[PPT] - Synthesis of Optimal Strategies Using H Y T ECH Patricia Bouyer 1 , PowerPoint Presentation

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Synthesis of Optimal Strategies Using HYTECH

Patricia Bouyer1, Franck Cassez2, Emmanuel Fleury3 & Kim Guldstrand Larsen3

1 LSV, ENS-Cachan, France 2 IRCCyN, Nantes, France 3 Comp. Science. Dept., Aalborg University, Danemark

Games in Design and Verification Boston, Massachusetts, USA July 18, 2004

http://www.lsv.ens-cachan.fr/aci-cortos/ptga

Synthesis of Optimal Strategies Using HYTECH page 1/35

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Context

Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal a2 a3 x ≥ 2; a4 x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5 Timed Automata + Reachability [AD94]

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Context

Timed Game Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; c2 x ≤ 2 ; c1 y := 0 x ≥ 2 ; c2 Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98]

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Context

As soon As Possible in Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; a4 1 ≤ x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5 Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99]

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Context

Reachability in Priced Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal a2 a3 x ≥ 2; a4

cost = 1

x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5

cost = 7 cost(ℓ2) = 10 cost(ℓ0) = 5 cost(ℓ3) = 1

Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99] Priced (or Weighted) Timed Automata [LBB+01, ALTP01]

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Context

Priced Timed Game Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; c2

cost = 1

x ≤ 2 ; c1 y := 0 x ≥ 2 ; c2

cost = 7 cost(ℓ2) = 10 cost(ℓ0) = 5 cost(ℓ3) = 1

Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99] Priced (or Weighted) Timed Automata [LBB+01, ALTP01]

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

Model = Game = Controller vs. Environment What is the best cost whatever the environment does ?

Synthesis of Optimal Strategies Using HYTECH page 4-a/35

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

What is the best cost whatever the environment does ? inf

0≤t≤2 max{5t + 10(2 − t) + 1, 5t + (2 − t) + 7}

Synthesis of Optimal Strategies Using HYTECH page 4-b/35

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

What is the best cost whatever the environment does ? inf

0≤t≤2 max{5t+10(2−t)+1, 5t+(2−t)+7} at t = 4

3 inf = 141 3

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

What is the best cost whatever the environment does ? = ⇒ 141

3 at t = 4 3

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

What is the best cost whatever the environment does ? = ⇒ 141

3 at t = 4 3

Is there a strategy to achieve this optimal cost ?

Yes: because see later

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A Simple Example

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

What is the best cost whatever the environment does ? = ⇒ 141

3 at t = 4 3

Is there a strategy to achieve this optimal cost ?

Yes: because see later

Can we compute such a strategy ?

Yes: in ℓ0, x < 4

3 wait then do c1; in ℓ2,3 do c2 when x ≥ 2

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Optimal Control Problems

ℓ0

cost(ℓ0) = 5

ℓ1 [y = 0] ℓ2

cost(ℓ2) = 10

ℓ3

cost(ℓ3) = 1

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

Can we find algorithms for these problems on PTGA:

1. Compute the optimal cost
2. Decide if there is an optimal strategy
3. Compute an optimal strategy (if ∃)

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Related Work

La Torre et al. [LTMM02] (IFIP TCS’02)

Acyclic Priced Timed Game Automata
Recursive definition of optimal cost

[= ⇒ La Torre et al. Def.]

Computation of the infimum of the optimal cost

OptCost = 2 could be 2 or 2 + ε

No strategy synthesis

Synthesis of Optimal Strategies Using HYTECH page 5-a/35

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Related Work

La Torre et al. [LTMM02] (IFIP TCS’02)

Acyclic Games, infimum, no strategy synthesis

Alur et al. [ABM04] (ICALP’04)

bounded optimality: optimal cost within k steps
complexity bound: exponential in k and #states of the

PTGA

no bound for the more general optimal problem
Computation of the infimum of the optimal cost
no strategy synthesis

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Related Work

La Torre et al. [LTMM02] (IFIP TCS’02)

Acyclic Games, infimum, no strategy synthesis

Alur et al. [ABM04] (ICALP’04)

bounded optimality, complexity bound, infimum, no strategy synthesis

Our work [BCFL04]:

Run-based definition of optimal cost
We can decide whether ∃ an optimal strategy
We can synthesize an optimal strategy (if ∃)
We can prove structural properties of optimal strategies
Applies to Linear Hybrid Game (Automata)

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Priced Timed Game Automata

A Timed Game Automaton (PTGA) G is a tuple (L, ℓ0, Act, X,

E, inv, cost) where: L is a finite set of locations; ℓ0 ∈ L is the initial location; Act = Actc ∪ Actu is the set of actions (partitioned into

controllable and uncontrollable actions);

X is a finite set of real-valued clocks; E ⊆ L × B(X) × Act × 2X × L is a finite set of transitions; inv : L − → B(X) associates to each location its invariant;

Synthesis of Optimal Strategies Using HYTECH page 7-a/35

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Priced Timed Game Automata

A Priced Timed Game Automaton (PTGA) G is a tuple

(L, ℓ0, Act, X, E, inv, cost) where: L is a finite set of locations; E ⊆ L × B(X) × Act × 2X × L is a finite set of transitions; Priced Version: cost : L ∪ E − → N associates to each

location a cost rate and to each discrete transition a cost value.

