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A vector J:T —> Q is a T-invariant iff J — ( x ... x ) for some x. • Chapter 3. 17 T-systems Let (TV, M ) be a strongly connected T-system. equivalent: 0 The following statements are (a) (TV, M ) is live. 0 (b) (TV, M ) is deadlock-free. 0 (c) (TV, M ) has an infinite occurrence sequence. 0 Proof: As was seen in Chapter 2, (a) implies (b). It follows easily from the definition of deadlock-freeness that (b) implies (c). We prove that (c) implies (a). Assume that (TV, Mo) has an infinite occurrence sequence Mo -^+.

So we can apply the Exchange Lemma, and conclude that the sequence a\u is also enabled at MoBy definition, t G U. Let s be the unique place in t*. Since no circuit contains only transitions of A(a), no transition in s* belongs to U. So no transition of s* occurs in a\u- Since (TV, Mo) is 6-bounded and t G *s, we have that t occurs at most b times • in a\u- Therefore, t occurs at most b times in cr, which finishes the proof. 2 T-systems In T-systems places have exactly one input and one output transition.

6 Reachability Theorem Let (TV, Afo) be a live S-system and let Af be a marking of TV. Af is reachable iff Mo{S) = Af ( 5 ) , where S is the set of places of TV. Proof: (=>): Follows from the fundamental property of S-systems. (<=): By the Liveness Theorem, TV is strongly connected. 4. • The reachable markings of live S-systems can also be characterized in terms of S-invariants. So we first study the S-invariants of S-nets. Chapter 3. 7 S-invariants of S-nets Let N = (5, T, F) be a connected S-net.

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