Optimal Semicomputable Approximations to Reachable and Invariant Sets

Abstract

Abstract. In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are uppersemicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.

The Structure of Differential Invariants and Differential Cut Elimination

“…These approaches are very successful for linear systems. Even though the overall theory is interesting, it is purely semantical and defined in terms of limit properties of general functions, which are not computable, even in rich computation frameworks [Col07]. Similarly, working with solutions of differential equations, which are defined in terms of limits of functions, leads to sound and mathematically well-defined but generally incomputable approaches (except for simple cases like nilpotent linear systems).…”

“…For other differential equations, numerous approximation techniques have been considered to obtain approximate answers [GM99,ADG03,GP07,RS07,Fre08]. It is generally surprisingly difficult to get them formally sound, however, due to inherent numerical approximation that make the numerical image computation problem itself neither semidecidable nor co-semidecidable [PC07,Col07], even when tolerating arbitrarily large error bounds on the decision.…”

“…Differential equations enjoy various universal computation properties, hence verification is not even semidecidable [Bra95,GCB07,BCGH07,Col07]. Consequently, every complete proof rule is unsound or ineffective.…”

The Complete Proof Theory of Hybrid Systems

“…The theory of hybrid systems concluded that very limited classes of systems are undecidable [AM98,Hen96,CL00]. Most hybrid systems research since focused on practical approaches for efficient approximate reachability analysis for classes of hybrid systems [ADG03,Col07,PC07,GG09]. Undecidability also did not stop researchers in program verification from making impressive progress.…”

Computable CTL * for Discrete-Time and Continuous-Space Dynamic Systems

“…Moreover, in [7] it was shown that our semantics for the henceforth operator is optimal. For the negation operator, let us notice that in the standard semantics ¬¬Φ and Φ are equivalent whereas in the given topological semantics we generally have I |= ¬¬Φ ⇔ I |= Φ.…”