, following p t n+1 such that ? i goes from corner v i to v i , and ? i+1 from corner v i to v i+1 , for all i 0. Moreover, all corner plays ? i have weight w and all corner plays ? i have weight ?w. Consider the first index l such that v l = v k for some k < l, which exists because the number of corners is finite. We apply the induction to find a corner play following t 1 · · · t n p tn · · · p t 1 , going through the corner v k in the middle: more formally, there exists w ? , a corner play ? ? following ? , of weight w ? + w ? w ? w ? = w ? ? w ? , follows the cycle t 1 · · · t n t n+1 p t n+1 p tn · · · p t 1 . Since the game is almost-divergent, and those two corner plays are in the same SCC, both have branch: the end of each branch is called a stopped leaf of the semi-unfolding, particular, the depth of T (G) is bounded by |R(G)|?, and thus is polynomial in |R(G)|, w max and w

, We can define a semi-unfolding T (G) with initial state (? 0 , r 0 ) (a copy of state ( 0 , r 0 )) which is equivalent to G, i.e. for all ? 0 ? r 0, Proposition 11.2. Let G be an almost-divergent WTG, and let

, If ( , r) is in K, we let K ,r be the part of K accessible from ( , r) (note that K ,r is an SCC as K is a disjoint set of SCCs)

. Formally, we define a tree T whose nodes will either be labelled by region graph states ( , r) ? S\S K or by kernels K ,r , and whose edges will be labelled by

, Edges of T are replaced by the edges labelling them, and have a similar notion of inheritance. Every non-leaf node labelled by a kernel K ,r is replaced by a copy of the WTG K ,r , output edges being plugged in the expected way. We deal with stopped leaves labelled by a kernel K ,r by replacing them with a single node copy of ( , r), like we dealt with node labelled by a state ( , r). State partition between players and weights are inherited from the copied states of R(G). The only initial state of T (G) is the state denoted by

, Proof of Claim. By contradiction, consider a strategy ? Max of Max ensuring weight A ?|R(G)|w max ? w t max ? 1 in R(G). Then, for all ? Min , the cumulated weight of play R(G) (((? 0 , r 0 ), ? 0 ), ? Min , ? Max ) (reaching target configuration ( , ?)) is at least A ? wt t ( , ?) ?|R(G)|w max ? 2w t max ? 1, and by Lemma 11, SCC, the final weight function wt t is inherited from R(G) on target leaves and set to +? (resp. ??) on stopped leaves

, It is a winning strategy, so every play derived from ? Min in T (G) reaches a target leaf, and can be mimicked in R(G) by Lemma 11.4. Therefore, ? Min can be mimicked in R(G), where it is also winning, with the same value, If R(G) is non-negative, for all ? > 0 we can fix an ?-optimal strategy ? Min for Min in T (G)

, Let us now show the reverse inequality. If R(G) is non-negative, let us fix 0 < ? < 1, an ?-optimal strategy ? Min for Min in R(G), and a strategy ? Max of Max in R(G)

, In their setting, all runs in kernels had weight 0, allowing a reduction to a finite weighted game. In our setting, we have to approximate the timed dynamics of runs, and therefore resort to the corner-point abstraction (as shown to the right of Figure 11.1)

, In particular, at each step the configurations ( , ?) in ? and ( , r , v ) in ? (with v a corner of the 1/N -region v ) satisfy = and ? ? r , and the edges taken in both plays have the same discrete weights. Close plays have close weights, in the following sense: Lemma 11

, Consider the region path p of the run ?: p can be decomposed into a simple path with maximal cycles in it. The number of such maximal cycles is bounded by |L × Reg(X , M )| and the remaining simple path has length at most |L × Reg(X , M )|. Since all cycles of a kernel are 0-cycles, the parts of ? that follow the maximal cycles have weight exactly 0. Consider the same decomposition for the play ? . Cycles of p do not necessarily map to cycles over locations of ? N (G), since the 1/N -regions could be distinct. However, Lemma 8.3 shows that, for all those cycles of p, there exists a sequence of finite plays of G whose weight tends to the weight of ? . Since all those finite plays follow a cycle of the region game R(G) (with G being a kernel), Proof. Since ? and ? follow the same locations of G, one reaches a target location if and only if the other does

