Optimal Reachability in Divergent Weighted Timed Games

Optimal Reachability in Divergent Weighted Timed Games

Benjamin Monmege

Aix-Marseille Université, LIF, CNRS, France with Damien Busatto-Gaston and Pierre-Alain Reynier

May 19, 2017

published at FoSSaCS 2017

Motivation: quantitative aspects of real-time synthesis Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller

Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller

Two-player timed game

Among all valid controller, choose a good/cheap/efficient one

Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis

Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller

Two-player timed game

Among all valid controller, choose a good/cheap/efficient one

Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis

Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one

Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis

Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Motivation: quantitative aspects of real-time synthesis

Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

= ⇒ model, e.g., production/consumption of resources

Weighted timed games

s1

s2 s3

s4 s5

s6

x>0 x:= 0

x61

x62

x<1 x:= 0

x>1 x>1

x:= 0

x>1 x:= 0

x>1

Timed automaton with state partition between 2 players + reachability objective

(s1,0)−−−−→(s^0.4,& 4,0.4)−−−−→(s^0.6,→ 5,0)−−−−→(s^1.5,← 4,0)−−−−→(s^1.1,→ 5,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)−−−−→(s^0.6,→ ₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)

0.4,&

−−−−→(s₄,0.4)−−−−→(s^0.6,→ ₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)

0.6,→

−−−−→(s₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)−−−−→(s^0.6,→ ₅,0)

1.5,←

−−−−→(s₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)−−−−→(s^0.6,→ ₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2)

1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)−−−−→(s^0.6,→ ₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8

(s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · · 1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

total weight until otherwise

Weighted timed games

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions

(s1,0)−−−−→(s^0.4,& ₄,0.4)−−−−→(s^0.6,→ ₅,0)−−−−→(s^1.5,← ₄,0)−−−−→(s^1.1,→ ₅,0)−^2,%−−→(,2) 1×0.4+1 −3×0.6+0 +1×1.5+0 −3×1.1+0 +1×2+2 = 1.8 (s1,0)−−−−→(s^0.2,% 2,0)−−−−→(s^0.9,→ 3,0.9)−^0.2,−−−→(s3,0)−^0.9,−−−→(s3,0) · · ·

1×0.2+0 +2×0.9+0 −1×0.2+0 −1×0.9+0 · · · = +∞

Weight of an execution :

(+∞ ifnot reached

Strategies and objectives

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Strategy for a player: map finite executions to a delay and a transition

Objective of player#: reachandminimise the weight

Objective of player2: avoidor, if not possible, maximise the weight Main object of interest:

Val(`, ν) = inf

σ_Min∈Strat^Min sup

σ_Max∈Strat^Max

Weight(Exec(`, ν,σMin,σMax))∈R What weight can players guarantee? Find (almost) optimal strategy?

Strategies and objectives

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Strategy for a player: map finite executions to a delay and a transition Objective of player#: reachandminimise the weight

Objective of player2: avoidor, if not possible, maximise the weight

Main object of interest: Val(`, ν) = inf

σ_Min∈Strat^Min sup

σ_Max∈Strat^Max

Weight(Exec(`, ν,σMin,σMax))∈R What weight can players guarantee? Find (almost) optimal strategy?

Strategies and objectives

1 s1

2 s2

−1 s3

−3 s4

1 s5

s6

x>0 x:= 0 0 x61 1

x62 0

x<1 x:= 0 0

x>1 x>1 1

x:= 0 0 x>1 x:= 0 0

x>1 2

Strategy for a player: map finite executions to a delay and a transition Objective of player#: reachandminimise the weight

Objective of player2: avoidor, if not possible, maximise the weight Main object of interest:

Val(`, ν) = inf

σ_Min∈Strat^Min

sup

σ_Max∈Strat^Max

Weight(Exec(`, ν,σMin,σMax))∈R

The real motivation!

Environment k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Controller?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Customer?? | = Spec

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Customer?? | = Beer!

Two-player game

Real-time requirements/environment = ⇒ real-time controller Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Customer?? | = Beer!

Two-player game

“I want a beer, and I want it fast! But barman needs time...”

Two-player timed game

Among all valid controller, choose a good/cheap/efficient one Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Customer?? | = Beer!

Two-player game

“I want a beer, and I want it fast! But barman needs time...”

Two-player timed game

“I want a good beer, but I want it cheap!”

Two-player weighted timed game

Main motivation since my postdoc here: negative weights

The real motivation!

Bar k Customer?? | = Beer!

Two-player game

“I want a beer, and I want it fast! But barman needs time...”

Two-player timed game

“I want a good beer, but I want it cheap!”

Two-player weighted timed game

Main motivation since my postdoc here: drink tasty beers!

