Timed basic parallel processes (TBPP) extend communication-free Petri nets (aka. BPP or commutative context-free grammars) by a global notion of time. TBPP can be seen as an extension of timed automata (TA) with context-free branching rules, and as such may be used to model networks of independent timed automata with process creation. We show that the coverability and reachability problems (with unary encoded target multiplicities) are PSPACE-complete and EXPTIME-complete, respectively. For the

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... special case of 1-clock TBPP, both are NP-complete and hence not more complex than for untimed BPP. This contrasts with known super-Ackermannian-completeness and undecidability results for general timed Petri nets. As a result of independent interest, and basis for our NP upper bounds, we show that the reachability relation of 1-clock TA can be expressed by a formula of polynomial size in the existential fragment of linear arithmetic, which improves on recent results from the literature. ACM Subject Classification Theory of computation → Timed and hybrid models free Petri nets (a.k.a. BPP [14, 24] ). The general picture that we obtain is that extending communication-free Petri nets by a global notion of time comes at no extra cost in the complexity of safety checking, and it improves on the prohibitive complexities of TdPN. Our contributions. We show that the TBPP coverability problem is PSPACE-complete, matching same complexity for TA [7, 25] , and that the more general TBPP reachability problem is EXPTIME-complete, thus improving on the undecidability of TdPN. The lower bounds already hold for TBPP with two clocks if constants are encoded in binary; EXPTIMEhardness for reachability with no restriction on the number of clocks holds for constants in {0, 1}. The upper bounds are obtained by reduction to TA reachability and reachability games [30] , and assume that process multiplicities in target configurations are given in unary. In the single-clock case, we show that both TBPP coverability and reachability are NP-complete, matching the same complexity for (untimed) BPP [24]. This paves the way for the automatic verification of unbounded networks of 1-clock timed processes, which is currently lacking in mainstream verification tools such as UPPAAL [34] and KRONOS [46] . The NP lower bound already holds when the target configuration has size 2; when it has size one, 1-clock TBPP coverability becomes NL-complete, again matching the same complexity for 1-clock TA [33] (and we conjecture that 1-clock reachability is in PTIME under the same restriction). As a contribution of independent interest, we show that the ternary reachability relation of 1-clock TA can be expressed by a formula of existential linear arithmetic (∃LA) of polynomial size. By ternary reachability relation we mean the family of relations {→ pq } s.t. µ δ → pq ν holds if from control location p and clock valuation µ ∈ R k ≥0 it is possible to reach control location q and clock valuation ν ∈ R k ≥0 in exactly δ ∈ R ≥0 time. This should be contrasted with analogous results (cf. [27] ) which construct formulas of exponential size, even in the case of 1-clock TA. Since the satisfiability problem for ∃LA is decidable in NP, we obtain a NP upper bound to decide ternary reachability → pq . We show that the logical approach is optimal by providing a matching NP lower bound for the same problem. Our NP upper bounds for the 1-clock TBPP coverability and reachability problems are obtained as an application of our logical expressibility result above, and the fact that ∃LA is in NP; as a further technical ingredient we use polynomial bounds on the piecewise-linear description of value functions in 1-clock priced timed games [29] . Related research. Starting from the seminal PSPACE-completeness result of the nonemptiness problem for TA [7] (cf. also [25] ), a rich literature has emerged considered more challenging verification problems, including the symbolic description of the reachability relation [20, 22, 31, 23, 27] . There are many natural generalizations of TA to add extra modelling capabilities, including time Petri Nets [36, 38] (which associate timing constraints to transitions) the already mentioned timed Petri nets (TdPN) [41, 6, 28] (where tokens carry clocks which are tested by transitions), networks of timed processes [5] , several variants of timed pushdown automata [12, 21, 8, 43, 2, 40, 9, 19, 18] , timed communicating automata [32, 16, 4, 15] , and their lossy variant [1], and timed process calculi based on Milners CCS (e.g. [10]). While decision problems for TdPN have prohibitive complexity/are undecidable, it has recently been shown that structural safety properties are PSPACE-complete using forward accelerations [3] . Outline. In Section 2 we define TBPP and their reachability and coverability decision problems. In Section 3 we show that the reachability relation for 1-clock timed automata can be expressed in polynomial time in an existential formula of linear arithmetic, and that C O N C U R 2 0 1 9 * − −→ (Y, ν) + γ for some γ in the original TBPP.