Region Graphs

For the purpose of this chapter, we collectively refer to region graph and boundary region graph as region graphs. In Subsections 4.3.1 and 4.3.2 we introduce region graph and boundary region graph, respectively, of a concavely-priced timed automaton. In Subsection 4.3.3 we define reachability price, discounted price, average price, price-per-time average, and price-per-reward average minimisation problems on these graph.

4.3.1. Region Graph

Let us recall the definition of the region graph of a timed automaton introduced in Chapter 3. The region graph of a timed automaton _T = (L,C,S,A,E,δ,ξ,F) is a labelled transition

systemTe = (Se,Ee,Se_F), where:

– Seis the set ofstatesdefined asSe={(s,R)∈S× R : s∈ R}; – Eeis thelabelled transition relationdefined as

e

E=_{((s,R),(t,R00,a),(s0,R0))_∈ Se×(R_⊕× R ×A)×Se

: R−→∗ R00 −→a R0ands0 =Succ(s,t,a)ands+t ∈R00}; and – Se_F ⊆Seis the set offinal statesdefined asSe_F={(s,R)∈Se : R⊆ F}.

We say that (s,R),(t,R00,a),(s0,R0)

is a transition in Te = (Se,Ee,Se_F)if we have that

(s,R),(t,R00,a),(s0,R0)

∈ Ee. We then also say that there is an (t,R00,a)-transition from state(s,R)inTe.

A run ofTe is a sequenceh(s₀,R₀),(t₁,R₁0,a₁),(s₁,R₁),(t₂,R0₂,a₂),(s₂,R₂), . . .i, such that for everyi_∈Nwe have that (si,Ri),(ti+1,R00_i+1,ai+1),(si0+1,R0i+1)

is a transition inTe. We write RunsTe

for the set of runs ofTe, and for(s,R)∈ Se, we write Runs

e

T_{(s,R)for the set of} runs ofTe whose initial state is(s,R).

The timed automaton_T and the region graphTe are equivalent in the following sense. PROPOSITION4.3.1. For everys∈ Sand(t,a)∈R_⊕×A, there is a(t,a)-transition froms

in_T if and only if there is a(t,[s+t],a)-transition from(s,[s])inTe.

Let (_T,π,K) be a concavely-priced timed automaton. We define the price function

e

π: Se×(R⊕× R ×A)→Rin the following way. For(s,R)∈Seand(t,R00,a)∈R⊕× R ×

A, such that there is a(t,R00,a)-transition from(s,R)inTe, we defineπ_e (s,R),(t,R00,a)

= π(s,t,a). For a concave price-reward automaton(T,π,$,K), we define functionsπeand$e in an analogous way.

4.3.2. Boundary Region Graph

Let us recall the definition of aboundary region graph from Chapter 3. Theboundary region graphTb = (Sb,Eb,Sb_F)of a timed automatonT = (L,C,S,A,E,δ,ξ,F)is a labelled transition system, where:

– Sbis the set ofstatesdefined as b

S={(s,R)∈ Q× R : s∈clos(R)}; – Ebis thelabelled transition relationdefined as

b

E=_{((s,R),(t,R00,a),(s0,R0))_∈ Sb×(R_⊕× R ×A)×Sb

– Sb_Fis the set offinal verticesdefined asSb_F ={(s,R)∈Sb : R⊆F}.

For(s,R),(s0,R0)∈Sband(t,R00,a)∈R⊕× R ×A, we say that (s,R),(t,R00,a),(s0,R0)

is a transition inTb = (Sb,Eb,Sb_F)if (s,R),(t,R00,a),(s0,R0)

∈ Eb.

A run of Tb is a sequence h(s0,R0),(t₁,R0₁,a₁),(s₁,R₁),(t2,R0₂,a2), . . .i, such that for every i _∈ N, we have that (si,Ri),(ti+1,Ri00+1,ai+1),(s0i+1,R0i+1)

is a transition inTb. We write RunsTb

for the set of runs ofTb, and for(s,R)∈ Sb, we write RunsTb(s,R)for the set of runs ofTb whose initial state is(s,R).

Let (_T,π,K) be a concavely-priced timed automaton. We define the price function

b

π : Sb×(R_⊕ × R × A) → R in the following way. Recall that for a ∈ A and

R,R00 _{∈ R}, the function πa_R,R00 : (s,t) 7→ π(s,a,t) defined on the set DR,R00 = {(s,t) :

s∈ Rand(s+t)∈ R00}is continuous. We writeπa_R,R00 for the unique continuous extension ofπa_R,R00 to the closureDR,R00 of the set D_R,R00. For (s,R) ∈ Sband(t,R00,a) ∈ R⊕× R ×A, such that there is an(t,R00,a)-transition from(s,R)inTb, we define

b

π (s,R),(t,R00,a)= π_Ra_,R00(s,t).