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly. time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp. algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013)

I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly. time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp. algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly. time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp. algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly. time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp. algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly.

time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp. algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly.

time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp.

algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004)

F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly.

time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp.

algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

State of the art

Value problem: decide whether Val(`, ν)6K?

I 1 player (weighted timed automaton): PSPACE-complete

I Algorithm based on regions(Bouyer, Brihaye, Bruyère, and Raskin 2007);

I PSPACE-hard with at least 2 clocks(Fearnley and Jurdziński 2013) I 2 players: undecidableeven with non-negative weights and 3 clocks

(Brihaye, Bruyère, and Raskin 2005; Bouyer, Brihaye, and Markey 2006)

I 1 clockandnon-negative weights: exp. algo. (Bouyer, Larsen, Markey, and Rasmussen 2006)

I 1 clockand62 location-weights from{−d,0,+d}: pseudo-poly.

time algo. (Brihaye, Geeraerts, Narayanan Krishna, Manasa, Monmege, and Trivedi 2014)(corner-point abstraction)

I non-negative weights and strictly non-Zeno-cost cycles: 2-exp.

algo. (Bouyer, Cassez, Fleury, and Larsen 2004; Alur, Bernadsky, and Madhusudan 2004) F(x)_(`,ν)=

(_sup

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x(`⁰,ν⁰) if`∈LMax

inf

(`,ν)−−→<sup>d,t</sup> (`⁰,ν⁰)

d×Weight(`) + Weight(t) +x<sub>(`</sub>0,ν⁰) if`∈LMin

Missing: decidable fragment of games without constraints on number of clocks, and with negative weights

A special case first: Weighted (untimed) games

v1

v2 v3

v4 v5

v6

0

1

0

0

1

0 1

2

Weighted oriented graph with vertices partition between 2 players + reachability objective

A special case first: Weighted (untimed) games

v1

v2 v3

v4 v5

v6

0

1

0

0

1

0 1

2

Weighted oriented graph with vertices partition between 2 players + reachability objective

v1

&

−→v4

−→v→ 5

−→v← 4

−→v→ 5

−→%

1 +1 +2 = 4

v1

−→v% 2

−→v→ 3−→v3−→v3 · · ·

· · · = +∞(not reached)

Weight of a path:

(+∞ if not reached

total weight until otherwise

Strategies and objectives

v1

v2 v3

v4 v5

v6

0

1

0

0

1

0 1

2

Player#’s objective: reachandminimise the cost

Player2’s objective: avoidor, if not possible, maximise the weight Strategy for a player: map finite paths to edges

Main object of interest:

Val(v) = inf

σ_Min∈Strat^Min

sup

σ_Max∈Strat^Max

Weight(Exec(v,σMin,σMax))∈Z What weight can players guarantee? Find optimal strategy?

State of the art

Value computation

I non-negative weights: polynomial algo. (Khachiyan, Boros, Borys, Elbassioni, Gurvich, Rudolf, et al. 2008)

"Dijkstra on 2 players games"

I with negative weights : pseudo-poly algo. (Brihaye, Geeraerts, Haddad, and Monmege 2016)

"value iteration algorithm"

F(x)v =





 min

e=(v,a,v⁰)∈EWeight(e) +x_v⁰ ifv ∈V_Min max

e=(v,a,v⁰)∈EWeight(e) +xv⁰ ifv ∈VMax.

I Value−∞: detection is as hard as solving parity games (NP∩co-NP)

State of the art

Value computation

I non-negative weights: polynomial algo. (Khachiyan, Boros, Borys, Elbassioni, Gurvich, Rudolf, et al. 2008)

"Dijkstra on 2 players games"

I with negative weights : pseudo-poly algo. (Brihaye, Geeraerts, Haddad, and Monmege 2016)

"value iteration algorithm"

F(x)v =





 min

e=(v,a,v⁰)∈EWeight(e) +x_v⁰ ifv ∈V_Min max

e=(v,a,v⁰)∈EWeight(e) +xv⁰ ifv ∈VMax.

I Value−∞: detection is as hard as solving parity games (NP∩co-NP)

State of the art

Value computation

I non-negative weights: polynomial algo. (Khachiyan, Boros, Borys, Elbassioni, Gurvich, Rudolf, et al. 2008)

"Dijkstra on 2 players games"

I with negative weights : pseudo-poly algo. (Brihaye, Geeraerts, Haddad, and Monmege 2016)

"value iteration algorithm"

F(x)v =





 min

e=(v,a,v⁰)∈EWeight(e) +x_v⁰ ifv ∈V_Min max

e=(v,a,v⁰)∈EWeight(e) +xv⁰ ifv ∈VMax.

I Value−∞: detection is as hard as solving parity games (NP∩co-NP)

Contribution 1: divergent weighted games

Property of the underlying graph:

Divergence

Every cycle have total weight either6−1 or>1, i.e., no cycles of weight = 0.

