Synchronized Product of Semi-Algebraic O-Minimal Hybrid Automata ⋆ A. Casagrande 1,2 , P. Corvaja 1 , C. Piazza 1 , and B. Mishra 3,4 1 DIMI, Universit ` a di Udine, Via delle Scienze, 206, 33100 Udine, Italy 2 Istituto di Genomica Applicata, Via J.Linussio, 51, 33100 Udine, Italy 3 Courant Institute of Mathematical Science, NYU, New York, U.S.A. 4 NYU School of Medicine, 550 First Avenue, New York, 10016 U.S.A. Abstract. This paper addresses questions regarding the decidability of hybrid automata that may be constructed hierarchically and in a modular way, as is the case in many exemplar systems, be it natural or engineered. Since the basic fundamental step in such constructions is a product op- eration, which constructs a new product hybrid automaton by combin- ing two simpler component hybrid automata, an essential property that would be desired is that the reachability property of the product hybrid automaton be decidable, provided that the component hybrid automata belong to a suitably restricted family of automata, for which the reacha- bility property is provably decidable. Somewhat surprisingly, it does not appear that, under product operation, closure of decidability property for reachability condition could be guaranteed, in an arbitrarily general set- ting. Nonetheless, this paper establishes the decidability of the reachability condition over automata which are obtained by synchronizing two semi- algebraic o-minimal systems. Such hybrid automata appear in systems biological modeling, and hence could be applied when one is interested in understanding a complex biological systems composed of smaller self- organizing systems. 1 Introduction Classically two disparate mathematical traditions have incongruously coex- isted, while studying the same natural phenomena and their dynamics using two different elementary mechanistic representations. In one setting, the sys- tems remain unchanged during a resting period, interrupted discretely and intermittently by modification of its current state to a non-neighboring distant state. In the other, the system only makes continuous changes, while metic- ulously avoiding any perceptibly significant changes over any infinitesimally small time interval. Common sense and intuition dictate that neither of the ap- proaches should suffice to capture the substantial details of natural phenomena ⋆ This work is developed within the framework of the HYCON Network of Excellence, contract number FP6-IST-511368 and partially supported by the project PRIN 2005 project 2005015491. B.M. has been supported by funding from two NSF ITR grants and one NSF EMT grant.