t ⇒ ∃cap

But: every time energy level increases above threshold t on average, it drops below t later Crossings are characterized by vertex v and energy level in range t + 1, . . . , t + W For every such combination play like in situation with smallest maximal energy level before next drop below t t EL This strategy bounds the energy level by some cap.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 7/14

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SLIDE 45

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

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  • 1

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

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A play with cap = 5: (v0, 5)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 47

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 48

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 49

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1) (v0, 5)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 50

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1) (v0, 5) (v2, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 51

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1) (v0, 5) (v2, 4) (v0, 4)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 52

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1) (v0, 5) (v2, 4) (v0, 4) (v1, 3)

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 53

Recharge Games

Previsouly: positive and negative weights Now: only negative weights and recharge edges that recharge to a fixed capacity cap. v2 v1 v0

  • 3

R

  • 1
  • 1
  • 1

A play with cap = 5: (v0, 5) (v2, 4) (v1, 1) (v0, 5) (v2, 4) (v0, 4) (v1, 3) · · ·

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 8/14

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SLIDE 54

Objectives

RE(cap) = {v0v1 · · · | ∀n. ELcap(v0 · · · vn) ≥ 0} AR(cap, t) = RE(cap) ∩ {v0v1 · · · | lim sup

n→∞ 1 n

n−1

i=0 ELcap(v0 · · · vi) ≤ t}

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 9/14

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SLIDE 55

Objectives

RE(cap) = {v0v1 · · · | ∀n. ELcap(v0 · · · vn) ≥ 0} AR(cap, t) = RE(cap) ∩ {v0v1 · · · | lim sup

n→∞ 1 n

n−1

i=0 ELcap(v0 · · · vi) ≤ t}

Theorem

The problem Input: Weighted arena A, cap ∈ N, and t ∈ N. Question: Does Player 0 win (A, AR(cap, t))? is ExpTime-complete.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 9/14

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SLIDE 56

Objectives

RE(cap) = {v0v1 · · · | ∀n. ELcap(v0 · · · vn) ≥ 0} AR(cap, t) = RE(cap) ∩ {v0v1 · · · | lim sup

n→∞ 1 n

n−1

i=0 ELcap(v0 · · · vi) ≤ t}

Theorem

The problem Input: Weighted arena A, cap ∈ N, and t ∈ N. Question: Does Player 0 win (A, AR(cap, t))? is ExpTime-complete. Proof: Upper bound: Reduction to mean-payoff games. Lower bound: Reduction from countdown games.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 9/14

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SLIDE 57

Proof Sketch

A countdown game. Objective: reach v⊥ with energy-level −cap for some given cap ∈ N. v⊥

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  • 2

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 10/14

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SLIDE 58

Proof Sketch

A countdown game. Objective: reach v⊥ with energy-level −cap for some given cap ∈ N. v⊥

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Theorem (Jurdzi´ nski, Sproston, Laroussini ’08)

Solving countdown games is ExpTime-complete.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 10/14

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SLIDE 59

Proof Sketch

A countdown game. Objective: reach v⊥ with energy-level −cap for some given cap ∈ N. v⊥

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Theorem (Jurdzi´ nski, Sproston, Laroussini ’08)

Solving countdown games is ExpTime-complete. Turn countdown game into average bounded recharge game: capacity cap and threshold 0.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 10/14

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SLIDE 60

Who is to Blame?

Theorem

Solving average-bounded recharge games with existentially quantified capacity and a given threshold is ExpTime-hard. −c −1 G

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 11/14

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SLIDE 61

Who is to Blame?

Theorem

Solving average-bounded recharge games with existentially quantified capacity and a given threshold is ExpTime-hard. −c −1 G

Theorem

The problem Input: Weighted arena A Question: Exists a capacity cap s.t. Player 0 wins (A, RE(cap))? is in PTime.

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 11/14

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SLIDE 62

Tradeoffs: Capacity vs. Average

Available loops depend on capacity Tradeoff not monotonic Cause of tradeoff: recharge to cap at recharge-edges

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 12/14

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SLIDE 63

Tradeoffs: Average vs. Memory

With n memory states, use self-loop n − 1 times Then, recharge to level cap

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 13/14

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SLIDE 64

Future Work

We started the investigation of average-energy and recharge games with existentially quantified bounds. Many problems remain open: Show that games with winning condition AEL are decidable..

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 14/14

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SLIDE 65

Future Work

We started the investigation of average-energy and recharge games with existentially quantified bounds. Many problems remain open: Show that games with winning condition AEL are decidable.. maybe by a refinement of our technique for lower-bounded average-energy games with existentially quantified threshold

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 14/14

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SLIDE 66

Future Work

We started the investigation of average-energy and recharge games with existentially quantified bounds. Many problems remain open: Show that games with winning condition AEL are decidable.. maybe by a refinement of our technique for lower-bounded average-energy games with existentially quantified threshold Give a lower bound on solving lower-bounded average-energy games with existentially quantified threshold

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 14/14

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SLIDE 67

Future Work

We started the investigation of average-energy and recharge games with existentially quantified bounds. Many problems remain open: Show that games with winning condition AEL are decidable.. maybe by a refinement of our technique for lower-bounded average-energy games with existentially quantified threshold Give a lower bound on solving lower-bounded average-energy games with existentially quantified threshold Study tradeoffs, in particular upper bounds

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 14/14

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SLIDE 68

Future Work

We started the investigation of average-energy and recharge games with existentially quantified bounds. Many problems remain open: Show that games with winning condition AEL are decidable.. maybe by a refinement of our technique for lower-bounded average-energy games with existentially quantified threshold Give a lower bound on solving lower-bounded average-energy games with existentially quantified threshold Study tradeoffs, in particular upper bounds Multi-dimensional games

Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 14/14