[= ⇒ Example]

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Priced Timed Game Automata

A Priced Timed Game Automaton (PTGA) G is a tuple

(L, ℓ0, Act, X, E, inv, cost) where: L is a finite set of locations; E ⊆ L × B(X) × Act × 2X × L is a finite set of transitions; Priced Version: cost : L ∪ E − → N associates to each

location a cost rate and to each discrete transition a cost value.

[= ⇒ Example]

we assume that PTGA are deterministic w.r.t. controllable

actions (+ time-deterministic)

A reachability PTGA (RPTGA) = PTGA with distinguished

set of states Goal ⊆ L.

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Configurations, Runs, Costs

configuration: (ℓ, v) in L × RX

≥0

step: ti = (ℓi, vi) αi − → (ℓi+1, vi+1)

αi ∈ R>0 =

⇒ ℓi+1 = ℓi ∧ vi+1 = vi + αi αi ∈ Act = ⇒ ∃(ℓi, g, αi, Y, ℓi+1) ∈ E ∧ vi | = g ∧ vi+1 = vi[Y ] run ρ = t0t1t2 · · · tn−1 · · · finite or infinite sequence of ti cost of a transition:

Cost(ti) = αi.cost(ℓi) if αi ∈ R>0

Cost(ti) = cost((ℓi, g, αi, Y, ℓi+1)) if αi ∈ Act if ρ finite Cost(ρ) =

0≤i≤n−1 Cost(ti)

winning run if States(ρ) ∩ Goal = ∅

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Strategies

strategy f over G = partial function from Runs(G) to Actc ∪ {λ}. Outcome((ℓ, v), f) (outcomes) of f from configuration (ℓ, v) = a subset of Runs((ℓ, v), G)

[= ⇒ Formal Definition of Outcome]

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Strategies

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2 x ≥ 2; c2

Example:

         f(ℓ0, x < 4

3) = λ

f(ℓ0, 4

3 ≤ x ≤ 2) = c1

f(ℓ1, −) undefined f(ℓ2, x < 2) = λ f(ℓ2, x ≥ 2) = c2 f(ℓ3, x < 2) = λ f(ℓ3, x ≥ 2) = c2

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Strategies

strategy f over G = partial function from Runs(G) to Actc ∪ {λ}. Outcome((ℓ, v), f) = outcomes of f from configuration (ℓ, v);

[= ⇒ Formal Definition of Outcome]

a strategy f is winning from (ℓ, v) if Outcome((ℓ, v), f) ⊆ WinRuns((ℓ, v), G) The cost of f from (ℓ, v) is Cost((ℓ, v), f) = sup{Cost(ρ) | ρ ∈ Outcome((ℓ, v), f)}

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(Formal) Optimal Control Problems

Optimal Cost Computation Problem: compute the optimal cost

ne can expect from s0 = (ℓ0,

0) OptCost(s0, G) = inf{Cost(s0, f) | f ∈ WinStrat(s0, G)}

Optimal Strategy Existence Problem: determine whether the

ptimal cost can actually be reached

∃?f ∈ WinStrat(s0, G) s.t. Cost(s0, f) = OptCost(s0, G)

Optimal Strategy Synthesis Problem: in case an optimal strategy

exists, compute a witness.

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(Formal) Optimal Control Problems

Optimal Cost Computation Problem: compute the optimal cost

ne can expect from s0 = (ℓ0,

0) OptCost(s0, G) = inf{Cost(s0, f) | f ∈ WinStrat(s0, G)}

Optimal Strategy Existence Problem: determine whether the

ptimal cost can actually be reached

∃?f ∈ WinStrat(s0, G) s.t. Cost(s0, f) = OptCost(s0, G)

Optimal Strategy Synthesis Problem: in case an optimal strategy

exists, compute a witness. Relation to La Torre et al. work [LTMM02] (acyclic game): Theorem 1: OptCost(s0, G) = O(s0)

[= ⇒ Definition of O(q)]

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Example: No Optimal Strategy

ℓ0

cost(ℓ0) = 1

ℓ1

cost(ℓ1) = 2

Goal x < 1; c x = 1; c x < 1 x ≤ 1 define fε with 0 < ε < 1 by:

in ℓ0: f(ℓ0, x < 1 − ε) = λ, f(ℓ0, 1 − ε ≤ x < 1) = c in ℓ1: f(ℓ1, x ≤ 1) = c

Cost(fε) = 1 + ε. there are RPTGA for which no optimal strategy exists In this case there is a family of strategies fε such that |Cost((ℓ0, 0), fε) − OptCost((ℓ0, 0), G)| < ε new problem: given ε, compute such an fε strategy.

