761 research outputs found
A Lazy SMT-Solver for a Non-Linear Subset of Real Algebra
There are several methods for the synthesis and analysis of hybrid
systems that require efficient algorithms and tools for satisfiability
checking. For analysis, e.g., bounded model checking describes
counterexamples of a fixed length by logical formulas, whose
satisfiability corresponds to the existence of such a counterexample.
As an example for parameter synthesis, we can state the correctness of
a parameterized system by a logical formula; the solution set of
the formula gives us possible safe instances of the parameters.
For discrete systems, which can be described by propositional logic
formulas, SAT-solvers can be used for the satisfiability checks. For
hybrid systems, having mixed discrete-continuous behavior, SMT-solvers
are needed. SMT-solving extends SAT with theories, and has its main
focus on linear arithmetic, which is sufficient to handle, e.g.,
linear hybrid systems. However, there are only few solvers for
more expressive but still decidable logics like the
first-order theory of the reals with addition and multiplication --
real algebra. Since the synthesis and analysis of non-linear
hybrid systems requires such a powerful logic, we need efficient
SMT-solvers for real algebra. Our goal is to develop such an
SMT-solver for the real algebra, which is both complete and
efficient
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
Adapting Real Quantifier Elimination Methods for Conflict Set Computation
The satisfiability problem in real closed fields is decidable. In the context
of satisfiability modulo theories, the problem restricted to conjunctive sets
of literals, that is, sets of polynomial constraints, is of particular
importance. One of the central problems is the computation of good explanations
of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the
input constraints whose conjunction is already unsatisfiable. We adapt two
commonly used real quantifier elimination methods, cylindrical algebraic
decomposition and virtual substitution, to provide such conflict sets and
demonstrate the performance of our method in practice
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
A CDCL-style calculus for solving non-linear constraints
In this paper we propose a novel approach for checking satisfiability of
non-linear constraints over the reals, called ksmt. The procedure is based on
conflict resolution in CDCL style calculus, using a composition of symbolical
and numerical methods. To deal with the non-linear components in case of
conflicts we use numerically constructed restricted linearisations. This
approach covers a large number of computable non-linear real functions such as
polynomials, rational or trigonometrical functions and beyond. A prototypical
implementation has been evaluated on several non-linear SMT-LIB examples and
the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at
<http://informatik.uni-trier.de/~brausse/ksmt/
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Satisfiability Modulo Transcendental Functions via Incremental Linearization
In this paper we present an abstraction-refinement approach to Satisfiability
Modulo the theory of transcendental functions, such as exponentiation and
trigonometric functions. The transcendental functions are represented as
uninterpreted in the abstract space, which is described in terms of the
combined theory of linear arithmetic on the rationals with uninterpreted
functions, and are incrementally axiomatized by means of upper- and
lower-bounding piecewise-linear functions. Suitable numerical techniques are
used to ensure that the abstractions of the transcendental functions are sound
even in presence of irrationals. Our experimental evaluation on benchmarks from
verification and mathematics demonstrates the potential of our approach,
showing that it compares favorably with delta-satisfiability /interval
propagation and methods based on theorem proving
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