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We study the complexity of reachability problems on branching extensions of vector addition systems, which allows us to derive new non-elementary complexity bounds for fragments and variants of propositional linear logic. We show that provability in the multiplicative exponential fragment is Tower-hard already in the affine case -- and hence non-elementary. We match this lower bound for the full propositional affine linear logic, proving its Tower-completeness. We also show that provability in propositional contractive linear logic is Ackermann-complete., Comment: Fixed Fig. 3 thanks to Hiromi Tanaka

International audience; We consider regular languages of labeled trees. We give an effective characterization of the regular languages over such trees that are definable in first-order logic in the language of labeled graphs. These languages are the analog on trees of the “locally threshold testable” languages on strings. We show that this characterization yields a decision procedure for determining whether a regular tree language is first-order definable: The procedure is polynomial time in the minimal automaton presenting the regular language. We also provide an algorithm for deciding whether a regular language is definable in first-order logic supplemented with modular quantifiers.