Propuesta de tratamiento de riesgos disergonómicos

So far, we showed that if there exists a solution (T,D) of _OERT_MinMax(ΓBR) such that T is

regionally simple and D is regionally constant, then(Te,De)is a solution of OERT_MinMax(Γ). In this section, using a strategy improvement algorithm, we give a constructive proof of the fact that for every boundary region automaton _TBR there exists a solution (T,D) |=

OERTMinMax(ΓBR)such thatTis regionally simple andDis regionally constant.

We begin by defining a special class of strategies in boundary region automata which we call regionally constant positional strategies. These strategies are of interest as all strategies appearing in our strategy improvement algorithm are regionally constant positional strategies. In Subsection 5.4.2 we define optimality equationsOERT

Max(ΓBR) and

OERT

Min(ΓBR) characterising maximum and minimum, respectively, reachability-price in a

boundary region automata. In Subsection 5.4.3 we present strategy improvement algorithm to solve_OERT_Max(ΓBR), which is called as a subroutine in the strategy improvement algorithm

presented in Subsection 5.4.4 to solveOERT

MinMax(ΓBR).

5.4.1. Positional Strategies in Boundary Region Automata

A positional strategy for player Max in a boundary region automaton ΓBR is a function χ : SMax → M, such that for every s ∈ SMax, we haveχ(s) = ([s],α,R), for some α ∈ A

andR ∈ R. A strategyχ : SMax → M isregionally constantif for alls,s0 ∈ SMax, we have

that[s] = [s0]impliesχ(s) = χ(s0); we can then write χ([s])forχ(s). Positional strategies for player Min are defined analogously. We write∆Max and∆Minfor the sets of positional

strategies for players Max and Min, respectively.

If χ ∈ ∆Max is regionally constant then we define the strategy subgraphΓBRχ to be

the subgraph(_R,_Mχ)where Mχ ⊆ Mconsists of: all moves(R,α,R0) ∈ M, such that

R_{∈ R}Min; and of all movesm= (R,α,R0), such thatR∈ RMaxandχ(R) =m. The strategy

subgraph ΓBRµfor a regionally constant positional strategy µ ∈ ∆Min for player Min is

We say thatR _{∈ R}ischoicelessin a boundary region automatonΓBR ifRhas a unique

successor inΓBR. We say thatΓBR is 0-player if allR∈ Rare choiceless inΓBR; we say that

ΓBR is 1-player if either allR ∈ RMinor allR ∈ RMaxare choiceless inΓBR; every boundary

region automatonΓBR is 2-player. Note that ifχandµare positional strategies in ΓBR for

players Max and Min, respectively, then ΓBRχ and ΓBRµare 1-player and (ΓBRχ)µ is

0-player.

For functions T : R → [S → R] and D : R → [S → R], and s ∈ SMax, we

define sets M∗(s,(T,D))and M_∗(s,(T,D)), respectively, of moves enabled in s which are (lexicographically)(T,D)-optimal for player Max and Min, respectively:

M∗(s,(T,D)) = argmaxlex m∈M T(R0)⊕_α(s),D(R0)_α(s) : m= ([s],α,R0) , and M_∗(s,(T,D)) = argminlex m∈M T(R0)⊕_α(s),D(R0)_α(s) : m= ([s],α,R0) .

LetChoose : 2M _{→ M}be a function such that for every non-empty set of movesM _{⊆ M}, we have Choose(M) _∈ M. For regional functions T : _{R →} [S + R] and D : _{R →}

[S+N], the canonical(T,D)-optimal strategiesχ(T,D)andµ(T,D)for player Max and Min, respectively, are defined by: χ(T,D)(s) = Choose(M∗(s,(T,D))), for everys ∈ SMax; and µ_(T,D)(s) =Choose(M_∗(s,(T,D))), for everys ∈SMin.

5.4.2. Optimality Equations

_OE

(

)

OE

(

)

OE

(

)

Optimality equations sets OERT

Max(ΓBR), OERTMax(ΓBR), and OERT(ΓBR) are introduced.

Intuitively,_OERT_Max(ΓBR)(OERTMax(ΓBR)) characterises optimal reachability-price when player

Min (Max) is choiceless, while _OERT(ΓBR) characterises optimal reachability-price when

both players are choiceless. We also define the sets OERT

≥ (ΓBR) and OERT_≤ (ΓBR) that

are useful technical tools in showing strict improvement in every iteration of strategy improvement algorithm.

LetT :R →[S _→R]andD:R →[S _→N]. We write(T,D)_|= _OERT

Max(ΓBR)if for all s_∈ Swe have the following:

e T(s),De(s) = ( 0, 0 ifs _∈F maxlex m∈M T(R0)⊕α(s),D(R0)α(s) : m= ([s],α,R0) , otherwise. Similarly we write(T,D)_|=_OERT

Min(ΓBR), if for alls∈ F, we have the following:

e T(s),De(s) = ( 0, 0 ifs _∈F minlexm∈M T(R0)⊕α(s),D(R0)α(s) : m= ([s],α,R0) , otherwise.

