3 Timed Automata
3.5. Competitive Optimisation on Timed Automata
In this section we define competitive optimisation problems on timed automata which are of interest in this thesis. Since some of these problems are known by different names in the literature, in this section we set a uniform terminology for these problems.
3.5.1. The Noncompetitive Case
In this thesis we consider minimisation problems for various cost functions on concavely- priced timed automata. Henceforth we reserve the term optimisation, optimum, and optimal to refer to minimisation, minimum, and minimal, respectively. Fix a timed automaton T, price and reward functions π,$ : S×R⊕× A → R, and an initial state s∈ S.
A cost function is a function which takes an infinite run and returns its cost. Given a cost functionCost : Runs → Rwe define the optimal cost functionCost∗ : S → Rin the following way: Cost∗(s) = inf r∈Runs(s) Cost(r) = inf σ∈Σ Cost(Run(s,σ)) .
Theoptimisation problemfor the cost functionCostis to compute the optimum costCost∗(s)
of a given states∈S. The decision version of the optimisation problem is as follows: “given a timed automatonT, a states ∈Sand a numberD∈ Q, determine whetherCost∗(s)≤ D.” We say that a strategy σ∗ ∈ Σ is optimal for the cost function Cost if we have that
Cost(Run(s,σ∗)) = Cost∗(s). For a givenε> 0, we say that a strategyσ∈ Σisε-optimal if
we have thatCost(Run(s,σ))≤Cost∗(s) +ε.
Letr =hs0,τ1,s1,τ2, . . .i ∈Runs be a run of the timed automatonT, whereτi = (ti,ai)
for every positive integer i. Moreover, for π and $, the price and reward functions,
respectively, of a priced (or price-reward) timed automaton, and for every positive integern, we define:Tn(r) =∑ni=1ti,πn(r) =∑ni=1π(si−1,τi), and$n(r) =∑in=1$(si−1,τi).
The following list of cost functions gives rise to a number of corresponding optimisa- tion problems.
(1) Reachability time. The reachability-time cost function is defined in the following way: for every runr∈Runs we have
RT(r) =
(
TN(r), ifN =Stop(r)<∞,
∞, otherwise.
(2) Reachability price. The reachability-price cost function is defined in the following way: for every runr∈Runs we have
RP(r) =
(
πN(r), ifN=Stop(r)<∞,
∞, otherwise.
(3) Discounted time. The discounted-time cost function is defined in the following way: for every runr∈Runs and everydiscount factorλ∈(0, 1)we have
DT(λ)(r) = (1−λ)
∞
∑
i=1 λi−1ti.
(4) Discounted price. The discounted-price cost function is defined in the following way: for every runr∈Runs and everydiscount factorλ∈(0, 1)we have
DP(λ)(r) = (1−λ)
∑
∞i=1
λi−1π(si−1,τi).
(5) Average time. The average-time cost function is defined in the following way: for every runr∈Runs we have
AT(r) =lim sup
n→∞
Tn
n .
(6) Average price. The average-price cost function is defined in the following way: for every runr∈Runs we have
AP(r) =lim sup
n→∞
πn(r)
n .
(7) Price-per-time average. The price-per-time average cost function is defined in the following way: for every runr∈Runs we have
PTAvg(r) =lim sup
n→∞
πn(r)
Tn(r)
.
(8) Price-per-reward average. The price-per-reward average cost function is defined in the following way: for every runr∈Runs we have
PRAvg(r) =lim sup
n→∞
πn(r)
$n(r)
.
In Chapter 4 we show (Theorem 4.2.1) that for arbitrary initial state s ∈ S, the optimisation problem for all these cost functions on a concavely-price timed automata or concave price-reward timed automata, as appropriate, is PSPACE-complete.
3.5.2. Competitive Optimisation
Fix a timed game automaton Γ = (T,LMin,LMax), price and reward functions π,$ : S×
R⊕×A→R, and an initial states∈S. LetΣMinandΣMax be the set of strategies for player
Min and player Max, respectively. In the context of games we sometimes refer to a run as a
playof the game.
DEFINITION3.5.1(Cost Game on a Timed Game Automaton). Acost gameon a timed game automaton is a tuple(Γ,CostMin,CostMax), where:
– Γ= (T,LMin,LMax)is a timed game automaton,
– CostMin : Runs → R is a payoff function for player Min which gives the amount Min loses in a play, and
– CostMax : Runs→Ris the payoff function for player Max, which gives the amount Max wins in a play.
A cost game is calledzero-sumif we have thatCostMin(r) =CostMax(r)for every playr
of the game.
Payoff functionsCostMax : Runs →RandCostMin : Runs →Rnaturally gives rise to the payoff functionsCostMax : S×ΣMin×ΣMax → RandCostMin : S×ΣMin×ΣMax → R
in the following way: for strategies µ ∈ ΣMin and χ ∈ ΣMax of respective players and
a state s ∈ S we have CostMin(s,µ,χ) = CostMin(Run(s,µ,χ)) and CostMax(s,µ,χ) =
CostMax(Run(s,µ,χ)).