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From Optimal Control to Control

ℓ0 ℓ1 [y = 0] ℓ2 Goal ℓ3 x ≤ 2; c1 y := 0 u u

cost(ℓ2) = 10 cost(ℓ0) = 5 cost(ℓ3) = 1 cost(ℓ2) = 10

x ≥ 2; c2

cost = 1

x ≥ 2; c2

cost = 7

A RPTGA A

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From Optimal Control to Control

ℓ0 ℓ1 [y = 0] ℓ2 Goal ℓ3 x ≤ 2; c1 y := 0 u u

dCost dt

= −5

dCost dt

= −10

dCost dt

= −1 x ≥ 2; c2 Cost′ = Cost − 1 x ≥ 2; c2 Cost′ = Cost − 7 A Linear Hybrid Game Automaton H Reachability Game for H with goal = Goal ∧ Cost ≥ 0

Optimal Cost for RPTGA ⇐

⇒ Reachability Control on LHA

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From Optimal Control to Control

ℓ0 ℓ1 [y = 0] ℓ2 Goal ℓ3 x ≤ 2; c1 y := 0 u u

dCost dt

= −5

dCost dt

= −10

dCost dt

= −1 x ≥ 2; c2 Cost′ = Cost − 1 x ≥ 2; c2 Cost′ = Cost − 7

Assume ∃ semi-algorithm CompWin s.t. WH = CompWin(H) and WH = largest set of winning states Theorem 2: If CompWin terminates for H then:

it terminates for A and WA

def

= CompWin(A) = ∃Cost.WH (q, c) ∈ WH ⇐ ⇒ ∃f ∈ WinStrat(q, WA) with Cost(q, f) ≤ c

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Known Results for Reachability Games

Controllable Predecessors [MPS95, DAHM01] π(X) = Predt

X ∪ cPred(X), uPred(X)
[=

⇒ Formal Def. of π]

W (largest) set of winning states, goal = X0 W = µX.X0 ∪ π(X)

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Known Results for Reachability Games

Controllable Predecessors [MPS95, DAHM01] π(X) = Predt

X ∪ cPred(X), uPred(X)
[=

⇒ Formal Def. of π]

W (largest) set of winning states, goal = X0 W = µX.X0 ∪ π(X) π preserves Cost upward-closed sets π(R ∧ Cost ≻ h) = R′ ∧ Cost ≻′ h′ semi-algorithm CompWin (preserves upwards closure) result of CompWin of the form ∪n∈N((ℓn, Rn ∧ Cost ≻n hn))

where hn is a piece-wise affine function

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Computing the Optimal Cost for PTGA

1. ∃ semi-algorithm CompWin for LHG
2. W = CompWin(H, Goal ∧ Cost ≥ 0)
3. W0 = W ∩ (ℓ0,

0)

4. projection on Cost: ∃(All \ {Cost}).W0

if Cost ≥ k, OptCost = k and ∃ an optimal strategy if Cost > k, OptCost = k and ∃ a family of sub-optimal

strategies Semi-algorithm for Priced Timed Hybrid Automata Termination ???

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Termination for RPTGA

A a RPTGA s.t. non-zeno cost: ∃κ s.t. every cycle in the

region automaton has cost at least κ

A is bounded i.e. all clocks in A are bounded

Theorem 4 CompWin terminates for H, where H is the LHG associated with A

[= ⇒ Sketch of the Proof]

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Termination for RPTGA

A a RPTGA s.t. non-zeno cost: ∃κ s.t. every cycle in the

region automaton has cost at least κ

A is bounded i.e. all clocks in A are bounded

Theorem 4 CompWin terminates for H, where H is the LHG associated with A

[= ⇒ Sketch of the Proof]

Non zeno cost really needed ? Complexity ???

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Optimal Strategy Synthesis

S algorithm for synthetizing strategies for reachability

timed games ? see [BCFL04] . . .

use S on the LHG H: strategies are cost-dependent

Theorem 5 If S terminates and ∃ an optimal strategy we can compute a witness (cost-dependent)

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Optimal Strategy Synthesis

S algorithm for synthetizing strategies for reachability

timed games ? see [BCFL04] . . .

use S on the LHG H: strategies are cost-dependent

Theorem 5 If S terminates and ∃ an optimal strategy we can compute a witness (cost-dependent)

assume a RPTGA A is bounded, non zeno cost W is the set of winning states in the LHG H W = ∪n∈N((ℓn, Rn ∧ Cost ≥ hn)) (hn piece-wise lin. aff.)

