Sicun Gao

dReal: an SMT solver for nonlinear theories over the reals

We describe the open-source tool dReal, an SMT solver for nonlinear formulas over the reals. The tool can handle various nonlinear real functions such as polynomials, trigonometric functions, exponential functions, etc. dReal implements the framework of δ-complete decision procedures: It returns either unsat or δ-sat on input formulas, where δ is a numerical error bound specified by the user. dReal also produces certificates of correctness for both δ-sat (a solution) and unsat answers (a proof of unsatisfiability).

A non-prenex, non-clausal QBF solver with game-state learning

We describe a DPLL-based solver for the problem of quantified boolean formulas (QBF) in non-prenex, non-CNF form. We make two contributions. First, we reformulate clause/cube learning, extending it to non-prenex instances. We call the resulting technique game-state learning. Second, we introduce a propagation technique using ghost literals that exploits the structure of a non-CNF instance in a manner that is symmetric between the universal and existential variables. Experimental results on the QBFLIB benchmarks indicate our approach outperforms other state-of-the-art solvers on certain benchmark families, including the tipfixpoint and tipdiam families of model checking problems.

dReach: δ-Reachability Analysis for Hybrid Systems

dReach is a bounded reachability analysis tool for nonlinear hybrid systems. It encodes reachability problems of hybrid systems to first-order formulas over real numbers, which are solved by delta-decision procedures in the SMT solver dReach. In this way, dReach is able to handle a wide range of highly nonlinear hybrid systems. It has scaled well on various realistic models from biomedical and robotics applications.

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 δ-complete algorithms for SMT formulas that are purely existentially quantified, as well as ∃∀-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.

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 sentence A containing only bounded quantifiers and functions in F, and any positive rational number delta, decides either “A is true”, or “a delta-strengthening of A is false”. Moreover, if F can be computed in complexity class C, then under mild assumptions, this “delta-decision problem” for bounded Sigma k-sentences resides in Sigma k(C). The results stand in sharp contrast to the well-known undecidability of the general first-order theories with these functions, and serve as a theoretical basis for the use of numerical methods in decision procedures for formulas over the reals.

Towards personalized prostate cancer therapy using delta-reachability analysis

Recent clinical studies suggest that the efficacy of hormone therapy for prostate cancer depends on the characteristics of individual patients. In this paper, we develop a computational framework for identifying patient-specific androgen ablation therapy schedules for postponing the potential cancer relapse. We model the population dynamics of heterogeneous prostate cancer cells in response to androgen suppression as a nonlinear hybrid automaton. We estimate personalized kinetic parameters to characterize patients and employ δ-reachability analysis to predict patient-specific therapeutic strategies. The results show that our methods are promising and may lead to a prognostic tool for prostate cancer therapy.

Parameter Synthesis for Cardiac Cell Hybrid Models Using δ-Decisions

A central problem in systems biology is to identify parameter values such that a biological model satisfies some behavioral constraints (e.g., time series). In this paper we focus on parameter synthesis for hybrid (continuous/discrete) models, as many biological systems can possess multiple operational modes with specific continuous dynamics in each mode. These biological systems are naturally modeled as hybrid automata, most often with nonlinear continuous dynamics. However, hybrid automata are notoriously hard to analyze — even simple reachability for hybrid systems with linear differential dynamics is an undecidable problem. In this paper we present a parameter synthesis framework based on δ-complete decision procedures that sidesteps undecidability. We demonstrate our method on two highly nonlinear hybrid models of the cardiac cell action potential. The results show that our parameter synthesis framework is convenient and efficient, and it enabled us to select a suitable model to study and identify crucial parameter ranges related to cardiac disorders.

SReach: A Probabilistic Bounded Delta-Reachability Analyzer for Stochastic Hybrid Systems

In this paper, we present a new tool SReach, which solves probabilistic bounded reachability problems for two classes of models of stochastic hybrid systems. The first one is (nonlinear) hybrid automata with parametric uncertainty. The second one is probabilistic hybrid automata with additional randomness for both transition probabilities and variable resets. Standard approaches to reachability problems for linear hybrid systems require numerical solutions for large optimization problems, and become infeasible for systems involving both nonlinear dynamics over the reals and stochasticity. SReach encodes stochastic information by using a set of introduced random variables, and combines \(\delta \)-complete decision procedures and statistical tests to solve \(\delta \)-reachability problems in a sound manner. Compared to standard simulation-based methods, it supports non-deterministic branching, increases the coverage of simulation, and avoids the zero-crossing problem. We demonstrate SReach’s applicability by discussing three representative biological models and additional benchmarks for nonlinear hybrid systems with multiple probabilistic system parameters.

SMT-Based Analysis of Virtually Synchronous Distributed Hybrid Systems

This paper presents general techniques for verifying virtually synchronous distributed control systems with interconnected physical environments. Such cyber-physical systems (CPSs) are notoriously hard to verify, due to their combination of nontrivial continuous dynamics, network delays, imprecise local clocks, asynchronous communication, etc. To simplify their analysis, we first extend the PALS methodology---that allows to abstract from the timing of events, asynchronous communication, network delays, and imprecise clocks, as long as the infrastructure guarantees bounds on the network delays and clock skews---from real-time to hybrid systems. We prove a bisimulation equivalence between Hybrid PALS synchronous and asynchronous models. We then show how various verification problems for synchronous Hybrid PALS models can be reduced to SMT solving over nonlinear theories of the real numbers. We illustrate the Hybrid PALS modeling and verification methodology on a number of CPSs, including a control system for turning an airplane.

APEX: Autonomous Vehicle Plan Verification and Execution

Autonomous vehicles (AVs) have already driven millions of miles on public roads, but even the simplest scenarios have not been certified for safety. Current methodologies for the verification of AVs decision and control systems attempt to divorce the lower level, short-term trajectory planning and trajectory tracking functions from the behavioral rules-based framework that governs mid-term actions. Such analysis is typically predicated on the discretization of the state space and has several limitations. First, it requires that a conservative buffer be added around obstacles such that many feasible plans are classified as unsafe. Second, the discretized controllers modeled in this analysis require several refinement steps before being implementable on an actual AV, and typically do not allow the specification of comfort-related properties on the trajectories. In contrast, consumer-ready AVs use motion planning algorithms that generate smooth trajectories. While viable algorithms exist for the generation of smooth trajectories originating from a single state, analysis should consider that the AV faces state estimation errors and disturbances. Third, verification is restricted to a discretized state space with fixed-size cells; this assumption can artificially limit the set of available trajectories if the discretization is too coarse. Conversely, too fine of a discretization renders the problem intractable for automated analysis. This work presents a new verification tool, APEX, which investigates the combined action of a behavioral planner and state lattice-based motion planner to guarantee a safe vehicle trajectory is chosen. In APEX, decisions made at the behavioral layer can be traced through to the spatio-temporal evolution of the AV and verified. Thus, there is no need to create abstractions of the AVs controllers, and aggressive trajectories required for evasive maneuvers can be accurately investigated.