1,428 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
Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks
Deep neural networks have emerged as a widely used and effective means for
tackling complex, real-world problems. However, a major obstacle in applying
them to safety-critical systems is the great difficulty in providing formal
guarantees about their behavior. We present a novel, scalable, and efficient
technique for verifying properties of deep neural networks (or providing
counter-examples). The technique is based on the simplex method, extended to
handle the non-convex Rectified Linear Unit (ReLU) activation function, which
is a crucial ingredient in many modern neural networks. The verification
procedure tackles neural networks as a whole, without making any simplifying
assumptions. We evaluated our technique on a prototype deep neural network
implementation of the next-generation airborne collision avoidance system for
unmanned aircraft (ACAS Xu). Results show that our technique can successfully
prove properties of networks that are an order of magnitude larger than the
largest networks verified using existing methods.Comment: This is the extended version of a paper with the same title that
appeared at CAV 201
Learning Tuple Probabilities
Learning the parameters of complex probabilistic-relational models from
labeled training data is a standard technique in machine learning, which has
been intensively studied in the subfield of Statistical Relational Learning
(SRL), but---so far---this is still an under-investigated topic in the context
of Probabilistic Databases (PDBs). In this paper, we focus on learning the
probability values of base tuples in a PDB from labeled lineage formulas. The
resulting learning problem can be viewed as the inverse problem to confidence
computations in PDBs: given a set of labeled query answers, learn the
probability values of the base tuples, such that the marginal probabilities of
the query answers again yield in the assigned probability labels. We analyze
the learning problem from a theoretical perspective, cast it into an
optimization problem, and provide an algorithm based on stochastic gradient
descent. Finally, we conclude by an experimental evaluation on three real-world
and one synthetic dataset, thus comparing our approach to various techniques
from SRL, reasoning in information extraction, and optimization
The First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
The intersection of two halfspaces has high threshold degree
The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least
degree of a real polynomial p such that f(x)=sgn p(x). We construct two
halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an
exponential improvement on previous lower bounds. This solves an open problem
due to Klivans (2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central unresolved challenge
in computational learning. We also prove that the intersection of two majority
functions has threshold degree Omega(log n), which is tight and settles a
conjecture of O'Donnell and Servedio (2003).
Our proof consists of two parts. First, we show that for any nonconstant
Boolean functions f and g, the intersection f(x)^g(y) has threshold degree O(d)
if and only if ||f-F||_infty + ||g-G||_infty < 1 for some rational functions F,
G of degree O(d). Second, we settle the least degree required for approximating
a halfspace and a majority function to any given accuracy by rational
functions.
Our technique further allows us to make progress on Aaronson's challenge
(2008) and contribute strong direct product theorems for polynomial
representations of composed Boolean functions of the form F(f_1,...,f_n). In
particular, we give an improved lower bound on the approximate degree of the
AND-OR tree.Comment: Full version of the FOCS'09 pape
Three-valued logics, uncertainty management and rough sets
This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets
- …