Theorem 6 [State-based Strategies] Let WA = CompWin(A).

∃f state-based s.t. ∀(ℓ, v) ∈ WA Cost((ℓ, v), f) = OptCost(ℓ, v)

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Synthesis of Cost-Dependent Strategies

for LHG winning states = fixed point of π operator W0 = Goal and Wi+1 = Predt

Wi ∪ cPred(Wi), uPred(Wi)
Synthesis of Optimal Strategies Using HYTECH

page 20-a/35

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Synthesis of Cost-Dependent Strategies

for LHG winning states = fixed point of π operator W0 = Goal and Wi+1 = Predt

Wi ∪ cPred(Wi), uPred(Wi)
synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi

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Synthesis of Cost-Dependent Strategies

for LHG winning states = fixed point of π operator W0 = Goal and Wi+1 = Predt

Wi ∪ cPred(Wi), uPred(Wi)
synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi

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Synthesis of Cost-Dependent Strategies

for LHG winning states = fixed point of π operator W0 = Goal and Wi+1 = Predt

Wi ∪ cPred(Wi), uPred(Wi)
synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}

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Synthesis of Cost-Dependent Strategies

for LHG winning states = fixed point of π operator W0 = Goal and Wi+1 = Predt

Wi ∪ cPred(Wi), uPred(Wi)
synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}
on Yt = Y \ Yc define fi+1 = {λ}

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Synthesis of Cost-Dependent Strategies

synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}
on Yt = Y \ Yc define fi+1 = {λ}

Problem ? ℓ0 Goal 0 < x < 1; c1

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Synthesis of Cost-Dependent Strategies

synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}
on Yt = Y \ Yc define fi+1 = {λ}

Problem ? ℓ0 Goal 0 < x < 1; c1

W1 = {Goal} ∪ {(ℓ0, 0 ≤ x < 1)} and Y = (ℓ0, 0 ≤ x < 1)
f1(ℓ0, 0 < x < 1) = {c1} and f1(ℓ0, x = 0) = {λ}
blocking strategy

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Synthesis of Cost-Dependent Strategies

synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}
on Yt = Y \ Yc define fi+1 = {λ}

Problem ? ℓ0 Goal 0 < x < 1; c1

Choose ε > 0
f1(ℓ0, ε ≤ x < 1) = {c1} and f1(ℓ0, 0 ≤ x < ε) = {λ}
new winning, state-based strategy

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Synthesis of Cost-Dependent Strategies

synthesis of cost-dependent (state-based on LHG)

strategy:

assume fi is a winning, state-based strategy on Wi
compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
on Wi define fi+1 = fi
on Yc = cPred(Wi) ∩ Y define fi+1 = {some c action}
on Yt = Y \ Yc define fi+1 = {λ}

Computation of a winning state-based strategy:

if guards of u actions are strict and guards on c actions

are large then fi+1 is winning (Yt is future-open)

otherwise fi+1 can be altered to be made winning
consequence: if π∗(W0) = Wk for some k ∈ N there is a

winning state-based (cost-dependent) strategy

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Optimal Cost-Independent Strategy

compute a cost-dependent winning strategy f; f(q, cost) ∈ Actc ∪ {λ} Optimal cost-independent winning strategy f∗:

take the best action in each state: f∗(q) = e if
1. e = f(q, cost)
2. ∀e′ = e, f(q, cost′) = e′ =

⇒ cost′ ≥ cost result: under strictness assumptions, we can build a

uniform optimal strategy i.e. optimal in each state (non blocking)

[= ⇒ Algorithm & HYTECH ]

Synthesis of Optimal Strategies Using HYTECH page 21/35

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No Optimal Cost-Independent Strategy

ℓ0

cost(ℓ0) = 2

x < 1 ℓ1

cost(ℓ1) = 1

Goal x < 1;u y := x := 0 y > 0; c2 Optimal cost is 2 An optimal winning cost-dependent strategy f: f(ℓ1, −, cost < 2) = λ and f(ℓ1, −, cost = 2) = c2

assume u taken at time (1 − δ0):

Cost(f, (ℓ0, 0)) = 2 · (1 − δ0) + δ1

and according to f we have δ1 = 2 · δ2

Synthesis of Optimal Strategies Using HYTECH page 22-a/35

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No Optimal Cost-Independent Strategy

