arXiv subjects

Sicun Gao

Publications and source records attributed to Sicun Gao.

60 records · Page 4Linked to original sources

arXiv2014

Revisiting the Complexity of Stability of Continuous and Hybrid Systems

We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using first-order formulas over the real numbers, and reduce stability problems to the delta-decision problems of these formulas. The framework allows us to obtain a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the stability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we give upper bounds on their degrees of unsolvability.

eess.SY↗

arXiv2014

Delta-Complete Analysis for Bounded Reachability of Hybrid Systems

We present the framework of delta-complete analysis for bounded reachability problems of general hybrid systems. We perform bounded reachability checking through solving delta-decision problems over the reals. The techniques take into account of robustness properties of the systems under numerical perturbations. We prove that the verification problems become much more mathematically tractable in this new framework. Our implementation of the techniques, an open-source tool dReach, scales well on several highly nonlinear hybrid system models that arise in biomedical and robotics applications.

eess.SY↗

arXiv2013

Satisfiability Modulo ODEs

We study SMT problems over the reals containing ordinary differential equations. They are important for formal verification of realistic hybrid systems and embedded software. We develop delta-complete algorithms for SMT formulas that are purely existentially quantified, as well as exists-forall formulas whose universal quantification is restricted to the time variables. We demonstrate scalability of the algorithms, as implemented in our open-source solver dReal, on SMT benchmarks with several hundred nonlinear ODEs and variables.

cs.LO↗

arXiv2012

Delta-Complete Decision Procedures for Satisfiability over the Reals

We introduce the notion of "δ-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem φand a positive rational number δ, a δ-complete decision procedure determines either that φis unsatisfiable, or that the "δ-weakening" of φis satisfiable. Here, the δ-weakening of φis a variant of φthat allows δ-bounded numerical perturbations on φ. We prove the existence of δ-complete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded δ-SMT problem is in NP^C. δ-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL framework, which integrates Interval Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient conditions for its δ-completeness. We discuss practical applications of δ-complete decision procedures for correctness-critical applications including formal verification and theorem proving.

cs.LO↗

arXiv2012

Delta-Decidability over the Reals

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any L_F-sentence φcontaining only bounded quantifiers, and any positive rational number δ, decides either "φis true", or "a δ-strengthening of φis false". Under mild assumptions, for a C-computable signature F, the δ-decision problem for bounded Σ_k-sentences in L_F resides in (Σ_k^P)^C. The results stand in sharp contrast to the well-known undecidability results, and serve as a theoretical basis for the use of numerical methods in decision procedures for nonlinear first-order theories over the reals.

cs.LO↗

arXiv2011

Quantifier Elimination over Finite Fields Using Gröbner Bases

We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gröbner basis methods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite fields. We analyze the theoretical complexity of the algorithm and show its application in the formal analysis of a biological controller model.

cs.SC↗