If ΓBR is 0-player then OERTMax(ΓBR) and OERTMin(ΓBR) are equivalent to each other and

denoted by_OERT(ΓBR).

We write(T,D)|=OERT

e T(s),De(s) ≥lex ( 0, 0 ifs ∈F maxlexm∈M T(R0)⊕α(s),D(R0)α(s) : m= ([s],α,R0) , otherwise. Similarly we write(T,D)|=OERT

≤ (ΓBR)if for alls ∈S, we have:

e T(s),De(s) ≤lex ( 0, 0 ifs ∈F minlexm∈M T(R0)⊕α(s),D(R0)α(s) : m= ([s],α,R0) , otherwise.

PROPOSITION 5.4.1 (Relaxations of optimality equations). If (T,D) |= OERT

Max(ΓBR) then (T,D)_|=_OERT

≥ (ΓBR), and if(T,D)|=OERTMin(ΓBR)then(T,D)|=OERT_≤ (ΓBR).

LEMMA5.4.2(Solution ofOERT(ΓBR)is regionally simple). LetΓBRbe a 0-player boundary

region automaton. If(T,D) |= OERT_(Γ

BR)then Tis regionally simple and Dis regionally

constant.

PROOF. In a 0-player boundary region automatonΓBR, for every regionR, there is at most

one outgoing labelled edge(R,α,R0)∈ M, and hence for every region R, there is a unique M-path fromRin ΓBR. For every regionR ∈ R, we define the distanced(R) ∈ N to be

the smallest number of edges in the unique_M-path fromR, that one needs to reach a final region. It is easy to show that for every states _∈ S, we have thatD([s])(s) = d([s]), and henceDis regionally constant.

We prove that for every region R _{∈ R}, the function T(R) : R _→ R is simple, by induction ond(R). Ifd(R) =0 thenT(R)(s) =0 for alls_∈ R, and henceT(R)is simple on

R.

Let d(R) = n+1 and let (R,α,R0) ∈ M be the unique edge going out of Rin ΓBR.

Observe that T(R) = T(R0)⊕_α because for every s ∈ R, we have T(R)(s) = T([s])(s) = T(R0)⊕_α(s), where the second equality follows from(T,D)|=OERT_(Γ

BR). Moreover, by the

induction hypothesis the functionT(R0): R0 →Ris simple, and hence by Proposition 5.3.1 we get thatT(R0)⊕_α =T(R)is simple.

Ifd(R) =∞, i.e., if the uniqueM-path fromRinΓBRnever reaches a final region, then

we setT(R0)(s) = ∞, for alls _∈ R. Therefore T(R0) : R _→ R is a constant function and hence it is simple.

5.4.3. Solving 1-Player Reachability-Time Optimality Equations

_OE

(

)

In this section we give a strategy improvement algorithm for solving maximum reachability- time optimality equations_OERT_Max(ΓBR)for a 1-player boundary region automatonΓBR.

We define the following strategy improvement operator Improve_Max:

Improve_Max(χ,(T,D))(s) =

(

χ(s) ifχ(s)_∈ M∗(s,(T,D)),

Note that Improve_Max(χ,(T,D))(s) may differ from the canonical (T,D)-optimal choice

χ(T,D)(s)only ifχ(s)is itself(T,D)-optimal in states, i.e., ifχ(s)∈ M∗(s,(T,D)).

LEMMA 5.4.3 (Improvement preserves regional constancy of strategies). If χ ∈ ∆Max is

regionally constant, T : _{R →} [S _→ R] is regionally simple, and D : _{R →} [S _→ N] is regionally constant, then Improve_Max(χ,(T,D))is regionally constant.

PROOF. We need to prove that for s,s0 ∈ S, if[s] = [s0]then χ0(s) = χ0(s0), where χ0 =

Improve_Max(χ,(T,D)). By regionality of χ it is sufficient to prove that M∗(s,(T,D)) = M∗(s0,(T,D)). By regional simplicity ofT, and by Proposition 5.3.1, we have that functions

T(R)⊕_α :[s]→R, for allm= ([s],α,R)∈ M, are simple. Then we have M∗(s,(T,D)) = argmaxlex m∈M T(R)⊕_α(s),D(R)_α(s) : m= ([s],α,R) = argmaxlex m∈M T(R)⊕_α(s0),D(R)_α(s0) : m= ([s0],α,R) = M∗(s0,(T,D)),

where the second equality follows from [s] = [s0], regional constancy of D, and by the application of Lemma 5.2.2 to the (finite) set of functions{T(R)⊕_α : ([s],α,R)∈ M}.

Input: Boundary Region AutomatonΓBR

Output: A solution ofOERT

Max(ΓBR)

begin 1

(Initialisation). Choose a regionally constant positional strategyχ0 ∈∆Maxfor

2