ℓ0

cost(ℓ0) = 2

x < 1 ℓ1

cost(ℓ1) = 1

Goal x < 1;u y := x := 0 y > 0; c2 Optimal cost is 2 assume ∃ f∗ cost-independent: f∗ must wait in ℓ1 at least ε

assume u taken at time (1 − δ):

Cost(f∗, (ℓ0, 0)) = 2 · (1 − δ) + ε

Take δ = ε

4: Cost(f∗, (ℓ0, 0)) = 2 + ε 2 and OptCost(f∗) = 2 + ε

Synthesis of Optimal Strategies Using HYTECH page 22-b/35

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Experiment

computation of optimal cost and optimal strategies (if ∃)

implemented in HYTECH (Demo ?)

a cyclic example:

[= ⇒ See the strategy]

Antenna 1 lowx

cost = 1

highx

cost = 10

Goalx jamx?; x := 0 x ≥ 5; y := 0 x ≥ 10;

cost = 7

jamx?; x := 0 Antenna 2 lowy

cost = 2

highy

cost = 20

Goaly jamy?; y := 0 y ≥ 2; x := 0 y ≥ 10;

cost = 1

jamy?; y := 0 Jammer X Y x > 6; jamy! y > 6; jamx! x > 6; jamx! y > 6; jamy!

Synthesis of Optimal Strategies Using HYTECH page 24/35

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Optimal Strategy for the Mobile Phone

Optimal cost is 109

Synthesis of Optimal Strategies Using HYTECH page 25/35

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Conclusion & Future Work

Current State of Our Work

Semi-algorithm for computing the optimal cost for LHG in case it terminates:

decide if ∃ optimal strategy
compute an optimal strategy

Implementation in HYTECH

Open Problems

Optimal Control – Decidability issues (non zeno cost) maximal class for which CompWin terminates

Future Work

compute fε strategies safety games . . .

Synthesis of Optimal Strategies Using HYTECH page 26/35

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References (1)

[AD94]

R. Alur and D. Dill. A theory of timed automata. Theoretical Computer Science

(TCS), 126(2):183–235, 1994. 3 [ALTP01]

R. Alur, S. La Torre, and G. J. Pappas. Optimal paths in weighted timed automata. In
Proc. 4th Int. Work. Hybrid Systems: Computation and Control (HSCC’01), volume

2034 of LNCS, pages 49–62. Springer, 2001. 3 [ABM04]

R. Alur, M, Bernadsky, and P

. Madhusudan. Optimal reachability in weighted timed

games. In Proc. 31st International Colloquium on Automata, Languages and

Programming (ICALP’04), Lecture Notes in Computer Science. Springer, 2004. To

appear. 5

[AM99]

E. Asarin and O. Maler. As soon as possible: Time optimal control for timed
automata. In Proc. 2nd Int. Work. Hybrid Systems: Computation and Control

(HSCC’99), volume 1569 of LNCS, pages 19–30. Springer, 1999. 3 [AMPS98] E. Asarin, O. Maler, A. Pnueli, and J. Sifakis. Controller synthesis for timed

automata. In Proc. IFAC Symposium on System Structure and Control, pages

469–474. Elsevier Science, 1998. 3 [BCFL04] P . Bouyer, F. Cassez, E. Fleury, and K. G. Larsen. Optimal strategies in priced timed game automata. BRICS Report Series, Basic Research In Computer Science, Denmark, February 2004. 5, 19

Synthesis of Optimal Strategies Using HYTECH page 27/35

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References (2)

[DAHM01] L. De Alfaro, T. A. Henzinger, and R. Majumdar. Symbolic algorithms for infinite-state

games. In Proc. 12th International Conference on Concurrency Theory

(CONCUR’01), volume 2154 of LNCS, pages 536–550. Springer, 2001. 14 [LBB+01]

K. G. Larsen, G. Behrmann, E. Brinksma, A. Fehnker, T. Hune, P

. Pettersson, and

J. Romijn. As cheap as possible: Efficient cost-optimal reachability for priced timed
automata. In Proc. 13th International Conference on Computer Aided Verification

(CAV’01), volume 2102 of LNCS, pages 493–505. Springer, 2001. 3 [LTMM02] S. La Torre, S. Mukhopadhyay, and A. Murano. Optimal-reachability and control for acyclic weighted timed automata. In Proc. 2nd IFIP International Conference on Theoretical Computer Science (TCS 2002), volume 223 of IFIP Conference Proceedings, pages 485–497. Kluwer, 2002. 5, 10 [MPS95]

O. Maler, A. Pnueli, and J. Sifakis. On the synthesis of discrete controllers for timed
systems. In Proc. 12th Annual Symposium on Theoretical Aspects of Computer

Science (STACS’95), volume 900, pages 229–242. Springer, 1995. 3, 14

Synthesis of Optimal Strategies Using HYTECH page 28/35

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Recursive Definition of Optimal Cost

