Parametric timed automata (PTA) have been introduced by Alur, Henzinger, and Vardi as an extension of timed automata in which clocks can be compared against parameters. The reachability problem asks for the existence of an assignment of the parameters to the non-negative integers such that reachability holds in the underlying timed automaton. The reachability problem for PTA is long known to be undecidable, already over three parametric clocks. A few years ago, Bundala and Ouaknine proved that for PTA over two parametric clocks and one parameter the reachability problem is decidable and also showed a lower bound for the complexity class PSPACE^NEXP. Our main result is that the reachability problem for two-parametric timed automata with one parameter is EXPSPACE-complete. Our contribution is two-fold. For the EXPSPACE lower bound, inspired by [13, 14], we make use of deep results from complexity theory, namely a serializability characterization of EXPSPACE (in turn based on Barrington’s Theorem) and a logspace translation of numbers in Chinese remainder representation to binary representation due to Chiu, Davida, and Litow. It is shown that with small PTA over two parametric clocks and one parameter one can simulate serializability computations. For the EXPSPACE upper bound, we first give a careful exponential time reduction from PTA over two parametric clocks and one parameter to a (slight subclass of) parametric one-counter automata over one parameter based on a minor adjustment of a construction due to Bundala and Ouaknine. For solving the reachability problem for parametric one-counter automata with one parameter, we provide a series of techniques to partition a fictitious run into several carefully chosen subruns that allow us to prove that it is sufficient to consider a parameter value of exponential magnitude only. This allows us to show a doubly-exponential upper bound on the value of the only parameter of a PTA over two parametric clocks and one parameter. We hope that extensions of our techniques lead to finally establishing decidability of the long-standing open problem of reachability in parametric timed automata with two parametric clocks (and arbitrarily many parameters) and, if decidability holds, determinining its precise computational complexity.

Alur、Henzinger 和 Vardi 引入了参数化时间自动机 (PTA) 作为时间自动机的扩展，其中时钟可以与参数进行比较。可达性问题要求存在将参数分配给非负整数，以便可达性在底层时间自动机中成立。众所周知，PTA 的可达性问题是不可判定的，已经超过三个参数时钟。几年前，Bundala 和 Ouaknine 证明了对于超过两个参数时钟和一个参数的 PTA，可达性问题是可判定的，并且还显示了复杂性类P S P A C E ^{N E X P的下界}. 我们的主要结果是，具有一个参数的双参数时间自动机的可达性问题是E X P S P A C E -complete。我们的贡献是双重的。对于E X P S P A C E的下界，受 [13, 14] 的启发，我们利用了复杂性理论的深层结果，即E X P S P A C E的可串行化特征（反过来基于 Barrington 定理）和由 Chiu、Davida 和 Litow 将中文余数表示形式的数字对数空间转换为二进制表示形式。结果表明，使用超过两个参数时钟和一个参数的小型 PTA 可以模拟可串行化计算。对于E X P S P A C E上界，我们首先根据 Bundala 和 Ouaknine 对构造的微小调整，从基于两个参数时钟和一个参数的 PTA 到基于一个参数的参数单计数器自动机（的轻微子类）进行仔细的指数时间减少。为了解决具有一个参数的参数单计数器自动机的可达性问题，我们提供了一系列技术来将虚拟运行划分为几个精心选择的子运行，这使我们能够证明仅考虑指数大小的参数值就足够了。这使我们能够在两个参数时钟和一个参数上显示 PTA 唯一参数值的双指数上限。