52 research outputs found
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-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 \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-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 \delta-SMT problem is in NP^C.
\delta-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 \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
Proof Generation from Delta-Decisions
We show how to generate and validate logical proofs of unsatisfiability from
delta-complete decision procedures that rely on error-prone numerical
algorithms. Solving this problem is important for ensuring correctness of the
decision procedures. At the same time, it is a new approach for automated
theorem proving over real numbers. We design a first-order calculus, and
transform the computational steps of constraint solving into logic proofs,
which are then validated using proof-checking algorithms. As an application, we
demonstrate how proofs generated from our solver can establish many nonlinear
lemmas in the the formal proof of the Kepler Conjecture.Comment: Appeared in SYNASC'1
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
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.Comment: Published in FMCAD 201
Quantifier Elimination over Finite Fields Using Gr\"obner Bases
We give an algebraic quantifier elimination algorithm for the first-order
theory over any given finite field using Gr\"obner 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.Comment: A shorter version is to appear in International Conference on
Algebraic Informatics 201
Neural Lyapunov Control
We propose new methods for learning control policies and neural network
Lyapunov functions for nonlinear control problems, with provable guarantee of
stability. The framework consists of a learner that attempts to find the
control and Lyapunov functions, and a falsifier that finds counterexamples to
quickly guide the learner towards solutions. The procedure terminates when no
counterexample is found by the falsifier, in which case the controlled
nonlinear system is provably stable. The approach significantly simplifies the
process of Lyapunov control design, provides end-to-end correctness guarantee,
and can obtain much larger regions of attraction than existing methods such as
LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality
solutions for challenging control problems.Comment: NeurIPS 201
Monte Carlo Tree Descent for Black-Box Optimization
The key to Black-Box Optimization is to efficiently search through input
regions with potentially widely-varying numerical properties, to achieve
low-regret descent and fast progress toward the optima. Monte Carlo Tree Search
(MCTS) methods have recently been introduced to improve Bayesian optimization
by computing better partitioning of the search space that balances exploration
and exploitation. Extending this promising framework, we study how to further
integrate sample-based descent for faster optimization. We design novel ways of
expanding Monte Carlo search trees, with new descent methods at vertices that
incorporate stochastic search and Gaussian Processes. We propose the
corresponding rules for balancing progress and uncertainty, branch selection,
tree expansion, and backpropagation. The designed search process puts more
emphasis on sampling for faster descent and uses localized Gaussian Processes
as auxiliary metrics for both exploitation and exploration. We show empirically
that the proposed algorithms can outperform state-of-the-art methods on many
challenging benchmark problems.Comment: 17 pages, published in NeurIPS 202
Faster Constraint Solving Using Learning Based Abstractions
This work addresses the problem of scalable constraint solving. Our
technique combines traditional constraint-solving approaches with
machine learning techniques to propose abstractions that simplify the
problem. First, we use a collection of heuristics to learn sets of constraints
that may be well abstracted as a single, simpler constraint. Next, we
use an asymmetric machine learning procedure to abstract the set of clauses, using
satisfying and falsifying instances as training data. Next, we solve a
reduced constraint problem to check that the learned formula is indeed a
consequent (or antecedent) of the formula we sought to abstract, and
finally we use the learned formula to check the original property.
Our experiments show that our technique allows improved handling of
constraint solving instances that are slow to complete on a conventional
solver. Our technique is complementary to existing constraint solving
approaches, in the sense that it can be used to improve the scalability
of any existing tool
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