Let G be a RPTG. Let O be the function from Q to R≥0 ∪ {+∞} that is the least fixed point of the following functional: O(q)? q

t,p

− − → q′ max                           

Synthesis of Optimal Strategies Using HYTECH page 29-a/35

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Recursive Definition of Optimal Cost

Let G be a RPTG. Let O be the function from Q to R≥0 ∪ {+∞} that is the least fixed point of the following functional: O(q)? q

t,p

− − → q′ max                            min               min

q′ c,p′

− − →q′′

c∈Actc

p + p′ + O(q′′)        , p + O(q′)        Controllable actions in q′

Synthesis of Optimal Strategies Using HYTECH page 29-b/35

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Recursive Definition of Optimal Cost

Let G be a RPTG. Let O be the function from Q to R≥0 ∪ {+∞} that is the least fixed point of the following functional: O(q)? q

t,p

− − → q′ max                            min               min

q′ c,p′

− − →q′′

c∈Actc

p + p′ + O(q′′)        , p + O(q′)        sup

q

t′,p′

− − − →q′′

t′≤t

max

q′′ u,p′′

− − − →q′′′

u∈Actu

p′ + p′′ + O(q′′′) Controllable actions in q′ Uncontrollable actions before q′

Synthesis of Optimal Strategies Using HYTECH page 29-c/35

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Recursive Definition of Optimal Cost

Let G be a RPTG. Let O be the function from Q to R≥0 ∪ {+∞} that is the least fixed point of the following functional: O(q) = inf

q

t,p

− − →q′

t∈R≥0

max                            min               min

q′ c,p′

− − →q′′

c∈Actc

p + p′ + O(q′′)        , p + O(q′)        sup

q

t′,p′

− − − →q′′

t′≤t

max

q′′ u,p′′

− − − →q′′′

u∈Actu

p′ + p′′ + O(q′′′) Controllable actions in q′ Uncontrollable actions before q′ Minimize over t

Synthesis of Optimal Strategies Using HYTECH page 29-d/35

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Outcome

Let G = (L, ℓ0, Act, X, E, inv, cost) be a (R)PTGA and f a strategy over G. The outcome Outcome((ℓ, v), f) of f from configuration (ℓ, v) in G is the subset of Runs((ℓ, v), G) defined inductively by: (ℓ, v) ∈ Outcome((ℓ, v), f), if ρ ∈ Outcome((ℓ, v), f) then ρ′ = ρ

e

− − → (ℓ′, v′) ∈ Outcome((ℓ, v), f) if ρ′ ∈ Runs((ℓ, v), G) and one of the following three conditions hold:

1. e ∈ Actu,
2. e ∈ Actc and e = f(ρ),
3. e ∈ R≥0 and ∀0 ≤ e′ < e, ∃(ℓ′′, v′′) ∈ (L × RX

≥0) s.t. last(ρ) e′

− − → (ℓ′′, v′′) ∧ f(ρ

e′

− − → (ℓ′′, v′′)) = λ. an infinite run ρ is in ∈ Outcome((ℓ, v), f) if all the finite prefixes of ρ are in Outcome((ℓ, v), f).

[= ⇒ Back to Strategies]

Synthesis of Optimal Strategies Using HYTECH page 30/35

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A Tricky Example

ℓ0

cost(ℓ0) = 1

x ≤ 1 ℓ1

cost(ℓ1) = 2

x ≤ 1 Goal y > 0; u ; y := 0 y > 0; c ; y := 0 x = 1; c

Synthesis of Optimal Strategies Using HYTECH page 31-a/35

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A Tricky Example

ℓ0

cost(ℓ0) = 1

x ≤ 1 ℓ1

cost(ℓ1) = 2

x ≤ 1 Goal y > 0; u ; y := 0 y > 0; c ; y := 0 x = 1; c what is the optimal cost? Is there an optimal strategy?

Synthesis of Optimal Strategies Using HYTECH page 31-b/35

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A Tricky Example

ℓ0

cost(ℓ0) = 1

x ≤ 1 ℓ1

cost(ℓ1) = 2

x ≤ 1 Goal y > 0; u ; y := 0 y > 0; c ; y := 0 x = 1; c what is the optimal cost? Is there an optimal strategy? . . . assume you start with 2 . . . start with less than 2 (2 − ǫ)

Synthesis of Optimal Strategies Using HYTECH page 31-c/35

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π Operator

(Un)Controllable Predecessors Preda(X) = {q ∈ Q | q

a

− − → q′, q′ ∈ X} cPred(X) =

c∈Actc Predc(X)

uPred(X) =

u∈Actu Predu(X)

