The complete-information ω-regular game

(S, p) ,→P (hS, Oi, p) hS, Oi,→ Sβ 0 _{∧ hS, Oi ∈ V} env ∧ S ∈ Vctr ∧ p0 = δ(p, β) (hS, Oi, p),→β P (S0, p0) hS, Oi ,→ v ∧ hS, Oi ∈ Venv ∧ v ∈ Vterm ∧ p ∈ P (v, p) ,_→P (v, p)

Figure 7.1: Rules for the edge relation ,→P of the product game, where ,→ is the edge

relation as in Figure 5.1.

7.2

The complete-information ω-regular game

Given a transition system(A, Fair[A]) with the additional information about visibility and controllability as before, let _{G = (V, ,→) be the complete-information game obtained} from A# _{as in Section 5.1 and let P = (P, Obs, δ, p}

0, Acc, Accstop, Accdiv) be the de-

terministic, observation-based Streett automaton for the objective, with acceptance pairs Acc =_{(R1, G1), . . . , (Rk, Gk)}, i.e., a Streett condition with k acceptance pairs.

We perform a product construction between the gameG and P to obtain the complete- information gameGP = (VP, ,→P) whereP serves as a monitor to determine acceptance

of the plays in the game. The set of vertices in the product game is_VP =VctrP∪ VenvP ∪ VtermP ,

withVP

ctr = Vctr× P , VenvP = Venv× P and VtermP = Vterm× P , i.e., each vertex in the

complete-information game consists of a vertex of the underlying gameG and a state of P. The edge relation ,→P in the complete-information game is given by the three rules in

Figure 7.1, where,_{→ is the edge relation as in Figure 5.1.}

The first two rules ensure that the state ofP embedded in the vertices of VP tracks the ob-

servables occurring in the plays. The third rule introduces self-loops on each of the terminal vertices of the underlying game. This ensures that all maximal plays in_GP are infinite and

simplifies the handling of these terminal states later on.

The initial vertex for plays inGP is the vertex([Q0]∗, p0)∈ VctrP, where[Q0]∗ is the initial

vertex of_{G and p}0 is the initial state ofP. As P is total and deterministic, the product

construction preserves the structure of the underlying game G and there exists a one-to- one relation between maximal plays inG and GP by adding or stripping the state ofP and

extending or contracting the plays reaching vertices in_Vterm.

We can now treat_GP as a standard two-player turn-based game, with the verticesVctrP be-

longing to player 1, while the vertices of player 2 consist ofVP

env. TheVtermP can either be

treated as belonging to player 1 or to player 2, as they only have a single self-looping edge and thus provide only a single option to the player.

7.2. The complete-information ω-regular game Chapter 7. Most general controllers for ω-regular objectives

We now consider how to encode the constraints on decision functions that enforce the ob- jective into a winning condition for the gameGP, i.e., a Streett condition over the vertices of

the game. The winning condition encodes three constraints for player 1. The first condition, ΩP, ensures that the infinite observations of winning plays in the game are accepted by the

automatonP. The second condition, Ωterm, ensures that plays that reach a vertex inVtermP

are correctly classified as winning or losing depending onAccdivandAccstopas well as the

failure to ensure admissibility. The third Streett condition,Ωfair, ensures that winning plays

have to satisfy the fairness conditionFair[A] of the systemA. The first part of the Streett winning conditionΩP =



(RP₁, GP₁), . . . , (RP_k, GP_k) for the game_GP is obtained by lifting the acceptance conditionAcc =



(R1, G1), . . . , (Rk, Gk)

ofP from states of P to the corresponding vertices in VP

ctrofGP, i.e., for16 i 6 k,

RP

i = {(S, p) ∈ VctrP : p∈ Ri},

GP_i = {(S, p) ∈ VP

ctr : p∈ Gi}.

The second part of the Streett winning condition for the game, Ωterm = (Rterm, Gterm) is

obtained fromAccstopandAccdiv inP, the accepting states in case of termination and di-

vergence, with

Rterm ₌ _{(stop(s), p)∈ V}P

term : p /∈ Accstop

∪ (div(s), p)_{∈ Vterm}P : p /_{∈ Acc}div

∪ (fail(s), p)_{∈ Vterm}P and

Gterm _{= ∅}

Hence Rterm _{contains exactly the vertices from V}P

term that represent either non-accepting

termination, non-accepting divergence or failure to ensure admissibility.

Let Fair[_{A] = {F}1, . . . , F`} be the fairness condition of the system A. As in the pre-

vious chapter, we require that Fair[A] is history-independent (cf. Definition 5.1). Re- call that this allows us to relate the vertices S in the game to the corresponding fair choices as specified byFair[_{A]. The third part of the Streett winning condition for G}P,

Ωfair =

 (Rfair

1 , Gfair1 ), . . . , (Rfair` , Gfair` )

is then obtained by lifting the fairness condition to the corresponding vertices, i.e., for16 i 6 `,

Rfair i =  (S, p)_{∈ Vctr}P : Fi(S|A)6= ∅ , Gfair i =  (hS, Oi, p) ∈ VP

env : β∈ O implies β ∈ O0for someO0 ∈ Fi(S|A) .

A vertexv _{∈ V}P

ctr ofGP is thus contained inRfair_i if the observations that can be used to

reachv require a fair choice according to FiofFair[A]. Similarly, a vertex v ∈ VenvP ofGP

is contained inGfair

i if the choiceO is a fair choice according to FiofFair[A].

The Streett winning condition for GP _{is then given by Ω}

GP = ΩP ∪ Ωterm ∪ Ωfair, the

conjunction ofΩP,ΩtermandΩfair.

The construction ofGP ensures that there are no vertices without outgoing edges, thus all