Soonho Kong

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).

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.

The Lean Theorem Prover (System Description)

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

Automatically inferring quantified loop invariants by algorithmic learning from simple templates

By combining algorithmic learning, decision procedures, predicate abstraction, and simple templates, we present an automated technique for finding quantified loop invariants. Our technique can find arbitrary first-order invariants (modulo a fixed set of atomic propositions and an underlying SMT solver) in the form of the given template and exploits the flexibility in invariants by a simple randomized mechanism. The proposed technique is able to find quantified invariants for loops from the Linux source, as well as for the benchmark code used in the previous works. Our contribution is a simpler technique than the previous works yet with a reasonable derivation power.

Deriving invariants by algorithmic learning, decision procedures, and predicate abstraction

By combining algorithmic learning, decision procedures, and predicate abstraction, we present an automated technique for finding loop invariants in propositional formulae. Given invariant approximations derived from pre- and post-conditions, our new technique exploits the flexibility in invariants by a simple randomized mechanism. The proposed technique is able to generate invariants for some Linux device drivers and SPEC2000 benchmarks in our experiments.

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.

Compositional Sequentialization of Periodic Programs

We advance the state-of-the-art in verifying periodic programs --- a commonly used form of real-time software that consists of a set of asynchronous tasks running periodically and being scheduled preemptively based on their priorities. We focus on an approach based on sequentialization generating an equivalent sequential program of a time-bounded periodic program. We present a new compositional form of sequentialization that improves on earlier work in terms of both scalability and completeness i.e., false warnings by leveraging temporal separation between jobs in the same hyper-period and across multiple hyper-periods. We also show how the new sequentialization can be further improved in the case of harmonic systems to generate sequential programs of asymptotically smaller size. Experiments indicate that our new sequentialization improves verification time by orders of magnitude compared to competing schemes.

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.