Safe Time Predecessors Predt(X, Y ) = {q ∈ Q | ∃δ ∈ R≥0 | q

δ

− → q′, q′ ∈ X ∧ Post[0,δ](q) ⊆ Y } Post[0,δ](q) = {q′ ∈ Q | ∃t ∈ [0, δ] | q

t

− → q′} π Operator (uncontrollable actions “cannot win”): π(X) = Predt

X ∪ cPred(X), uPred(X)
Synthesis of Optimal Strategies Using HYTECH

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π Operator

(Un)Controllable Predecessors Preda(X) = {q ∈ Q | q

a

− − → q′, q′ ∈ X} cPred(X) =

c∈Actc Predc(X)

uPred(X) =

u∈Actu Predu(X)

Safe Time Predecessors Predt(X, Y ) = {q ∈ Q | ∃δ ∈ R≥0 | q

δ

− → q′, q′ ∈ X ∧ Post[0,δ](q) ⊆ Y } Post[0,δ](q) = {q′ ∈ Q | ∃t ∈ [0, δ] | q

t

− → q′} π′: uncontrollable actions sometimes can win: π′(X) = π(X) ∪ Predt

uPred(X) ∩ STOP, uPred(X)
Synthesis of Optimal Strategies Using HYTECH

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π Operator

(Un)Controllable Predecessors Preda(X) = {q ∈ Q | q

a

− − → q′, q′ ∈ X} cPred(X) =

c∈Actc Predc(X)

uPred(X) =

u∈Actu Predu(X)

Safe Time Predecessors Predt(X, Y ) = {q ∈ Q | ∃δ ∈ R≥0 | q

δ

− → q′, q′ ∈ X ∧ Post[0,δ](q) ⊆ Y } Post[0,δ](q) = {q′ ∈ Q | ∃t ∈ [0, δ] | q

t

− → q′} π′′: uncontrollable actions bound to happen win: π′′(X) = π(X)∪Predt

Inv ∩ Predt(uPred(X) ∩ Inv), uPred(X)
Synthesis of Optimal Strategies Using HYTECH

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Termination Criterion for RPTGA

R is a (bounded) region of the region automaton (RA) every cycle in the RA costs at least κ m1

ℓ,R

M1

ℓ,R

cost ≥ f1

ℓ,R =↑f1 ℓ,R

κ κ κ κ κ κ

cost

R

[= ⇒ Back to Termination]

Synthesis of Optimal Strategies Using HYTECH page 33-a/35

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Termination Criterion for RPTGA

R is a (bounded) region of the region automaton (RA) every cycle in the RA costs at least κ m1

ℓ,R

M1

ℓ,R

cost ≥ f1

ℓ,R =↑f1 ℓ,R

κ f2

ℓ,R

m2

ℓ,R

κ κ κ κ κ

cost

R

[= ⇒ Back to Termination]

Synthesis of Optimal Strategies Using HYTECH page 33-b/35

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Termination Criterion for RPTGA

R is a (bounded) region of the region automaton (RA) every cycle in the RA costs at least κ m1

ℓ,R

M1

ℓ,R

cost ≥ f1

ℓ,R =↑f1 ℓ,R

κ f2

ℓ,R

m2

ℓ,R

κ κ κ κ κ m7

ℓ,R

↑f7

ℓ,R

cost

R

[= ⇒ Back to Termination]

Synthesis of Optimal Strategies Using HYTECH page 33-c/35

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Optimal Cost-Independent Strategy

ℓ0 (5) ℓ1 [y = 0] ℓ2 (10) ℓ3 (1) Goal x ≤ 2; c1; y := 0 u u x ≥ 2; c2 ; (1) x ≥ 2; c2 ; (7)

Synthesis of Optimal Strategies Using HYTECH page 34-a/35

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Optimal Cost-Independent Strategy

4 3

141

3

7 x cost ℓ0 (5) ℓ1 [y = 0] ℓ2 (10) ℓ3 (1) Goal x ≤ 2; c1; y := 0 u u x ≥ 2; c2 ; (1) x ≥ 2; c2 ; (7)

Synthesis of Optimal Strategies Using HYTECH page 34-b/35

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Optimal Cost-Independent Strategy

4 3

141

3

7 x cost W [λ]

4

ℓ0 (5) ℓ1 [y = 0] ℓ2 (10) ℓ3 (1) Goal x ≤ 2; c1; y := 0 u u x ≥ 2; c2 ; (1) x ≥ 2; c2 ; (7)

Synthesis of Optimal Strategies Using HYTECH page 34-c/35

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Optimal Cost-Independent Strategy

4 3

141

3

7 x cost W [λ]

4

W [c1]