Theupper valueof a cost game at the statesis defined as the maximum loss that player Min can secure, in a play starting at the states, irrespective of the strategies used by player Max. Similarly thelower valueof a cost game at the statesis defined as the minimum win that player Max can secure, in a play starting at the states, irrespective of the strategies used by player Min. Formally we define theupper valueValΓ(s)and the lower-valueValΓ(s)of the game at the statesby as follows:
ValΓ(s) = inf µ∈ΣMin
sup χ∈ΣMax
CostMin(s,µ,χ), andValΓ(s) = sup
χ∈ΣMax
inf µ∈ΣMin
CostMax(s,µ,χ).
From Proposition 1.2.4 we know that the lower value of a game at a state is always less than the upper value of the game at that state, i.e., for every states ∈ S the inequality
ValΓ(s)≤ValΓ(s)always holds.
A cost game is determined if for every states ∈ S, we haveValΓ(s) =ValΓ(s). We then say that the value of the game exists. We writeValΓ(s)for this number and we call it the
valueof the game at the states.
The strategies µ∗ ∈ ΣMin andχ∗ ∈ ΣMax are optimalfor the respective players, if for
every states ∈Swe have that sup
χ∈ΣMax
CostMin(s,µ∗,χ) =ValΓ(s), and inf
µ∈ΣMin
CostMin(s,µ∗,χ) =ValΓ(s).
We say that a game ispositionally determinedif players have positional optimal strate- gies.
If the game is determined then the competitive optimisation problem is to compute value of the gameVal(s)for a give states∈S. The corresponding decision problem is given a cost game(Γ,CostMin,CostMax), a state s ∈ Sand a number D ∈ Q, determine whether
ValΓ(s)≤ D.
The following cost games on timed automata are of interest.
(1) Reachability game. A reachability game (Γ,ReachMin,ReachMax)has the following payoff functions:
ReachMin(r) =ReachMax(r) =
(
N ifN=Stop(r)<∞
∞ otherwise.
(2) Reachability-time game. A reachability-time game(Γ,RTMin,RTMax)has the follow- ing payoff functions:
RTMin(r) =RTMax(r) =
(
TN(r) ifN=Stop(r)<∞
(3) Reachability-price game. A reachability-price game(Γ,RPMin,RPMax)has the follow- ing payoff functions:
RPMin(r) =RPMax(r) =
(
πN(r) ifN=Stop(r)<∞
∞ otherwise.
(4) Discounted-time game. For a given discount factor λ ∈ (0, 1), a discounted-time
game(Γ,DTMin(λ),DTMax(λ))has the following payoff functions:
DTMin(λ)(r) =DTMax(λ)(r) = (1−λ)
∞
∑
i=1 λi−1ti.
(5) Discounted-price game. For a given discount factor λ ∈ (0, 1), a discounted-price
game(Γ,DPMin(λ),DPMax(λ))has the following payoff functions:
DPMin(λ)(r) =DPMax(λ)(r) = (1−λ)
∑
∞i=1
λi−1π(si−1,τi).
(6) Average-time game. An average-time game (Γ,APMin,APMax) has the following payoff functions:
ATMin(r) =lim sup
n→∞
Tn(r)
n andATMax(r) =lim infn→∞
Tn(r)
n .
(7) Average-price game. An average-price game (Γ,APMin,APMax) has the following
payoff functions:
APMin(r) =lim sup
n→∞
πn(r)
n andAPMax(r) =lim infn→∞
πn(r)
n .
(8) Price-per-time average game. In a similar manner, a price-per-time average game
(Γ,PTAvgMin,PTAvgMax)has the following payoff functions:
PTAvgMin(r) =lim sup
n→∞
πn(r)
Tn(r) and
PTAvgMax(r) =lim inf
n→∞
πn(r)
Tn(r).
(9) Price-per-reward average game.In a similar manner, a price-per-reward average game
(Γ,PRAvgMin,PRAvgMax)has the following payoff functions:
PRAvgMin(r) =lim sup
n→∞
πn(r)
$n(r)
andPRAvgMax(r) =lim inf
n→∞
πn(r)
$n(r)
.
In this thesis we study reachability-time games and average-time games in Chapter 5 and Chapter 6, respectively, and show that these game are determined (Theorem 5.1.2 and Theorem 6.4.2). We also prove that decision problems corresponding to these games are EXPTIME-complete for timed automata with at least two clocks (Theorem 5.5.4 and Theorem 6.5.1).
Reachability games were shown to be EXPTIME-complete for timed automata by Henzinger and Kopke [HK99]. In Chapter 5 we strengthen this results by showing that reachability games are EXPTIME-hard for timed automata with at least two clocks (Theorem 5.5.3).
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FIGURE3.4. A Zeno Timed Automaton.