Composing Semi-algebraic O-Minimal Automata

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 an important step in such constructions is a product operation, which constructs a new product hybrid automaton by combining 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. Somewhat surprisingly, the product operation does not assure a closure of decidability for the reachability problem. Nonetheless, this paper establishes the decidability of the reachability condition over automata which are obtained by composing two semi-algebraic o-minimal systems. The class of semi-algebraic o-minimal automata is not even closed under composition, i.e., the product of two automata of this class is not necessarily a semi-algebraic o-minimal automaton. However, we can prove our decidability result combining the decidability of both semi-algebraic formulæ over the reals and linear Diophantine equations. All the proofs of the results presented in this paper can be found in [1]. 1 Semi-algebraic O-Minimal Automata and Composition Hybrid automata are systems in which discrete and continuous evolutions are mixed. In particular, their discrete nature is usually modeled through labeled directed graphs (called graphs in the rest of this paper), i.e., directed graphs with labels on the edges. On this kind of graphs we define: a path ph as sequence of edges; a cycle as a path in which the first and the last edges coincide; a simple cycle as a cycle without other repetitions. A hybrid automaton H = (Z, Z ′, V, E, Inv , F , Act , Res) of dimension k consists of the following components: This work is developed within HYCON, contract number FP6-IST-511368 and supported by the projects PRIN 2005 2005015491 and BIOCHECK. B.M. is supported by funding from two NSF ITR grants and one NSF EMT grant. A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, LNCS 4416, pp. 668–671, 2007. c © Springer-Verlag Berlin Heidelberg 2007 Composing Semi-algebraic O-Minimal Automata 669 1. Z = 〈Z1, . . ., Zk〉 and Z ′ = 〈Z ′ 1, . . ., Z ′ k〉 are two vectors of reals variables; 2. 〈V, E〉 is a labeled directed graph; the vertices, V, are called locations ; 3. Each vertex v ∈ V is labeled by the formulæ Inv(v)[Z] and Dyn(v)[Z,Z ′, T ] def = Z ′ = fv(Z, T ), where fv is the solution of the continuous vector field F ; 4. Each edge e ∈ E is labeled by the two formulæ Act(e)[Z] and Res(e)[Z,Z ′]. A state q of H is a pair 〈v, r〉, where v ∈ V is a location and r = 〈r1, . . . , rk〉 ∈ R k is an assignment of values for the variables of Z. A state 〈v, r〉 is said to be admissible if Inv(v)[r] is true. The semantics of hybrid automata is given in terms of continuous t −→C and discrete e −→D transitions over asmissible states in the standard way [1]. We use the notation q → q′ to denote that either q t −→C q′ or q e −→D q′. A trace tr = q0, q1, . . . , qnis a sequence of admissible states connected through transitions. The automaton H reaches a point s ∈ R (in time t) from a point r ∈ R if there exists a trace tr = q0, . . . , qn of H such that q0 = 〈v, r〉 and qn = 〈u, s〉, for some v, u ∈ V (and t is the sum of the continuous transitions elapsed times). Given a trace tr of H we can identify at least one path of 〈V,E〉 underlying tr. We call such paths corresponding paths of tr. A well-known class of hybrid automata is the class of o-minimal hybrid automata [2], defined by using formulæ taken over an ambient o-minimal theory [3] and by imposing the constraints of constant resets at discrete transitions. In the case of o-minimal automata defined by a decidable theory, reachability can be decided through bisimulation [2]. A theory which is both o-minimal and decidable is the first-order theory of (R, 0, 1,+, ∗, <) [4], also known as the theory of semialgebraic sets. In this paper we focus on semi-algebraic o-minimal hybrid automata, i.e., o-minimal hybrid automata built over the theory of (R, 0, 1,+, ∗, <). Let H1 = (Z1, Z1′,V1,E1, Inv1, F1, Act1, Res1) and H2 = (Z2, Z2′,V2,E2, Inv2, F2, Act2, Res2) be hybrid automata over distinct variables and let be a label not occurring in E1 ∪ E2. The product (see, e.g., [5,6]) of H1 and H2 is the hybrid automaton H1 ⊗H2 = (Z,Z ′,V,E, Inv , F , Act , Res), where: 1. Z (Z ′) is the concatenation of Z1 and Z2 (Z1′ and Z2′, respectively); 2. V = V1×V2 and E = Ex∪E∪E, where: Ex = {ee1,e2 |e1 ∈ E1 and e2 ∈ E2}, E1 = {ee,v | eE1 and v ∈ V2}, and E2 = {ev,e | v ∈ V1 and e ∈ E2}. 3. Inv(〈v1, v2〉)[Z] def = Inv(v1)[Z1] ∧ Inv(v2)[Z2]; 4. Dyn(〈v1, v2〉)[Z,Z ′, T ] def = Dyn(v1)[Z1, Z1′, T ] ∧Dyn(v2)[Z2, Z2′, T ]; 5. Act(ea,b)[Z] def = ⎧ ⎨ ⎩ Act(a)[Z1] ∧Act(b)[Z2] if ea,b ∈ Ex Act(a)[Z1] if ea,b ∈ E1 Act(b)[Z2] if ea,b ∈ E2 6. Res(ea,b)[Z,Z ′] def = ⎧ ⎨ ⎩ Res(a)[Z1] ∧ Res(b)[Z2] if ea,b ∈ Ex Res(a)[Z1] ∧ Z2′ = Z2 if ea,b ∈ E1 Z1′ = Z1 ∧ Res(b)[Z2] if ea,b ∈ E2 We study the reachability problem over H1⊗H2, where H1 and H2 are semialgebraic o-minimal hybrid automata, considering sets of points of the form I = I1×I2 and F = F1×F2. As noticed in [6] the decidability of reachability is not always preserved under product operations, i.e., it is possible that reachability is decidable over two classes of automata, but not over the product class. 670 A. Casagrande et al.