4

ℓ0 (5) ℓ1 [y = 0] ℓ2 (10) ℓ3 (1) Goal x ≤ 2; c1; y := 0 u u x ≥ 2; c2 ; (1) x ≥ 2; c2 ; (7)

Synthesis of Optimal Strategies Using HYTECH page 34-d/35

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Optimal Cost-Independent Strategy

4 3

141

3

7 x cost λ c1

ℓ0 (5)

ℓ1 [y = 0] ℓ2 (10) ℓ3 (1) Goal x ≤ 2; c1; y := 0 u u x ≥ 2; c2 ; (1) x ≥ 2; c2 ; (7)

Synthesis of Optimal Strategies Using HYTECH page 34-e/35

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Optimal Cost-Independent Strategy

tagged sets: keep information how to win on Wi+1

compute Wi+1 = π(Wi) and let Y = Wi+1 \ Wi
W [c]

i+1 can reach Wi doing a c

W [λ]

i+1 can reach Wi or cPred(Wi) by time-elapsing

ptimal state-based strategy:
on W [c]

i+1 ≤ W [λ] i+1 do c

on W [λ]

i+1 < W [c] i+1 do λ

Synthesis of Optimal Strategies Using HYTECH page 34-f/35

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How-To Cost-Independent Strategy

ℓ0

cost(ℓ0) = 2

ℓ1

cost(ℓ1) = 1

ℓ2

cost(ℓ2) = 3

Goal x < 1;u y := 0 x < 1;u y > 0; c1 x = 1; c2 Optimal cost is 3

Synthesis of Optimal Strategies Using HYTECH page 35-a/35

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How-To Cost-Independent Strategy

ℓ0

cost(ℓ0) = 2

ℓ1

cost(ℓ1) = 1

ℓ2

cost(ℓ2) = 3

Goal x < 1;u y := 0 x < 1;u y > 0; c1 x = 1; c2 Optimal cost is 3 Optimal move in (ℓ1, y > 0) = c1, in (ℓ1, 0) = λ

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How-To Cost-Independent Strategy

ℓ0

cost(ℓ0) = 2

ℓ1

cost(ℓ1) = 1

ℓ2

cost(ℓ2) = 3

Goal x < 1;u y := 0 x < 1;u y > 0; c1 x = 1; c2 Optimal cost is 3 Optimal move in (ℓ1, y > 0) = c1, in (ℓ1, 0) = λ Optimal strategy: f∗(ℓ1, 0 < y < 1

2) = λ, in (ℓ1, y ≥ 1 2) = c1

f∗(ℓ2, x < 1) = λ and f∗(ℓ2, x ≥ 1) = c2

Synthesis of Optimal Strategies Using HYTECH page 35-c/35

Synthesis of Optimal Strategies Using HYTECH

Contents

Computing The Optimal Cost Computing Optimal Strategies

Context

Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal a2 a3 x ≥ 2; a4 x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5 Timed Automata + Reachability [AD94]

Context

Timed Game Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; c2 x ≤ 2 ; c1 y := 0 x ≥ 2 ; c2 Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98]

Context

As soon As Possible in Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; a4 1 ≤ x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5 Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99]

Context

Reachability in Priced Timed Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal a2 a3 x ≥ 2; a4

x ≤ 2 ; a1 y := 0 x ≥ 2 ; a5

Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99] Priced (or Weighted) Timed Automata [LBB+01, ALTP01]

Context

Priced Timed Game Automata

ℓ0 ℓ1 [y = 0] ℓ2 ℓ3 Goal u u x ≥ 2; c2

x ≤ 2 ; c1 y := 0 x ≥ 2 ; c2

Timed Automata + Reachability [AD94] Timed Game Automata: Control [MPS95, AMPS98] Time Optimal Control (Reachability) [AM99] Priced (or Weighted) Timed Automata [LBB+01, ALTP01]

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

Model = Game = Controller vs. Environment What is the best cost whatever the environment does ?

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

What is the best cost whatever the environment does ? inf

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

What is the best cost whatever the environment does ? inf

3 inf = 141 3

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

What is the best cost whatever the environment does ? = ⇒ 141

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

What is the best cost whatever the environment does ? = ⇒ 141

Is there a strategy to achieve this optimal cost ?

Yes: because see later

A Simple Example

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

What is the best cost whatever the environment does ? = ⇒ 141

Is there a strategy to achieve this optimal cost ?

Yes: because see later

Can we compute such a strategy ?

Yes: in ℓ0, x < 4

Optimal Control Problems

ℓ0

ℓ1 [y = 0] ℓ2

ℓ3

Goal x ≤ 2; c1 y := 0 u u x ≥ 2; c2

x ≥ 2; c2

Can we find algorithms for these problems on PTGA:

Related Work

La Torre et al. [LTMM02] (IFIP TCS’02)