Contribution 1: divergent weighted games

Property of the underlying graph:

Divergence

Every cycle have total weight either6−1 or>1, i.e., no cycles of weight = 0.

Theorem:

We can compute the value of a divergent weighted game in poly. time.

Contribution 1: divergent weighted games

Property of the underlying graph:

Divergence

Every cycle have total weight either6−1 or>1, i.e., no cycles of weight = 0.

Theorem:

We can compute the value of a divergent weighted game in poly. time.

Theorem:

We can decide in PTIME if a weighted game is divergent.

Divergent weighted games analysis

divergence property

characterisation:

p>1

−q6−1

Divergent weighted games analysis

divergence property

characterisation: All the simple cycles in a SCC have the same sign

p>1

−q6−1 −q6−1 α β

p>1

Divergent weighted games analysis

divergence property

characterisation: All the simple cycles in a SCC have the same sign

class decision value computation

Value computation in a divergent weighted game

I Detect and remove +∞vertices (outside of the attractor of player #toward)

I SCC decomposition

I Value computation SCC by SCC, bottom-up

Value computation in a divergent weighted game

I Detect and remove +∞vertices (outside of the attractor of player #toward)

I SCC decomposition

I Value computation SCC by SCC, bottom-up

Value computation in a divergent weighted game

I Detect and remove +∞vertices (outside of the attractor of player #toward)

I SCC decomposition

I Value computation SCC by SCC, bottom-up

Value computation in a divergent weighted game

I Detect and remove +∞vertices (outside of the attractor of player #toward)

I SCC decomposition

I Value computation SCC by SCC, bottom-up

positive SCC

I The "value iteration" algorithm converges in linear time

Positive SCC: value iteration converges in linear time

I W := max_e∈E|Weight(e)|

I No negative cycles in the SCC =⇒no vertices of value−∞

I K := maxv|xⁿ_v|<∞, withn=|V|

I Let p>(2K +W(n−1))n. Values obtained after n+psteps are identical to those obtained after nsteps only!

I Ifx^n+pv <xⁿ_v ∃v⁰, π:v →^pv⁰ of weightx^n+pv −xⁿ_v0

I |π|>(2K+W(n−1))n=⇒ ∃v⁰⁰appearing>2K+W(n−1) times

I πcan be decomposed into>2K+W(n−1) cycles (positive!) and a finite playπ⁰ visiting each vertex at most once

I Weight(π)>2K+W(n−1)−(n−1)W = 2K

I x^n+pv −xⁿ_v0>2K, sox^n+pv >2K+xⁿ_v0 >K.

I ButK>xⁿ_v, sox^n+pv >xⁿ_v: contradiction.

I Therefore, stabilisation after nsteps only

Positive SCC: value iteration converges in linear time

I W := max_e∈E|Weight(e)|

I No negative cycles in the SCC =⇒no vertices of value−∞

I K := maxv|xⁿ_v|<∞, withn=|V|

I Let p>(2K +W(n−1))n. Values obtained after n+psteps are identical to those obtained after nsteps only!

I Ifx^n+pv <xⁿ_v ∃v⁰, π:v →^pv⁰ of weightx^n+pv −xⁿ_v0

I |π|>(2K+W(n−1))n=⇒ ∃v⁰⁰appearing>2K+W(n−1) times

I πcan be decomposed into>2K+W(n−1) cycles (positive!) and a finite playπ⁰ visiting each vertex at most once

I Weight(π)>2K+W(n−1)−(n−1)W = 2K

I x^n+pv −xⁿ_v0>2K, sox^n+pv >2K+xⁿ_v0 >K.

I ButK>xⁿ_v, sox^n+pv >xⁿ_v: contradiction.

I Therefore, stabilisation after nsteps only

Positive SCC: value iteration converges in linear time

I W := max_e∈E|Weight(e)|

I No negative cycles in the SCC =⇒no vertices of value−∞

I K := maxv|xⁿ_v|<∞, withn=|V|

I Let p>(2K +W(n−1))n. Values obtained after n+psteps are identical to those obtained after nsteps only!

I Ifx^n+pv <xⁿ_v ∃v⁰, π:v →^pv⁰ of weightx^n+pv −xⁿ_v0

I |π|>(2K+W(n−1))n=⇒ ∃v⁰⁰appearing>2K+W(n−1) times

I πcan be decomposed into>2K+W(n−1) cycles (positive!) and a finite playπ⁰ visiting each vertex at most once

I Weight(π)>2K+W(n−1)−(n−1)W = 2K

I x^n+pv −xⁿ_v0>2K, sox^n+pv >2K+xⁿ_v0 >K.

I ButK>xⁿ_v, sox^n+pv >xⁿ_v: contradiction.

I Therefore, stabilisation after nsteps only