, -close (since they stay in the same 1/N -regions). Moreover, the difference between the final weight functions is bounded by ?/N , since the final weight function wt t is ?-Lipschitzcontinuous and the final weight function of ? N (G) is obtained as limit of wt t, /N is the largest difference possible in time delays between plays that stay 1/N

, We already know that Val P ? N (G) (( , r, v), v) = Val ? N (G) (( , r, v), v) for all configurations (( , r, v), v) of ? N (G). Moreover, Section 11.3.1 ensures |Val G ( , ?)?Val ? N (G) (( , r, v), v)| ?/2 whenever ? is in the 1/N -region r. Therefore, we only need to prove that |Val P G ( , ?)? Val P ? N (G) (( , r, v), v)| ?/2 to conclude. This is done as for Lemma 11.6, since Lemma 11.5 (that we need to prove Lemma 11.6) does not depend on the length of the plays ? and ? , and both runs reach the target state in the same step, Lemma 11.8. If G is a kernel with no configurations of infinite value, then |Val G ( , ?) ? Val P G ( , ?)| ? for all configurations

, G ( , ?)| ? for all configurations ( , ?) of G. Proof. Consider a non-negative SCC G, a precision ?

, Let ? be the maximum number of kernels along a branch of T . Let P be an integer such that for all kernels K in T (G), |Val K ( , ?) ? Val P K ( , ?)| ?/? for all configurations ( , ?) of G. We can find such a P by using Lemma 11.8. Create T (G) from T by applying the method used to create T (G) but replace every kernel by its complete P -unfolding instead. This implies that T (G) is a tree, of bounded depth P (at most the depth of T times P ), Let T (G) be its finite semi-unfolding (obtained from the labelled tree T , as in Section 11.2.1), such that Val G ( 0 , ? 0 ) = Val T (G) ((? 0 , r 0 ), ? 0 )

, Consider now T (G) the (complete) unfolding of R(G) with unfolding depth P , where kernels are also unfolded, This holds because the value function is 1-Lipschitz-continuous with regard to the final weight function, so imprecision builds up additively

, Bringing everything together we obtain |Val P G ( 0 , ? 0 ) ? Val G ( 0 , ? 0 )| ?. The idea is to unfold every kernel of the semi-unfolding game T (G) according to its bound in Lemma 11.8. More precisely, let ? be the maximum number of kernels along one of the branches of T (G). In a bottom-up fashion, we can find for each kernel K in T (G) a bound P K such that, for all configurations ( , ?), |Val K ( , ?) ? Val P K K ( , ?)| ?/?. We thus unfold K in T (G) with depth up to P K . After each kernel has been replaced this P ? G ( , ?)| ? for all configurations, Then, we can prove that Val P T (G) ((? 0 , r 0 ), ? 0 ) = Val P G ( 0 , ? 0 ) (same strategies at bounded horizon P ), which implies Val R(G)) (( 0 , r 0 ), ? 0 ) Val T (G) ((? 0 , r 0 )

, We now change T , by adding a subtree under each stopped leaf: the complete unfolding of R(G), starting from the stopped leaf, Let T (G) be its finite semi-unfolding (obtained from the labelled tree T , as in Section 11.2.1)

, ? 0 , r 0 ), ? 0 ) still holds (the proof was based on branches being long enough, and we increased the lengths

, ? 0 ) Val T + (G) ((? 0 , r 0 ), ? 0 ) (we increased the final weight function)

, Let us define ? + Min , a strategy for Min in T + (G), by making the same choice as ? Min on the common prefix tree, and once a node that is a stopped leaf in T (G) is reached, we switch to a memoryless attractor strategy of Min towards target states. Consider any strategy ? + Max of Max in T + (G), and let ? Max be its projection in T (G)

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, Une approche classique consiste à se ramener à un jeu à somme nulle, où deux joueurs interagissent tour-à-tour dans un système de transitions, et à se demander si le joueur "contrôleur" peut garantir que son objectif sera rempli, et ce indépendamment des décisions du joueur "environnement". Nous étudions des spécifications temps-réel, modélisées par un automate temporisé équipé d'un objectif d'accessibilité ou de Büchi, et présentons des méthodes symboliques pour synthétiser des stratégies du contrôleur. Nos contributions concernent deux problématiques distinctes : on peut souhaiter que le contrôleur obtienne une stratégie robuste aux perturbations, Synthèse symbolique de contrôleurs pour systèmes temporisés: robustesse et optimalité Résumé : Le domaine de la synthèse réactive a pour objectif d