Thanks to the following proposition we can, and sometimes will, abuse notation by writingπ (s,R),(t,R,a)instead ofπe (s,R),(t,R,a)

orπb (s,R),(t,R,a)

.

PROPOSITION4.3.2. If(t,R,a)is a transition from the state(s,R)in both the region graph and the boundary region graph thenπe (s,R),(t,R,a)

=πb (s,R),(t,R,a)

.

For a concave price-reward automaton (T,π,$,K), we define functionsπb and$bin an analogous way.

4.3.3. Optimisation Problems on the Region Graphs

Before we discuss optimisation problems on region graph and boundary region graph, we define pre-runs of a timed automaton. The concept of pre-runs is a generalisation of the runs ofTe andTb, and it allows us to define cost functions on region graphs in a general setting.

DEFINITION4.3.3(Pre-Run). Apre-runof_T is a sequence

h(s0,R0),(t1,R10,a1),(s1,R1),(t1,R02,a2), . . .i ∈(Q× R)×((R⊕× R ×A)×(Q× R))ω, such thats_i+1 =Succ(si,(ti+1,ai+1))andRi −→∗ R0i+1

ai+1

−−→R_i+1for everyi∈N.

Observe that a pre-run follows the wait, the action, and the clock reset discipline, but not necessarily the guards. To be more precise, in a pre-run_h(s0,R0),(t1,R₁0,a1),(s1,R1), . . .i,

for every indexi, we do not require that the actionai+1 is enabled in the state si+ti+1, or

that the statesi is in the closure of regionRi.

We write PreRuns for the set of pre-runs and PreRuns(s,R) for the set of pre-runs starting from(s,R). We have the following relations among the sets of runs of region graphs and the set of pre-runs:

RunsTb

⊆PreRuns, and RunsTb

(s,R)⊆PreRuns(s,R) for all(s,R)∈Sb; RunsTe

⊆PreRuns, and RunsTe(

s,R)_⊆PreRuns(s,R) for all(s,R)_∈Se.

For a cost functionCost : PreRuns _→ R, theminimum costfunctionsCostTe

∗ : Se→ R andCostTb

∗ : Sb → R for region graphTe and boundary region graph Tb, respectively, are defined by:

CostTe

∗(s,R) = inf

r∈RunsTe₍

s,R)

Cost(r), and CostTb

∗(s,R) = inf

r∈RunsTb₍

s,R)

Cost(r).

The correspondingminimisation problemsforTe andTb are: given a states∈Sand a number

D_∈Q, determine whetherCostTe

∗(s,[s])≤ DandCost∗Tb(s,[s])≤D, respectively.

Let r = _h(s0,R0),(t1,R01,a1),(s1,R1),(t2,R02,a2), . . .i be a run of Te or Tb. We de- fine Stop(r) = inf_{i : Ri ⊆ F}. Also, for all n ∈ N we define Tn(r) =∑ni=1ti,

πn(r) =∑in=1π((si−1,Ri−1),(R0i,ti,ai)), and $n(r) =∑ni=1$((si−1,Ri−1),(R0i,ti,ai)). With

those notations, we define the reachability price, discounted price, average price, price-per- time average, and price-per-reward average cost functions, on the sets of runs ofTe andTb, in exactly the same way as for runs of the timed automatonT; see Section 4.2.

The following is an easy corollary of Proposition 4.3.1.

PROPOSITION4.3.4. IfCostis any of the reachability price, discounted price, average price, price-per-time average, or price-per-reward average cost functions, then for alls _∈ S, we haveCostT_∗(s) =CostTe

∗(s,[s]).

The following theorem is one of the main technical results of the chapter.

THEOREM 4.3.5. If Cost is any of the reachability price, discounted price, average time,

average price, price-per-time average, or price-per-reward average cost functions, then for alls_∈S, we haveCostTe

∗(s,[s]) =Cost∗Tb(s,[s]).

From Proposition 3.7.10 we know that for every state s _∈ S, the number of states reachable from s in the boundary region graph Tb is finite. By Theorem 2.2.1, it follows that optimal positional strategies exist in Tb for all the above-mentioned cost functions. Therefore, and since a run from a state according to a positional strategy in Tb can be guessed, and its cost computed, in PSPACE (with respect to the size of the input, i.e., a timed automatonT), it suffices to prove Theorem 4.3.5 in order to obtain the main Theorem 4.2.1. We dedicate the remaining two sections to the proof of Theorem 4.3.5.