Decidable Compositions of O-minimal Automata ? Alberto Casagrande 1,2,3 , Pietro Corvaja 2 , Carla Piazza 2 , and Bud Mishra 4,5 1 Istituto di Genomica Applicata, Via J. Linussio, 51, 33100 Udine, Italy 2 DIMI, Universit ` a di Udine, Via delle Scienze, 206, 33100 Udine, Italy 3 DISA, Universit ` a di Udine, Via delle Scienze, 208, 33100 Udine, Italy 4 Courant Institute of Mathematical Science, NYU, New York, U.S.A. 5 NYU School of Medicine, 550 First Avenue, New York, 10016 U.S.A. Abstract. We identify a new class of decidable hybrid automata: namely, parallel compositions of semi-algebraic o-minimal automata. The class we consider is fundamental to hierarchical modeling in many exemplar sys- tems, both natural and engineered. Unfortunately, parallel composition, which is an atomic operator in such constructions, does not preserve the decidability of reachability. Luckily, this paper is able to show that when one focuses on the composition of semi-algebraic o-minimal automata, it is possible to translate the decidability problem into a satisfiability prob- lem over formulæ involving both real and integer variables. While in the general case such formulæ would be undecidable, the particular format of the formulæ obtained in our translation allows combining decidability results stemming from both algebraic number theory and first-order logic over (R, 0, 1, +, *,<) to yield a novel decidability algorithm. From a more general perspective, this paper exposes many new open questions about decidable combinations of real/integer logics. Introduction We wish to suggest a novel algebraic framework for the purpose of study- ing composition of hybrid automata. In this framework, we exploit various algebraic techniques (both semi-algebraic geometric and algebraic-number the- oretic) to provide effective procedures to solve reachability problems for at least one important class, namely, semi-algebraic o-minimal hybrid automata. We believe that these techniques are applicable more generally and will mo- tivate further applications to other classes and subclasses of hybrid-automata. Our techniques show how to model state-space evolution (as quantified semi- algebraic formulae) separately from the temporal synchronization (modeled as a system of linear algebraic Diophantine equations and inequalities) and yet, seek a combined solution to represent simultaneous arrival at a point in the product state-space by each individual component automaton. In order to ob- tain this decidability result, we needed to innovate in at least three different ? This work is partially supported by PRIN ”BISCA” 2006011235, FIRB ”LIBI” RBLA039M7M, two NSF ITR grants, and one NSF EMT grant. Corresponding au- thor: carla.piazza@dimi.uniud.it.