, define d 2 = max(0, max x (µ ? w(x))) + 1. Then, for all clocks x and for all d ? d 2 , w(x) + d v(p + ) is false

, We build a TA T + (A) as follows: first remove all locations not in T and remove all transitions to and from those removed locations. Second, add self-loops to all locations in L D (A), TA without invariants A, and a subset T of its locations

, EG(T ) holds if and only if there exists an infinite run in T + (A), Lemma, vol.3

, Suppose EG(T ) holds. Then there is a maximal path in A that stays in T

, From Lemma 2, this means that some location in T ? L D (A) is reachable in A, by always staying in T. Consequently that location is still reachable in T + (A) and since it belongs to L D (A), it has a self-loop in T + (A), Otherwise, it is finite and therefore it is a deadlock

, Either the corresponding infinite path never uses any of the added self-loops and therefore it is possible as is in A, which implies EG(T ), or it goes through L D (A) at least once. The latter means that some location in L D (A) is reachable in A by staying in T , and by Lemma 2, this implies that there exists a finite maximal path in A, and finally that we have EG(T ) in A. Corollary 1. The EG-emptiness and EG, ? In the other direction, suppose that there is an infinite run in T + (A)

, PSPACE-hardness comes from the fact that an L/U-PTA that does not use parameters in guards is a TA and EG is PSPACE-hard for TAs, vol.94

. Let-a-be-an-l/u-pta, Remark that the construction of Lemma 3 is independant of the constants in the guards, and hence can be done in the same way for a PTA, giving another PTA T + (A) such that, for all parameter valuations v, T + (v(A)) = v(T + (A))

, This result is important as it is the first non-trivial subclass of PTAs for which EG-universality (equivalent by negation to AF-emptiness) is decidable. We already had the same complexity for EF-emptiness and EF-universality

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